CS 473: Algorithms. Chandra Chekuri 3228 Siebel Center. Fall University of Illinois, Urbana-Champaign
|
|
|
- Marian May
- 2 years ago
- Views:
Transcription
1 CS 473: Algorithms Chandra 3228 Siebel Center University of Illinois, Urbana-Champaign Fall 2009
2 Complementation Part I Complementation and Self-Reduction
3 The class P Complementation Recap Motivation co-np Definition Relationship between P, NP and co-np A language L (equivalently decision problem) is in the class P if there is a polynomial time algorithm A for deciding L; that is given a string x, A correctly decides if x L and running time of A on x is polynomial in x, the length of x.
4 The class NP Complementation Recap Motivation co-np Definition Relationship between P, NP and co-np Two equivalent definitions: Language L is in NP if there is a non-deterministic polynomial time algorithm A (Turing Machine) that decides L. For x L, A has some non-deterministic choice of moves that will make A accept x For x L, no choice of moves will make A accept x L has an efficient certifier C(, ). C is a polynomial time deterministic algorithm For x L there is a string y (proof) of length polynomial in x such that C(x, y) accepts For x L, no string y will make C(x, y) accept
5 Complementation Recap Motivation co-np Definition Relationship between P, NP and co-np Complementation Definition Given a decision problem X, its complement X is the collection of all instances s such that s X Equivalently, in terms of languages: Definition Given a language L over alphabet Σ, its complement L is the language Σ L.
6 Examples Complementation Recap Motivation co-np Definition Relationship between P, NP and co-np PRIME = {n n is an integer and n is prime} PRIME = {n n is an integer and n is not a prime} PRIME = COMPOSITE SAT = {ϕ ϕ is a SAT formula and ϕ is satisfiable} SAT = {ϕ ϕ is a SAT formula and ϕ is not satisfiable} SAT = UnSAT
7 Examples Complementation Recap Motivation co-np Definition Relationship between P, NP and co-np PRIME = {n n is an integer and n is prime} PRIME = {n n is an integer and n is not a prime} PRIME = COMPOSITE SAT = {ϕ ϕ is a SAT formula and ϕ is satisfiable} SAT = {ϕ ϕ is a SAT formula and ϕ is not satisfiable} SAT = UnSAT Technicality: SAT also includes strings that do not encode any valid SAT formula. Typically we ignore those strings because they are not interesting. In all problems of interest, we assume that it is easy to check whether a given string is a valid instance or not.
8 Complementation P is closed under complementation Recap Motivation co-np Definition Relationship between P, NP and co-np Proposition Decision problem X is in P if and only if X is in P.
9 Complementation Recap Motivation co-np Definition Relationship between P, NP and co-np P is closed under complementation Proposition Decision problem X is in P if and only if X is in P. Proof. If X is in P let A be a polynomial time algorithm for X. Construct polynomial time algorithm A for X as follows: given input x, A runs A on x and if A accepts x, A rejects x and if A rejects x then A accepts x. Only if direction is essentially the same argument.
10 Complementation Recap Motivation co-np Definition Relationship between P, NP and co-np Asymmetry of NP Definition Nondeterministic Polynomial Time (denoted by NP) is the class of all problems that have efficient certifiers. Observation To show that a problem is in NP we only need short, efficiently checkable certificates for yes -instances. What about no -instances?
11 Complementation Recap Motivation co-np Definition Relationship between P, NP and co-np Asymmetry of NP Definition Nondeterministic Polynomial Time (denoted by NP) is the class of all problems that have efficient certifiers. Observation To show that a problem is in NP we only need short, efficiently checkable certificates for yes -instances. What about no -instances? given CNF formulat ϕ, is ϕ unsatisfiable?
12 Complementation Recap Motivation co-np Definition Relationship between P, NP and co-np Asymmetry of NP Definition Nondeterministic Polynomial Time (denoted by NP) is the class of all problems that have efficient certifiers. Observation To show that a problem is in NP we only need short, efficiently checkable certificates for yes -instances. What about no -instances? given CNF formulat ϕ, is ϕ unsatisfiable? Easy to give a proof that ϕ is satisfiable (an assignment) but no easy (known) proof to show that ϕ is unsatisfiable!
13 Complementation Recap Motivation co-np Definition Relationship between P, NP and co-np Examples Some languages UnSAT: CNF formulas ϕ that are not satisfiable No-Hamilton-Cycle: graphs G that do not have a Hamilton cycle No-3-Color: graphs G that are not 3-colorable
14 Complementation Recap Motivation co-np Definition Relationship between P, NP and co-np Examples Some languages UnSAT: CNF formulas ϕ that are not satisfiable No-Hamilton-Cycle: graphs G that do not have a Hamilton cycle No-3-Color: graphs G that are not 3-colorable Above problems are complements of known NP problems (viewed as languages).
15 NP and co-np Complementation Recap Motivation co-np Definition Relationship between P, NP and co-np NP Decision problems with a polynomial certifier. Examples, SAT, Hamiltonian Cycle, 3-Colorability
16 Complementation Recap Motivation co-np Definition Relationship between P, NP and co-np NP and co-np NP Decision problems with a polynomial certifier. Examples, SAT, Hamiltonian Cycle, 3-Colorability Definition co-np is the class of all decision problems X such that X NP. Examples, UnSAT, No-Hamiltonian-Cycle, No-3-Colorable.
17 Complementation Recap Motivation co-np Definition Relationship between P, NP and co-np co-np L is a language in co-np implies that there is a polynomial time certifier/verifier C(, ) such that for s L there is a proof t of size polynomial in s such that C(s, t) correctly says NO for s L there is no proof t for which C(s, t) will say NO
18 Complementation Recap Motivation co-np Definition Relationship between P, NP and co-np co-np L is a language in co-np implies that there is a polynomial time certifier/verifier C(, ) such that for s L there is a proof t of size polynomial in s such that C(s, t) correctly says NO for s L there is no proof t for which C(s, t) will say NO co-np has checkable proofs for strings NOT in the language.
19 Complementation Recap Motivation co-np Definition Relationship between P, NP and co-np Natural Problems in co-np Tautology: given a Boolean formula (not necessarily in CNF form), is it true for all possible assignments to the variables? Graph expansion: given a graph G, is it an expander? A graph G = (V, E) is an expander iff for each S V with S V /2, N(S) S. Expanders are very important graphs in theoretical computer science and mathematics.
20 P, NP, co-np Complementation Recap Motivation co-np Definition Relationship between P, NP and co-np co-p: complement of P. Language X is in co-p iff X P
21 Complementation Recap Motivation co-np Definition Relationship between P, NP and co-np P, NP, co-np co-p: complement of P. Language X is in co-p iff X P Proposition P = co-p.
22 Complementation Recap Motivation co-np Definition Relationship between P, NP and co-np P, NP, co-np co-p: complement of P. Language X is in co-p iff X P Proposition P = co-p. Proposition P NP co-np.
23 Complementation Recap Motivation co-np Definition Relationship between P, NP and co-np P, NP, co-np co-p: complement of P. Language X is in co-p iff X P Proposition P = co-p. Proposition P NP co-np. Saw that P NP. Same proof shows P co-np
24 Complementation Recap Motivation co-np Definition Relationship between P, NP and co-np P, NP, and co-np NP co-np NP co-np P Open Problems: Does NP = co-np?
25 Complementation Recap Motivation co-np Definition Relationship between P, NP and co-np P, NP, and co-np NP co-np NP co-np P Open Problems: Does NP = co-np? Consensus opinion: No Is P = NP co-np?
26 Complementation Recap Motivation co-np Definition Relationship between P, NP and co-np P, NP, and co-np NP co-np NP co-np P Open Problems: Does NP = co-np? Consensus opinion: No Is P = NP co-np? No real consensus
27 Complementation Recap Motivation co-np Definition Relationship between P, NP and co-np P, NP, and co-np Proposition If P = NP then NP = co-np.
28 Complementation Recap Motivation co-np Definition Relationship between P, NP and co-np P, NP, and co-np Proposition If P = NP then NP = co-np. Proof. P = co-p If P = NP then co-np = co-p = P.
29 Complementation Recap Motivation co-np Definition Relationship between P, NP and co-np P, NP, and co-np Proposition If P = NP then NP = co-np. Proof. P = co-p If P = NP then co-np = co-p = P. Corollary If NP co-np then P NP.
30 Complementation Recap Motivation co-np Definition Relationship between P, NP and co-np P, NP, and co-np Proposition If P = NP then NP = co-np. Proof. P = co-p If P = NP then co-np = co-p = P. Corollary If NP co-np then P NP. Importance of corollary: try to prove P NP by proving that NP co-np.
