# Approximation Algorithms: LP Relaxation, Rounding, and Randomized Rounding Techniques. My T. Thai

Save this PDF as:

Size: px
Start display at page:

Download "Approximation Algorithms: LP Relaxation, Rounding, and Randomized Rounding Techniques. My T. Thai"

## Transcription

1 Approximation Algorithms: LP Relaxation, Rounding, and Randomized Rounding Techniques My T. Thai 1

2 Overview An overview of LP relaxation and rounding method is as follows: 1. Formulate an optimization problem as an integer program (IP) 2. Relax the integral constraints to turn the IP to an LP 3. Solve LP to obtain an optimal solution x. 4. Construct a feasible solution x I to IP by rounding x to integers. Rounding can be done deterministically or probabilistically (called randomized rounding. Let us define some notations: Let cost(x I ) and cost(x ) be the objective values of x I and x respectively Let OP T (IP ) and OP T (LP ) be the optimal values of the IP and LP respectively. Note that OP T (IP ) is also a cost of an optimal solution for the optimization problem whereas the OP T (LP ) = cost(x ). Hence the approximation ratio using the rounding techniques can be done as follows (suppose the optimization is a minimization problem): cost(x I ) = cost(xi ) OP T (LP ) OP T (LP ) cost(xi ) OP T (IP ) cost(x ) Therefore, any upper bound of cost(xi ) cost(x ) is an approximation ratio of the construction algorithm (which is to round x to x I ). If we follows this method as mentioned about, the best approximation ratio is cost(xi ) cost(x ) OP T (IP ) OP T (LP ) Note that the supremum of OP T (IP ) OP T (LP ) is called integrality gap. 2

3 Linear Integer Program and Examples Roughly speaking, linear integer program is similar to an LP with an additional constraint, that is, variables are integer. The integer program is the problem of determining if a given integer program has a feasible solution. Let s see how we can use IP to formulate many discrete optimization problems. VERTEX COVER (VC) Recall that the vertex cover is defined as follows: Definition 1 Given a graph G = (V, E) where V = n and E = m, find a