31 Complementation Recap Motivation co-np Definition Relationship between P, NP and co-np NP co-np Complexity Class NP co-np Problems in this class have Efficient certifiers for yes-instances Efficient disqualifiers for no-instances Problems have a good characterization property, since for both yes and no instances we have short efficiently checkable proofs
32 Complementation Recap Motivation co-np Definition Relationship between P, NP and co-np NP co-np: Example Example Bipartite Matching: Given bipartite graph G = (U V, E), does G have a perfect matching? Bipartite Matching NP co-np
33 Complementation Recap Motivation co-np Definition Relationship between P, NP and co-np NP co-np: Example Example Bipartite Matching: Given bipartite graph G = (U V, E), does G have a perfect matching? Bipartite Matching NP co-np If G is a yes-instance, then proof is just the perfect matching
34 Complementation Recap Motivation co-np Definition Relationship between P, NP and co-np NP co-np: Example Example Bipartite Matching: Given bipartite graph G = (U V, E), does G have a perfect matching? Bipartite Matching NP co-np If G is a yes-instance, then proof is just the perfect matching If G is a no-instance, then by Hall s Theorem, there is a subset of vertices A U such that N(A) < A
35 Complementation Recap Motivation co-np Definition Relationship between P, NP and co-np Good Characterization? = Efficient Solution Bipartite Matching has a polynomial time algorithm Do all problems in NP co-np have polynomial time algorithms? That is, is P = NP co-np?
36 Complementation Recap Motivation co-np Definition Relationship between P, NP and co-np Good Characterization? = Efficient Solution Bipartite Matching has a polynomial time algorithm Do all problems in NP co-np have polynomial time algorithms? That is, is P = NP co-np? Problems in NP co-np have been proved to be in P many years later Linear programming (Khachiyan 1979) Duality easily shows that it is in NP co-np Primality Testing (Agarwal-Kayal-Saxena 2002) Easy to see that PRIME is in co-np (why?) PRIME is in NP - not easy to show! (Vaughan Pratt 1975)
37 Complementation P? = NP co-np (contd) Recap Motivation co-np Definition Relationship between P, NP and co-np Some problems in NP co-np still cannot be proved to have polynomial time algorithms
38 Complementation P? = NP co-np (contd) Recap Motivation co-np Definition Relationship between P, NP and co-np Some problems in NP co-np still cannot be proved to have polynomial time algorithms Parity Games
39 Complementation P? = NP co-np (contd) Recap Motivation co-np Definition Relationship between P, NP and co-np Some problems in NP co-np still cannot be proved to have polynomial time algorithms Parity Games Other more specialized problems
40 Complementation Recap Motivation co-np Definition Relationship between P, NP and co-np co-np Completeness Definition A problem X is said to be co-np-complete if X co-np (Hardness) For any Y co-np, Y P X
41 Complementation Recap Motivation co-np Definition Relationship between P, NP and co-np co-np Completeness Definition A problem X is said to be co-np-complete if X co-np (Hardness) For any Y co-np, Y P X co-np-complete problems are the hardest problems in co-np.
42 Complementation Recap Motivation co-np Definition Relationship between P, NP and co-np co-np Completeness Definition A problem X is said to be co-np-complete if X co-np (Hardness) For any Y co-np, Y P X co-np-complete problems are the hardest problems in co-np. Lemma X is co-np-complete if and only if X is NP-Complete. Proof left as an exercise.
43 Complementation Recap Motivation co-np Definition Relationship between P, NP and co-np P, NP and co-np Possible scenarios: P = NP. Then P = NP = co-np. NP = co-np and P NP (and hence also P co-np). NP co-np. Then P NP and also P co-np. Most people believe that the last scenario is the likely one.
44 Complementation Recap Motivation co-np Definition Relationship between P, NP and co-np P, NP and co-np Possible scenarios: P = NP. Then P = NP = co-np. NP = co-np and P NP (and hence also P co-np). NP co-np. Then P NP and also P co-np. Most people believe that the last scenario is the likely one. Question: Suppose P NP. Is every problem in NP P NP-Complete?
45 Complementation Recap Motivation co-np Definition Relationship between P, NP and co-np P, NP and co-np Possible scenarios: P = NP. Then P = NP = co-np. NP = co-np and P NP (and hence also P co-np). NP co-np. Then P NP and also P co-np. Most people believe that the last scenario is the likely one. Question: Suppose P NP. Is every problem in NP P NP-Complete? Theorem (Ladner) If P NP then there is a problem/language X NP P such that X is not NP-Complete.
46 Complementation SAT is Self Reducible Back to Decision versus Search Recall, decision problems are those with yes/no answers, while search problems require an explicit solution for a yes instance
47 Complementation SAT is Self Reducible Back to Decision versus Search Example Recall, decision problems are those with yes/no answers, while search problems require an explicit solution for a yes instance
48 Complementation SAT is Self Reducible Back to Decision versus Search Example Recall, decision problems are those with yes/no answers, while search problems require an explicit solution for a yes instance Satisfiability
49 Complementation SAT is Self Reducible Back to Decision versus Search Example Recall, decision problems are those with yes/no answers, while search problems require an explicit solution for a yes instance Satisfiability Decision: Is the formula ϕ satisfiable?
50 Complementation SAT is Self Reducible Back to Decision versus Search Example Recall, decision problems are those with yes/no answers, while search problems require an explicit solution for a yes instance Satisfiability Decision: Is the formula ϕ satisfiable? Search: Find assignment that satisfies ϕ
51 Complementation SAT is Self Reducible Back to Decision versus Search Example Recall, decision problems are those with yes/no answers, while search problems require an explicit solution for a yes instance Satisfiability Decision: Is the formula ϕ satisfiable? Search: Find assignment that satisfies ϕ Graph coloring
52 Complementation SAT is Self Reducible Back to Decision versus Search Example Recall, decision problems are those with yes/no answers, while search problems require an explicit solution for a yes instance Satisfiability Decision: Is the formula ϕ satisfiable? Search: Find assignment that satisfies ϕ Graph coloring Decision: Is graph G 3-colorable?
53 Complementation SAT is Self Reducible Back to Decision versus Search Example Recall, decision problems are those with yes/no answers, while search problems require an explicit solution for a yes instance Satisfiability Decision: Is the formula ϕ satisfiable? Search: Find assignment that satisfies ϕ Graph coloring Decision: Is graph G 3-colorable? Search: Find a 3-coloring of the vertices of G
54 Complementation SAT is Self Reducible Decision reduces to Search Efficient algorithm for search implies efficient algorithm for decision
55 Complementation SAT is Self Reducible Decision reduces to Search Efficient algorithm for search implies efficient algorithm for decision If decision problem is difficult then search problem is also difficult
56 Complementation SAT is Self Reducible Decision reduces to Search Efficient algorithm for search implies efficient algorithm for decision If decision problem is difficult then search problem is also difficult Can an efficient algorithm for decision imply an efficient algorithm for search?
57 Complementation SAT is Self Reducible Decision reduces to Search Efficient algorithm for search implies efficient algorithm for decision If decision problem is difficult then search problem is also difficult Can an efficient algorithm for decision imply an efficient algorithm for search? Yes, for all the problems we have seen. In fact for all NP-Complete Problems.
58 Complementation SAT is Self Reducible Definition A problem is said to be self reducible if the search problem reduces (by Cook reduction) in polynomial time to decision problem. In other words, there is an algorithm to solve the search problem that has polynomially many steps, where each step is either A conventional computational step, or A call to subroutine solving the decision problem
59 Complementation SAT is Self Reducible Back to SAT Proposition SAT is self reducible In other words, there is a polynomial time algorithm to find the satisfying assignment if one can periodically check if some formula is satisfiable
60 Complementation SAT is Self Reducible Search Algorithm for SAT from a Decision Algorithm for SAT Input: SAT formula ϕ with n variables x 1, x 2,..., x n. set x 1 = 0 in ϕ and get new formula ϕ 1. check if ϕ 1 is satisfiable using decision algorithm. if ϕ 1 is satisfiable, recursively find assignment to x 2, x 3,..., x n that satisfy ϕ 1 and output x 1 = 0 along with the assignment to x 2,..., x n. if ϕ 1 is not satisfiable then set x 1 = 1 in ϕ to get formula ϕ 2. if ϕ 2 is satisfiable, recursively find assignment to x 2, x 3,..., x n that satisfy ϕ 2 and output x 1 = 1 along with the assignment to x 2,..., x n. if ϕ 1 and ϕ 2 are both not satisfiable then ϕ is not satisfiable. Algorithm runs in polynomial time if the decision algorithm for SAT runs in polynomial time. At most 2n calls to decision algorithm.
61
62 Complementation SAT is Self Reducible Self-Reduction for NP-Complete Problems Theorem Every NP-Complete problem/language L is self-reducible. Proof out of scope. Note that proof is only for complete languages, not for all languages in NP. Otherwise Factoring would be in polynomial time and we would not rely on it for our current security protocols. Easy and instructive to prove self-reducibility for specific NP-Complete problems such as Independent Set, Vertex Cover, Hamiltonian Cycle etc. See HBS problems and finals prep.
63 Part II Coping with Intractability
64 Intractability Many important and useful problems are hard (NP-hard, co-np-hard, undecidable,...). No efficient algorithm possible or known.
65 Intractability Many important and useful problems are hard (NP-hard, co-np-hard, undecidable,...). No efficient algorithm possible or known. Nevertheless, need to address problems. Example Problems: build tools to check for bugs in programs and hardware (undecidable in general) assign frequencies to cell phone towers (NP-hard via coloring) solve integer programs to minimize cost of manufacturing goods (NP-hard)
66 Coping with Intractability Compromise! Some general things that people do. Consider special cases of the problem which may be tractable.
67 Coping with Intractability Compromise! Some general things that people do. Consider special cases of the problem which may be tractable. Run inefficient algorithms (for example exponential time algorithms for NP-hard problems) augmented with (very) clever heuristics
68 Coping with Intractability Compromise! Some general things that people do. Consider special cases of the problem which may be tractable. Run inefficient algorithms (for example exponential time algorithms for NP-hard problems) augmented with (very) clever heuristics stop algorithm when time/resources run out
69 Coping with Intractability Compromise! Some general things that people do. Consider special cases of the problem which may be tractable. Run inefficient algorithms (for example exponential time algorithms for NP-hard problems) augmented with (very) clever heuristics stop algorithm when time/resources run out use massive computational power
70 Coping with Intractability Compromise! Some general things that people do. Consider special cases of the problem which may be tractable. Run inefficient algorithms (for example exponential time algorithms for NP-hard problems) augmented with (very) clever heuristics stop algorithm when time/resources run out use massive computational power Exploit properties of instances that arise in practice which may be much easier. Give up on hard instances, which is ok.
71 Coping with Intractability Compromise! Some general things that people do. Consider special cases of the problem which may be tractable. Run inefficient algorithms (for example exponential time algorithms for NP-hard problems) augmented with (very) clever heuristics stop algorithm when time/resources run out use massive computational power Exploit properties of instances that arise in practice which may be much easier. Give up on hard instances, which is ok. Settle for sub-optimal solutions, especially for optimization problems
72 Heuristics heuristic: 1 Of or relating to a usually speculative formulation serving as a guide in the investigation or solution of a problem. 2 Of or constituting an educational method in which learning takes place through discoveries that result from investigations made by the student. 3 Computer Science. Relating to or using a problem-solving technique in which the most appropriate solution of several found by alternative methods is selected at successive stages of a program for use in the next step of the program.
73 Heuristics Heuristics work very well for many problems in practice! For example there are algorithms for SAT (SAT-solvers) that can solve restricted classes of SAT formulas even with thousands of variables and clauses! Heuristics need to be developed based on a good understanding of the underlying problem. Heuristics for some expressive problems such as SAT and Integer Programming are extensively studied because many other problems can be naturally reduced to them. Some generic heuristic methods: Intelligent exhaustive search methods via backtracking: Branch-and-Bound and Branch-and-Cut etc Local search and variants: Simulated Annealing, Random starts etc. Tabu serach, genetic algorithms,...
74 Load Balancing Part III Approximation Algorithms
75 Load Balancing Approximation Algorithms Approximation algorithms are heuristics with guarantees on their performance. Definition Let X is an optimization problem such that opt(s) is the optimum value on input instance s. An algorithm A is a ρ-approximation algorithm for X if
76 Load Balancing Approximation Algorithms Approximation algorithms are heuristics with guarantees on their performance. Definition Let X is an optimization problem such that opt(s) is the optimum value on input instance s. An algorithm A is a ρ-approximation algorithm for X if 1 On all inputs s, A(s) runs in polynomial time
77 Load Balancing Approximation Algorithms Approximation algorithms are heuristics with guarantees on their performance. Definition Let X is an optimization problem such that opt(s) is the optimum value on input instance s. An algorithm A is a ρ-approximation algorithm for X if 1 On all inputs s, A(s) runs in polynomial time 2 On all inputs s, the output A(s) is within a ρ-ratio of the optimal value opt(s)
78 Load Balancing Approximation Algorithms Approximation algorithms are heuristics with guarantees on their performance. Definition Let X is an optimization problem such that opt(s) is the optimum value on input instance s. An algorithm A is a ρ-approximation algorithm for X if 1 On all inputs s, A(s) runs in polynomial time 2 On all inputs s, the output A(s) is within a ρ-ratio of the optimal value opt(s), i.e., if X is a maximization problem then A(s) opt(s)/ρ and if X is a minimization problem then A(s) ρ opt(s)
79 Load Balancing Approximation Algorithms Approximation algorithms are heuristics with guarantees on their performance. Definition Let X is an optimization problem such that opt(s) is the optimum value on input instance s. An algorithm A is a ρ-approximation algorithm for X if 1 On all inputs s, A(s) runs in polynomial time 2 On all inputs s, the output A(s) is within a ρ-ratio of the optimal value opt(s), i.e., if X is a maximization problem then A(s) opt(s)/ρ and if X is a minimization problem then A(s) ρ opt(s) Thus, on all inputs A runs in polynomial time, and gives an answer that is guaranteed to be close to the optimal value.
80 Load Balancing The Problem Greedy Algorithm Analysis of Greedy Algorithm Analysis of Improved Algorithm Load Balancing Problem Input: m identical machines, n jobs with ith job having processing time t i Goal: Schedule jobs to computers such that Jobs run contiguously on a machine
81 Load Balancing The Problem Greedy Algorithm Analysis of Greedy Algorithm Analysis of Improved Algorithm Load Balancing Problem Input: m identical machines, n jobs with ith job having processing time t i Goal: Schedule jobs to computers such that Jobs run contiguously on a machine A machine processes only one job a time
82 Load Balancing The Problem Greedy Algorithm Analysis of Greedy Algorithm Analysis of Improved Algorithm Load Balancing Problem Input: m identical machines, n jobs with ith job having processing time t i Goal: Schedule jobs to computers such that Jobs run contiguously on a machine A machine processes only one job a time Makespan or maximum load on any machine is minimized
83 Load Balancing The Problem Greedy Algorithm Analysis of Greedy Algorithm Analysis of Improved Algorithm Load Balancing Problem Input: m identical machines, n jobs with ith job having processing time t i Goal: Schedule jobs to computers such that Jobs run contiguously on a machine A machine processes only one job a time Makespan or maximum load on any machine is minimized Definition Let A(i) be the set of jobs assigned to machine i. The load on i is T i = j A(i) t j.
84 Load Balancing The Problem Greedy Algorithm Analysis of Greedy Algorithm Analysis of Improved Algorithm Load Balancing Problem Input: m identical machines, n jobs with ith job having processing time t i Goal: Schedule jobs to computers such that Jobs run contiguously on a machine A machine processes only one job a time Makespan or maximum load on any machine is minimized Definition Let A(i) be the set of jobs assigned to machine i. The load on i is T i = j A(i) t j. The makespan of A is T = max i T i
85 Load Balancing The Problem Greedy Algorithm Analysis of Greedy Algorithm Analysis of Improved Algorithm Load Balancing: Example Example Consider 6 jobs whose processing times is given as follows Jobs t i
86 Load Balancing Load Balancing: Example The Problem Greedy Algorithm Analysis of Greedy Algorithm Analysis of Improved Algorithm Example Consider 6 jobs whose processing times is given as follows Jobs t i Consider the following schedule on 3 machines
87 Load Balancing Load Balancing: Example The Problem Greedy Algorithm Analysis of Greedy Algorithm Analysis of Improved Algorithm Example Consider 6 jobs whose processing times is given as follows Jobs t i Consider the following schedule on 3 machines The loads are: T 1 = 8, T 2 = 5, and T 3 = 6. So makespan of schedule is 8
88 Load Balancing Load Balancing is NP-Complete The Problem Greedy Algorithm Analysis of Greedy Algorithm Analysis of Improved Algorithm Decision version: given n, m and t 1, t 2,..., t n and a target T, is there a schedule with makespan at most T? Problem is NP-Complete: reduce from Subset Sum.
89 Load Balancing The Problem Greedy Algorithm Analysis of Greedy Algorithm Analysis of Improved Algorithm Greedy Algorithm 1 Consider the jobs in some fixed order 2 Assign job j to the machine with lowest load so far Example Jobs t i
90 Load Balancing The Problem Greedy Algorithm Analysis of Greedy Algorithm Analysis of Improved Algorithm Greedy Algorithm 1 Consider the jobs in some fixed order 2 Assign job j to the machine with lowest load so far Example Jobs t i
91 Load Balancing The Problem Greedy Algorithm Analysis of Greedy Algorithm Analysis of Improved Algorithm Greedy Algorithm 1 Consider the jobs in some fixed order 2 Assign job j to the machine with lowest load so far Example Jobs t i
92 Load Balancing The Problem Greedy Algorithm Analysis of Greedy Algorithm Analysis of Improved Algorithm Greedy Algorithm 1 Consider the jobs in some fixed order 2 Assign job j to the machine with lowest load so far Example Jobs t i
93 Load Balancing The Problem Greedy Algorithm Analysis of Greedy Algorithm Analysis of Improved Algorithm Greedy Algorithm 1 Consider the jobs in some fixed order 2 Assign job j to the machine with lowest load so far Example Jobs t i
94 Load Balancing The Problem Greedy Algorithm Analysis of Greedy Algorithm Analysis of Improved Algorithm Greedy Algorithm 1 Consider the jobs in some fixed order 2 Assign job j to the machine with lowest load so far Example Jobs t i
95 Load Balancing The Problem Greedy Algorithm Analysis of Greedy Algorithm Analysis of Improved Algorithm Greedy Algorithm 1 Consider the jobs in some fixed order 2 Assign job j to the machine with lowest load so far Example Jobs t i
96 Putting it together Load Balancing The Problem Greedy Algorithm Analysis of Greedy Algorithm Analysis of Improved Algorithm for each machine i T i = 0 (* initially no load *) A(i) = (* initially no jobs *) for each job j Let i be machine with smallest load A(i) = A(i) {j} (* schedule j on i *) T i = T i + t j (* compute new load *) Running Time
97 Load Balancing The Problem Greedy Algorithm Analysis of Greedy Algorithm Analysis of Improved Algorithm Putting it together for each machine i T i = 0 (* initially no load *) A(i) = (* initially no jobs *) for each job j Let i be machine with smallest load A(i) = A(i) {j} (* schedule j on i *) T i = T i + t j (* compute new load *) Running Time First loop takes O(m) time
98 Load Balancing The Problem Greedy Algorithm Analysis of Greedy Algorithm Analysis of Improved Algorithm Putting it together for each machine i T i = 0 (* initially no load *) A(i) = (* initially no jobs *) for each job j Let i be machine with smallest load A(i) = A(i) {j} (* schedule j on i *) T i = T i + t j (* compute new load *) Running Time First loop takes O(m) time Second loop has O(n) iterations
99 Load Balancing The Problem Greedy Algorithm Analysis of Greedy Algorithm Analysis of Improved Algorithm Putting it together for each machine i T i = 0 (* initially no load *) A(i) = (* initially no jobs *) for each job j Let i be machine with smallest load A(i) = A(i) {j} (* schedule j on i *) T i = T i + t j (* compute new load *) Running Time First loop takes O(m) time Second loop has O(n) iterations
100 Load Balancing The Problem Greedy Algorithm Analysis of Greedy Algorithm Analysis of Improved Algorithm Putting it together for each machine i T i = 0 (* initially no load *) A(i) = (* initially no jobs *) for each job j Let i be machine with smallest load A(i) = A(i) {j} (* schedule j on i *) T i = T i + t j (* compute new load *) Running Time First loop takes O(m) time Second loop has O(n) iterations Body of loop takes O(log m) time using a priority heap
101 Load Balancing The Problem Greedy Algorithm Analysis of Greedy Algorithm Analysis of Improved Algorithm Putting it together for each machine i T i = 0 (* initially no load *) A(i) = (* initially no jobs *) for each job j Let i be machine with smallest load A(i) = A(i) {j} (* schedule j on i *) T i = T i + t j (* compute new load *) Running Time First loop takes O(m) time Second loop has O(n) iterations Body of loop takes O(log m) time using a priority heap Total time is O(n log m + m)
102 Optimality Load Balancing The Problem Greedy Algorithm Analysis of Greedy Algorithm Analysis of Improved Algorithm Problem Is the greedy algorithm optimal?
103 Optimality Load Balancing The Problem Greedy Algorithm Analysis of Greedy Algorithm Analysis of Improved Algorithm Problem Is the greedy algorithm optimal? No! For example, on Jobs t i the greedy algorithm gives schedule with makespan 8, but optimal is 7
104 Optimality Load Balancing The Problem Greedy Algorithm Analysis of Greedy Algorithm Analysis of Improved Algorithm Problem Is the greedy algorithm optimal? No! For example, on Jobs t i the greedy algorithm gives schedule with makespan 8, but optimal is 7 In fact, the load balancing problem is NP-complete.
105 Quality of Solution Load Balancing The Problem Greedy Algorithm Analysis of Greedy Algorithm Analysis of Improved Algorithm
106 Load Balancing The Problem Greedy Algorithm Analysis of Greedy Algorithm Analysis of Improved Algorithm Quality of Solution Theorem (Graham 1966) The makespan of the schedule output by the greedy algorithm is at most 2 times the optimal make span. In other words, the greedy algorithm is a 2-approximation. Challenge How do we compare the output of the greedy algorithm with the optimal? How do we get the value of the optimal solution?
107 Load Balancing The Problem Greedy Algorithm Analysis of Greedy Algorithm Analysis of Improved Algorithm Quality of Solution Theorem (Graham 1966) The makespan of the schedule output by the greedy algorithm is at most 2 times the optimal make span. In other words, the greedy algorithm is a 2-approximation. Challenge How do we compare the output of the greedy algorithm with the optimal? How do we get the value of the optimal solution? We will obtain bounds on the optimal value
108 Load Balancing Bounding the Optimal Value The Problem Greedy Algorithm Analysis of Greedy Algorithm Analysis of Improved Algorithm Lemma T max j t j, where T is the optimal makespan
109 Load Balancing Bounding the Optimal Value The Problem Greedy Algorithm Analysis of Greedy Algorithm Analysis of Improved Algorithm Lemma T max j t j, where T is the optimal makespan Proof. Some machine will run the job with maximum processing time
110 Load Balancing Bounding the Optimal Value The Problem Greedy Algorithm Analysis of Greedy Algorithm Analysis of Improved Algorithm Lemma T max j t j, where T is the optimal makespan Proof. Some machine will run the job with maximum processing time Lemma T 1 m j t j, where T is the optimal makespan
111 Load Balancing Bounding the Optimal Value The Problem Greedy Algorithm Analysis of Greedy Algorithm Analysis of Improved Algorithm Lemma T max j t j, where T is the optimal makespan Proof. Some machine will run the job with maximum processing time Lemma T 1 m Proof. j t j, where T is the optimal makespan
112 Load Balancing The Problem Greedy Algorithm Analysis of Greedy Algorithm Analysis of Improved Algorithm Bounding the Optimal Value Lemma T max j t j, where T is the optimal makespan Proof. Some machine will run the job with maximum processing time Lemma T 1 m Proof. j t j, where T is the optimal makespan Total processing time is j t j
113 Load Balancing Bounding the Optimal Value The Problem Greedy Algorithm Analysis of Greedy Algorithm Analysis of Improved Algorithm Lemma T max j t j, where T is the optimal makespan Proof. Some machine will run the job with maximum processing time Lemma T 1 m Proof. j t j, where T is the optimal makespan Total processing time is j t j Some machine must do at least 1 m work (or average) of the total
114 Load Balancing Analysis of Greedy Algorithm The Problem Greedy Algorithm Analysis of Greedy Algorithm Analysis of Improved Algorithm Theorem The greedy algorithm is a 2-approximation
115 Load Balancing Analysis of Greedy Algorithm The Problem Greedy Algorithm Analysis of Greedy Algorithm Analysis of Improved Algorithm Theorem The greedy algorithm is a 2-approximation Proof. Let machine i have the maximum load T i, and let j be the last job scheduled on machine i
116 Load Balancing The Problem Greedy Algorithm Analysis of Greedy Algorithm Analysis of Improved Algorithm Analysis of Greedy Algorithm Theorem The greedy algorithm is a 2-approximation Proof. Let machine i have the maximum load T i, and let j be the last job scheduled on machine i At the time j was scheduled, machine i must have had the least load
117 Load Balancing The Problem Greedy Algorithm Analysis of Greedy Algorithm Analysis of Improved Algorithm Analysis of Greedy Algorithm Theorem The greedy algorithm is a 2-approximation Proof. Let machine i have the maximum load T i, and let j be the last job scheduled on machine i At the time j was scheduled, machine i must have had the least load; load on i before assigning job j is T i t j
118 Load Balancing The Problem Greedy Algorithm Analysis of Greedy Algorithm Analysis of Improved Algorithm Analysis of Greedy Algorithm Theorem The greedy algorithm is a 2-approximation Proof. Let machine i have the maximum load T i, and let j be the last job scheduled on machine i At the time j was scheduled, machine i must have had the least load; load on i before assigning job j is T i t j Since i has the least load, we know T i t j T k, for all k. Thus, m(t i t j ) k T k
119 Load Balancing Analysis of Greedy Algorithm The Problem Greedy Algorithm Analysis of Greedy Algorithm Analysis of Improved Algorithm Theorem The greedy algorithm is a 2-approximation Proof. Let machine i have the maximum load T i, and let j be the last job scheduled on machine i At the time j was scheduled, machine i must have had the least load; load on i before assigning job j is T i t j Since i has the least load, we know T i t j T k, for all k. Thus, m(t i t j ) k T k But k T k = l t l. So T i t j 1 m k T k = 1 m l t l T
120 Load Balancing Analysis of Greedy Algorithm The Problem Greedy Algorithm Analysis of Greedy Algorithm Analysis of Improved Algorithm Theorem The greedy algorithm is a 2-approximation Proof. Let machine i have the maximum load T i, and let j be the last job scheduled on machine i At the time j was scheduled, machine i must have had the least load; load on i before assigning job j is T i t j Since i has the least load, we know T i t j T k, for all k. Thus, m(t i t j ) k T k But k T k = l t l. So T i t j 1 m k T k = 1 m l t l T Finally, T i = (T i t j ) + t j T + T = 2T
121 Load Balancing The Problem Greedy Algorithm Analysis of Greedy Algorithm Analysis of Improved Algorithm Tightness of Analysis Proposition The analysis of the greedy algorithm is tight, i.e., there is an example on which the greedy schedule has twice the optimal makespan
122 Load Balancing The Problem Greedy Algorithm Analysis of Greedy Algorithm Analysis of Improved Algorithm Tightness of Analysis Proposition The analysis of the greedy algorithm is tight, i.e., there is an example on which the greedy schedule has twice the optimal makespan Proof. Consider problem of m(m 1) jobs with processing time 1 and the last job with processing time m.
123 Load Balancing The Problem Greedy Algorithm Analysis of Greedy Algorithm Analysis of Improved Algorithm Tightness of Analysis Proposition The analysis of the greedy algorithm is tight, i.e., there is an example on which the greedy schedule has twice the optimal makespan Proof. Consider problem of m(m 1) jobs with processing time 1 and the last job with processing time m. Greedy schedule: distribute first the m(m 1) jobs equally among the m machines, and then schedule the last job on machine 1, giving a makespan of (m 1) + m = 2m 1
124 Load Balancing The Problem Greedy Algorithm Analysis of Greedy Algorithm Analysis of Improved Algorithm Tightness of Analysis Proposition The analysis of the greedy algorithm is tight, i.e., there is an example on which the greedy schedule has twice the optimal makespan Proof. Consider problem of m(m 1) jobs with processing time 1 and the last job with processing time m. Greedy schedule: distribute first the m(m 1) jobs equally among the m machines, and then schedule the last job on machine 1, giving a makespan of (m 1) + m = 2m 1 The optimal schedule: run last job on machine 1, and then distribute remaining jobs equally among m 1 machines, giving a makespan of m
125 Load Balancing The Problem Greedy Algorithm Analysis of Greedy Algorithm Analysis of Improved Algorithm Improved Greedy Algorithm Modified Greedy Sort the jobs in descending order of processing time, and process jobs using greedy algorithm
126 Load Balancing The Problem Greedy Algorithm Analysis of Greedy Algorithm Analysis of Improved Algorithm Improved Greedy Algorithm Modified Greedy Sort the jobs in descending order of processing time, and process jobs using greedy algorithm for each machine i T i = 0 (* initially no load *) A(i) = (* initially no jobs *) for each job j in descending order of processing time Let i be machine with smallest load A(i) = A(i) {j} (* schedule j on i *) T i = T i + t j (* compute new load *)
127 Technical Lemma Load Balancing The Problem Greedy Algorithm Analysis of Greedy Algorithm Analysis of Improved Algorithm Lemma If there are more than m jobs then T 2t m+1
128 Technical Lemma Load Balancing The Problem Greedy Algorithm Analysis of Greedy Algorithm Analysis of Improved Algorithm Lemma If there are more than m jobs then T 2t m+1 Proof. Consider the first m + 1 jobs
129 Load Balancing The Problem Greedy Algorithm Analysis of Greedy Algorithm Analysis of Improved Algorithm Technical Lemma Lemma If there are more than m jobs then T 2t m+1 Proof. Consider the first m + 1 jobs Two of them must be scheduled on same machine, by pigeon-hole principle
130 Load Balancing The Problem Greedy Algorithm Analysis of Greedy Algorithm Analysis of Improved Algorithm Technical Lemma Lemma If there are more than m jobs then T 2t m+1 Proof. Consider the first m + 1 jobs Two of them must be scheduled on same machine, by pigeon-hole principle Both jobs have processing time at least t m+1, since we consider jobs according to processing time, and this proves the lemma
131 Load Balancing The Problem Greedy Algorithm Analysis of Greedy Algorithm Analysis of Improved Algorithm Analysis of Modified Greedy Theorem The modified greedy algorithm is a 3/2-approximation
132 Load Balancing The Problem Greedy Algorithm Analysis of Greedy Algorithm Analysis of Improved Algorithm Analysis of Modified Greedy Theorem The modified greedy algorithm is a 3/2-approximation Proof. Once again let i be the machine with highest load, and let j be the last job scheduled
133 Load Balancing The Problem Greedy Algorithm Analysis of Greedy Algorithm Analysis of Improved Algorithm Analysis of Modified Greedy Theorem The modified greedy algorithm is a 3/2-approximation Proof. Once again let i be the machine with highest load, and let j be the last job scheduled If machine i has only one job then schedule is optimal
134 Load Balancing The Problem Greedy Algorithm Analysis of Greedy Algorithm Analysis of Improved Algorithm Analysis of Modified Greedy Theorem The modified greedy algorithm is a 3/2-approximation Proof. Once again let i be the machine with highest load, and let j be the last job scheduled If machine i has only one job then schedule is optimal If i has at least 2 jobs, then it must be the case that j m + 1. This means t j t m T
135 Load Balancing The Problem Greedy Algorithm Analysis of Greedy Algorithm Analysis of Improved Algorithm Analysis of Modified Greedy Theorem The modified greedy algorithm is a 3/2-approximation Proof. Once again let i be the machine with highest load, and let j be the last job scheduled If machine i has only one job then schedule is optimal If i has at least 2 jobs, then it must be the case that j m + 1. This means t j t m T Thus, T i = (T i t j ) + t j T T
136 Load Balancing The Problem Greedy Algorithm Analysis of Greedy Algorithm Analysis of Improved Algorithm Tightness of Analysis Theorem (Graham) Modified greedy is a 4/3-approximation The 4/3-analysis is tight.
Computer Algorithms. NP-Complete Problems. CISC 4080 Yanjun Li
Computer Algorithms NP-Complete Problems NP-completeness The quest for efficient algorithms is about finding clever ways to bypass the process of exhaustive search, using clues from the input in order
Introduction to Logic in Computer Science: Autumn 2006
Introduction to Logic in Computer Science: Autumn 2006 Ulle Endriss Institute for Logic, Language and Computation University of Amsterdam Ulle Endriss 1 Plan for Today Now that we have a basic understanding
CSE 135: Introduction to Theory of Computation Decidability and Recognizability
CSE 135: Introduction to Theory of Computation Decidability and Recognizability Sungjin Im University of California, Merced 04-28, 30-2014 High-Level Descriptions of Computation Instead of giving a Turing
NP-complete? NP-hard? Some Foundations of Complexity. Prof. Sven Hartmann Clausthal University of Technology Department of Informatics
NP-complete? NP-hard? Some Foundations of Complexity Prof. Sven Hartmann Clausthal University of Technology Department of Informatics Tractability of Problems Some problems are undecidable: no computer
OHJ-2306 Introduction to Theoretical Computer Science, Fall 2012 8.11.2012
276 The P vs. NP problem is a major unsolved problem in computer science It is one of the seven Millennium Prize Problems selected by the Clay Mathematics Institute to carry a $ 1,000,000 prize for the
CSC 373: Algorithm Design and Analysis Lecture 16
CSC 373: Algorithm Design and Analysis Lecture 16 Allan Borodin February 25, 2013 Some materials are from Stephen Cook s IIT talk and Keven Wayne s slides. 1 / 17 Announcements and Outline Announcements
Introduction to Algorithms. Part 3: P, NP Hard Problems
Introduction to Algorithms Part 3: P, NP Hard Problems 1) Polynomial Time: P and NP 2) NP-Completeness 3) Dealing with Hard Problems 4) Lower Bounds 5) Books c Wayne Goddard, Clemson University, 2004 Chapter
Guessing Game: NP-Complete?
Guessing Game: NP-Complete? 1. LONGEST-PATH: Given a graph G = (V, E), does there exists a simple path of length at least k edges? YES 2. SHORTEST-PATH: Given a graph G = (V, E), does there exists a simple
1. Nondeterministically guess a solution (called a certificate) 2. Check whether the solution solves the problem (called verification)
Some N P problems Computer scientists have studied many N P problems, that is, problems that can be solved nondeterministically in polynomial time. Traditionally complexity question are studied as languages:
Algorithm Design and Analysis
Algorithm Design and Analysis LECTURE 27 Approximation Algorithms Load Balancing Weighted Vertex Cover Reminder: Fill out SRTEs online Don t forget to click submit Sofya Raskhodnikova 12/6/2011 S. Raskhodnikova;
Page 1. CSCE 310J Data Structures & Algorithms. CSCE 310J Data Structures & Algorithms. P, NP, and NP-Complete. Polynomial-Time Algorithms
CSCE 310J Data Structures & Algorithms P, NP, and NP-Complete Dr. Steve Goddard goddard@cse.unl.edu CSCE 310J Data Structures & Algorithms Giving credit where credit is due:» Most of the lecture notes
Applied Algorithm Design Lecture 5
Applied Algorithm Design Lecture 5 Pietro Michiardi Eurecom Pietro Michiardi (Eurecom) Applied Algorithm Design Lecture 5 1 / 86 Approximation Algorithms Pietro Michiardi (Eurecom) Applied Algorithm Design
CoNP and Function Problems
CoNP and Function Problems conp By definition, conp is the class of problems whose complement is in NP. NP is the class of problems that have succinct certificates. conp is therefore the class of problems
Approximation Algorithms
Approximation Algorithms or: How I Learned to Stop Worrying and Deal with NP-Completeness Ong Jit Sheng, Jonathan (A0073924B) March, 2012 Overview Key Results (I) General techniques: Greedy algorithms
Near Optimal Solutions
Near Optimal Solutions Many important optimization problems are lacking efficient solutions. NP-Complete problems unlikely to have polynomial time solutions. Good heuristics important for such problems.
! Solve problem to optimality. ! Solve problem in poly-time. ! Solve arbitrary instances of the problem. #-approximation algorithm.
Approximation Algorithms 11 Approximation Algorithms Q Suppose I need to solve an NP-hard problem What should I do? A Theory says you're unlikely to find a poly-time algorithm Must sacrifice one of three
Exponential time algorithms for graph coloring
Exponential time algorithms for graph coloring Uriel Feige Lecture notes, March 14, 2011 1 Introduction Let [n] denote the set {1,..., k}. A k-labeling of vertices of a graph G(V, E) is a function V [k].
Complexity Classes P and NP
Complexity Classes P and NP MATH 3220 Supplemental Presentation by John Aleshunas The cure for boredom is curiosity. There is no cure for curiosity Dorothy Parker Computational Complexity Theory In computer
Why? A central concept in Computer Science. Algorithms are ubiquitous.
Analysis of Algorithms: A Brief Introduction Why? A central concept in Computer Science. Algorithms are ubiquitous. Using the Internet (sending email, transferring files, use of search engines, online
Lecture 7: NP-Complete Problems
IAS/PCMI Summer Session 2000 Clay Mathematics Undergraduate Program Basic Course on Computational Complexity Lecture 7: NP-Complete Problems David Mix Barrington and Alexis Maciel July 25, 2000 1. Circuit
Lecture 1: Course overview, circuits, and formulas
Lecture 1: Course overview, circuits, and formulas Topics in Complexity Theory and Pseudorandomness (Spring 2013) Rutgers University Swastik Kopparty Scribes: John Kim, Ben Lund 1 Course Information Swastik
Notes on Complexity Theory Last updated: August, 2011. Lecture 1
Notes on Complexity Theory Last updated: August, 2011 Jonathan Katz Lecture 1 1 Turing Machines I assume that most students have encountered Turing machines before. (Students who have not may want to look
Welcome to... Problem Analysis and Complexity Theory 716.054, 3 VU
Welcome to... Problem Analysis and Complexity Theory 716.054, 3 VU Birgit Vogtenhuber Institute for Software Technology email: bvogt@ist.tugraz.at office hour: Tuesday 10:30 11:30 slides: http://www.ist.tugraz.at/pact.html
Chapter 11. 11.1 Load Balancing. Approximation Algorithms. Load Balancing. Load Balancing on 2 Machines. Load Balancing: Greedy Scheduling
Approximation Algorithms Chapter Approximation Algorithms Q. Suppose I need to solve an NP-hard problem. What should I do? A. Theory says you're unlikely to find a poly-time algorithm. Must sacrifice one
Factoring & Primality
Factoring & Primality Lecturer: Dimitris Papadopoulos In this lecture we will discuss the problem of integer factorization and primality testing, two problems that have been the focus of a great amount
! Solve problem to optimality. ! Solve problem in poly-time. ! Solve arbitrary instances of the problem. !-approximation algorithm.
Approximation Algorithms Chapter Approximation Algorithms Q Suppose I need to solve an NP-hard problem What should I do? A Theory says you're unlikely to find a poly-time algorithm Must sacrifice one of
Lecture 2: Universality
CS 710: Complexity Theory 1/21/2010 Lecture 2: Universality Instructor: Dieter van Melkebeek Scribe: Tyson Williams In this lecture, we introduce the notion of a universal machine, develop efficient universal
NP-Completeness and Cook s Theorem
NP-Completeness and Cook s Theorem Lecture notes for COM3412 Logic and Computation 15th January 2002 1 NP decision problems The decision problem D L for a formal language L Σ is the computational task:
P versus NP, and More
1 P versus NP, and More Great Ideas in Theoretical Computer Science Saarland University, Summer 2014 If you have tried to solve a crossword puzzle, you know that it is much harder to solve it than to verify
SIMS 255 Foundations of Software Design. Complexity and NP-completeness
SIMS 255 Foundations of Software Design Complexity and NP-completeness Matt Welsh November 29, 2001 mdw@cs.berkeley.edu 1 Outline Complexity of algorithms Space and time complexity ``Big O'' notation Complexity
Why Study NP- hardness. NP Hardness/Completeness Overview. P and NP. Scaling 9/3/13. Ron Parr CPS 570. NP hardness is not an AI topic
Why Study NP- hardness NP Hardness/Completeness Overview Ron Parr CPS 570 NP hardness is not an AI topic It s important for all computer scienhsts Understanding it will deepen your understanding of AI
Complexity Theory. IE 661: Scheduling Theory Fall 2003 Satyaki Ghosh Dastidar
Complexity Theory IE 661: Scheduling Theory Fall 2003 Satyaki Ghosh Dastidar Outline Goals Computation of Problems Concepts and Definitions Complexity Classes and Problems Polynomial Time Reductions Examples
NP-completeness and the real world. NP completeness. NP-completeness and the real world (2) NP-completeness and the real world
-completeness and the real world completeness Course Discrete Biological Models (Modelli Biologici Discreti) Zsuzsanna Lipták Imagine you are working for a biotech company. One day your boss calls you
How To Solve A K Path In Time (K)
What s next? Reductions other than kernelization Dániel Marx Humboldt-Universität zu Berlin (with help from Fedor Fomin, Daniel Lokshtanov and Saket Saurabh) WorKer 2010: Workshop on Kernelization Nov
Computational complexity theory
Computational complexity theory Goal: A general theory of the resources needed to solve computational problems What types of resources? Time What types of computational problems? decision problem Decision
A Working Knowledge of Computational Complexity for an Optimizer
A Working Knowledge of Computational Complexity for an Optimizer ORF 363/COS 323 Instructor: Amir Ali Ahmadi TAs: Y. Chen, G. Hall, J. Ye Fall 2014 1 Why computational complexity? What is computational
6.080/6.089 GITCS Feb 12, 2008. Lecture 3
6.8/6.89 GITCS Feb 2, 28 Lecturer: Scott Aaronson Lecture 3 Scribe: Adam Rogal Administrivia. Scribe notes The purpose of scribe notes is to transcribe our lectures. Although I have formal notes of my
8.1 Min Degree Spanning Tree
CS880: Approximations Algorithms Scribe: Siddharth Barman Lecturer: Shuchi Chawla Topic: Min Degree Spanning Tree Date: 02/15/07 In this lecture we give a local search based algorithm for the Min Degree
CSC2420 Spring 2015: Lecture 3
CSC2420 Spring 2015: Lecture 3 Allan Borodin January 22, 2015 1 / 1 Announcements and todays agenda Assignment 1 due next Thursday. I may add one or two additional questions today or tomorrow. Todays agenda
Generating models of a matched formula with a polynomial delay
Generating models of a matched formula with a polynomial delay Petr Savicky Institute of Computer Science, Academy of Sciences of Czech Republic, Pod Vodárenskou Věží 2, 182 07 Praha 8, Czech Republic
Outline. NP-completeness. When is a problem easy? When is a problem hard? Today. Euler Circuits
Outline NP-completeness Examples of Easy vs. Hard problems Euler circuit vs. Hamiltonian circuit Shortest Path vs. Longest Path 2-pairs sum vs. general Subset Sum Reducing one problem to another Clique
! X is a set of strings. ! Instance: string s. ! Algorithm A solves problem X: A(s) = yes iff s! X.
Decision Problems 8.2 Definition of NP Decision problem. X is a set of strings. Instance: string s. Algorithm A solves problem X: A(s) = yes iff s X. Polynomial time. Algorithm A runs in polytime if for
Quantum and Non-deterministic computers facing NP-completeness
Quantum and Non-deterministic computers facing NP-completeness Thibaut University of Vienna Dept. of Business Administration Austria Vienna January 29th, 2013 Some pictures come from Wikipedia Introduction
Nan Kong, Andrew J. Schaefer. Department of Industrial Engineering, Univeristy of Pittsburgh, PA 15261, USA
A Factor 1 2 Approximation Algorithm for Two-Stage Stochastic Matching Problems Nan Kong, Andrew J. Schaefer Department of Industrial Engineering, Univeristy of Pittsburgh, PA 15261, USA Abstract We introduce
The Classes P and NP
The Classes P and NP We now shift gears slightly and restrict our attention to the examination of two families of problems which are very important to computer scientists. These families constitute the
2.1 Complexity Classes
15-859(M): Randomized Algorithms Lecturer: Shuchi Chawla Topic: Complexity classes, Identity checking Date: September 15, 2004 Scribe: Andrew Gilpin 2.1 Complexity Classes In this lecture we will look
Tutorial 8. NP-Complete Problems
Tutorial 8 NP-Complete Problems Decision Problem Statement of a decision problem Part 1: instance description defining the input Part 2: question stating the actual yesor-no question A decision problem
CSC2420 Fall 2012: Algorithm Design, Analysis and Theory
CSC2420 Fall 2012: Algorithm Design, Analysis and Theory Allan Borodin November 15, 2012; Lecture 10 1 / 27 Randomized online bipartite matching and the adwords problem. We briefly return to online algorithms
Chapter 1. NP Completeness I. 1.1. Introduction. By Sariel Har-Peled, December 30, 2014 1 Version: 1.05
Chapter 1 NP Completeness I By Sariel Har-Peled, December 30, 2014 1 Version: 1.05 "Then you must begin a reading program immediately so that you man understand the crises of our age," Ignatius said solemnly.
CS 3719 (Theory of Computation and Algorithms) Lecture 4
CS 3719 (Theory of Computation and Algorithms) Lecture 4 Antonina Kolokolova January 18, 2012 1 Undecidable languages 1.1 Church-Turing thesis Let s recap how it all started. In 1990, Hilbert stated a
On the Unique Games Conjecture
On the Unique Games Conjecture Antonios Angelakis National Technical University of Athens June 16, 2015 Antonios Angelakis (NTUA) Theory of Computation June 16, 2015 1 / 20 Overview 1 Introduction 2 Preliminary
NP-Completeness. CptS 223 Advanced Data Structures. Larry Holder School of Electrical Engineering and Computer Science Washington State University
NP-Completeness CptS 223 Advanced Data Structures Larry Holder School of Electrical Engineering and Computer Science Washington State University 1 Hard Graph Problems Hard means no known solutions with
Introduction to NP-Completeness Written and copyright c by Jie Wang 1
91.502 Foundations of Comuter Science 1 Introduction to Written and coyright c by Jie Wang 1 We use time-bounded (deterministic and nondeterministic) Turing machines to study comutational comlexity of
NP-Completeness I. Lecture 19. 19.1 Overview. 19.2 Introduction: Reduction and Expressiveness
Lecture 19 NP-Completeness I 19.1 Overview In the past few lectures we have looked at increasingly more expressive problems that we were able to solve using efficient algorithms. In this lecture we introduce
Definition 11.1. Given a graph G on n vertices, we define the following quantities:
Lecture 11 The Lovász ϑ Function 11.1 Perfect graphs We begin with some background on perfect graphs. graphs. First, we define some quantities on Definition 11.1. Given a graph G on n vertices, we define
Computational Models Lecture 8, Spring 2009
Slides modified by Benny Chor, based on original slides by Maurice Herlihy, Brown Univ. p. 1 Computational Models Lecture 8, Spring 2009 Encoding of TMs Universal Turing Machines The Halting/Acceptance
On the Relationship between Classes P and NP
Journal of Computer Science 8 (7): 1036-1040, 2012 ISSN 1549-3636 2012 Science Publications On the Relationship between Classes P and NP Anatoly D. Plotnikov Department of Computer Systems and Networks,
1 Approximating Set Cover
CS 05: Algorithms (Grad) Feb 2-24, 2005 Approximating Set Cover. Definition An Instance (X, F ) of the set-covering problem consists of a finite set X and a family F of subset of X, such that every elemennt
5.1 Bipartite Matching
CS787: Advanced Algorithms Lecture 5: Applications of Network Flow In the last lecture, we looked at the problem of finding the maximum flow in a graph, and how it can be efficiently solved using the Ford-Fulkerson
1 Definition of a Turing machine
Introduction to Algorithms Notes on Turing Machines CS 4820, Spring 2012 April 2-16, 2012 1 Definition of a Turing machine Turing machines are an abstract model of computation. They provide a precise,
Theoretical Computer Science (Bridging Course) Complexity
Theoretical Computer Science (Bridging Course) Complexity Gian Diego Tipaldi A scenario You are a programmer working for a logistics company Your boss asks you to implement a program that optimizes the
Offline sorting buffers on Line
Offline sorting buffers on Line Rohit Khandekar 1 and Vinayaka Pandit 2 1 University of Waterloo, ON, Canada. email: rkhandekar@gmail.com 2 IBM India Research Lab, New Delhi. email: pvinayak@in.ibm.com
Universality in the theory of algorithms and computer science
Universality in the theory of algorithms and computer science Alexander Shen Computational models The notion of computable function was introduced in 1930ies. Simplifying (a rather interesting and puzzling)
Lecture 15 An Arithmetic Circuit Lowerbound and Flows in Graphs
CSE599s: Extremal Combinatorics November 21, 2011 Lecture 15 An Arithmetic Circuit Lowerbound and Flows in Graphs Lecturer: Anup Rao 1 An Arithmetic Circuit Lower Bound An arithmetic circuit is just like
School Timetabling in Theory and Practice
School Timetabling in Theory and Practice Irving van Heuven van Staereling VU University, Amsterdam Faculty of Sciences December 24, 2012 Preface At almost every secondary school and university, some
3. Mathematical Induction
3. MATHEMATICAL INDUCTION 83 3. Mathematical Induction 3.1. First Principle of Mathematical Induction. Let P (n) be a predicate with domain of discourse (over) the natural numbers N = {0, 1,,...}. If (1)
CS154. Turing Machines. Turing Machine. Turing Machines versus DFAs FINITE STATE CONTROL AI N P U T INFINITE TAPE. read write move.
CS54 Turing Machines Turing Machine q 0 AI N P U T IN TAPE read write move read write move Language = {0} q This Turing machine recognizes the language {0} Turing Machines versus DFAs TM can both write
CAD Algorithms. P and NP
CAD Algorithms The Classes P and NP Mohammad Tehranipoor ECE Department 6 September 2010 1 P and NP P and NP are two families of problems. P is a class which contains all of the problems we solve using
(67902) Topics in Theory and Complexity Nov 2, 2006. Lecture 7
(67902) Topics in Theory and Complexity Nov 2, 2006 Lecturer: Irit Dinur Lecture 7 Scribe: Rani Lekach 1 Lecture overview This Lecture consists of two parts In the first part we will refresh the definition
ON THE COMPLEXITY OF THE GAME OF SET. {kamalika,pbg,dratajcz,hoeteck}@cs.berkeley.edu
ON THE COMPLEXITY OF THE GAME OF SET KAMALIKA CHAUDHURI, BRIGHTEN GODFREY, DAVID RATAJCZAK, AND HOETECK WEE {kamalika,pbg,dratajcz,hoeteck}@cs.berkeley.edu ABSTRACT. Set R is a card game played with a
JUST-IN-TIME SCHEDULING WITH PERIODIC TIME SLOTS. Received December May 12, 2003; revised February 5, 2004
Scientiae Mathematicae Japonicae Online, Vol. 10, (2004), 431 437 431 JUST-IN-TIME SCHEDULING WITH PERIODIC TIME SLOTS Ondřej Čepeka and Shao Chin Sung b Received December May 12, 2003; revised February
Analysis of Approximation Algorithms for k-set Cover using Factor-Revealing Linear Programs
Analysis of Approximation Algorithms for k-set Cover using Factor-Revealing Linear Programs Stavros Athanassopoulos, Ioannis Caragiannis, and Christos Kaklamanis Research Academic Computer Technology Institute
Computability Theory
CSC 438F/2404F Notes (S. Cook and T. Pitassi) Fall, 2014 Computability Theory This section is partly inspired by the material in A Course in Mathematical Logic by Bell and Machover, Chap 6, sections 1-10.
20 Selfish Load Balancing
20 Selfish Load Balancing Berthold Vöcking Abstract Suppose that a set of weighted tasks shall be assigned to a set of machines with possibly different speeds such that the load is distributed evenly among
Diagonalization. Ahto Buldas. Lecture 3 of Complexity Theory October 8, 2009. Slides based on S.Aurora, B.Barak. Complexity Theory: A Modern Approach.
Diagonalization Slides based on S.Aurora, B.Barak. Complexity Theory: A Modern Approach. Ahto Buldas Ahto.Buldas@ut.ee Background One basic goal in complexity theory is to separate interesting complexity
Small Maximal Independent Sets and Faster Exact Graph Coloring
Small Maximal Independent Sets and Faster Exact Graph Coloring David Eppstein Univ. of California, Irvine Dept. of Information and Computer Science The Exact Graph Coloring Problem: Given an undirected
Notes on NP Completeness
Notes on NP Completeness Rich Schwartz November 10, 2013 1 Overview Here are some notes which I wrote to try to understand what NP completeness means. Most of these notes are taken from Appendix B in Douglas
MATH10040 Chapter 2: Prime and relatively prime numbers
MATH10040 Chapter 2: Prime and relatively prime numbers Recall the basic definition: 1. Prime numbers Definition 1.1. Recall that a positive integer is said to be prime if it has precisely two positive
11. APPROXIMATION ALGORITHMS
11. APPROXIMATION ALGORITHMS load balancing center selection pricing method: vertex cover LP rounding: vertex cover generalized load balancing knapsack problem Lecture slides by Kevin Wayne Copyright 2005
Dynamic TCP Acknowledgement: Penalizing Long Delays
Dynamic TCP Acknowledgement: Penalizing Long Delays Karousatou Christina Network Algorithms June 8, 2010 Karousatou Christina (Network Algorithms) Dynamic TCP Acknowledgement June 8, 2010 1 / 63 Layout
Ph.D. Thesis. Judit Nagy-György. Supervisor: Péter Hajnal Associate Professor
Online algorithms for combinatorial problems Ph.D. Thesis by Judit Nagy-György Supervisor: Péter Hajnal Associate Professor Doctoral School in Mathematics and Computer Science University of Szeged Bolyai
Introduction to computer science
Introduction to computer science Michael A. Nielsen University of Queensland Goals: 1. Introduce the notion of the computational complexity of a problem, and define the major computational complexity classes.
How To Find An Optimal Search Protocol For An Oblivious Cell
The Conference Call Search Problem in Wireless Networks Leah Epstein 1, and Asaf Levin 2 1 Department of Mathematics, University of Haifa, 31905 Haifa, Israel. lea@math.haifa.ac.il 2 Department of Statistics,
Approximated Distributed Minimum Vertex Cover Algorithms for Bounded Degree Graphs
Approximated Distributed Minimum Vertex Cover Algorithms for Bounded Degree Graphs Yong Zhang 1.2, Francis Y.L. Chin 2, and Hing-Fung Ting 2 1 College of Mathematics and Computer Science, Hebei University,
ARTICLE IN PRESS. European Journal of Operational Research xxx (2004) xxx xxx. Discrete Optimization. Nan Kong, Andrew J.
A factor 1 European Journal of Operational Research xxx (00) xxx xxx Discrete Optimization approximation algorithm for two-stage stochastic matching problems Nan Kong, Andrew J. Schaefer * Department of
Formal Languages and Automata Theory - Regular Expressions and Finite Automata -
Formal Languages and Automata Theory - Regular Expressions and Finite Automata - Samarjit Chakraborty Computer Engineering and Networks Laboratory Swiss Federal Institute of Technology (ETH) Zürich March
Chapter. NP-Completeness. Contents
Chapter 13 NP-Completeness Contents 13.1 P and NP......................... 593 13.1.1 Defining the Complexity Classes P and NP...594 13.1.2 Some Interesting Problems in NP.......... 597 13.2 NP-Completeness....................
Primality - Factorization
Primality - Factorization Christophe Ritzenthaler November 9, 2009 1 Prime and factorization Definition 1.1. An integer p > 1 is called a prime number (nombre premier) if it has only 1 and p as divisors.
Graph Security Testing
JOURNAL OF APPLIED COMPUTER SCIENCE Vol. 23 No. 1 (2015), pp. 29-45 Graph Security Testing Tomasz Gieniusz 1, Robert Lewoń 1, Michał Małafiejski 1 1 Gdańsk University of Technology, Poland Department of
14.1 Rent-or-buy problem
CS787: Advanced Algorithms Lecture 14: Online algorithms We now shift focus to a different kind of algorithmic problem where we need to perform some optimization without knowing the input in advance. Algorithms
A Note on Maximum Independent Sets in Rectangle Intersection Graphs
A Note on Maximum Independent Sets in Rectangle Intersection Graphs Timothy M. Chan School of Computer Science University of Waterloo Waterloo, Ontario N2L 3G1, Canada tmchan@uwaterloo.ca September 12,
Distributed Computing over Communication Networks: Maximal Independent Set
Distributed Computing over Communication Networks: Maximal Independent Set What is a MIS? MIS An independent set (IS) of an undirected graph is a subset U of nodes such that no two nodes in U are adjacent.
Fairness in Routing and Load Balancing
Fairness in Routing and Load Balancing Jon Kleinberg Yuval Rabani Éva Tardos Abstract We consider the issue of network routing subject to explicit fairness conditions. The optimization of fairness criteria
(IALC, Chapters 8 and 9) Introduction to Turing s life, Turing machines, universal machines, unsolvable problems.
3130CIT: Theory of Computation Turing machines and undecidability (IALC, Chapters 8 and 9) Introduction to Turing s life, Turing machines, universal machines, unsolvable problems. An undecidable problem
GRAPH THEORY LECTURE 4: TREES
GRAPH THEORY LECTURE 4: TREES Abstract. 3.1 presents some standard characterizations and properties of trees. 3.2 presents several different types of trees. 3.7 develops a counting method based on a bijection
Algebraic Computation Models. Algebraic Computation Models
Algebraic Computation Models Ζυγομήτρος Ευάγγελος ΜΠΛΑ 201118 Φεβρουάριος, 2013 Reasons for using algebraic models of computation The Turing machine model captures computations on bits. Many algorithms
Scheduling Shop Scheduling. Tim Nieberg
Scheduling Shop Scheduling Tim Nieberg Shop models: General Introduction Remark: Consider non preemptive problems with regular objectives Notation Shop Problems: m machines, n jobs 1,..., n operations
Mathematical Induction
Mathematical Induction (Handout March 8, 01) The Principle of Mathematical Induction provides a means to prove infinitely many statements all at once The principle is logical rather than strictly mathematical,
Topic: Greedy Approximations: Set Cover and Min Makespan Date: 1/30/06
CS880: Approximations Algorithms Scribe: Matt Elder Lecturer: Shuchi Chawla Topic: Greedy Approximations: Set Cover and Min Makespan Date: 1/30/06 3.1 Set Cover The Set Cover problem is: Given a set of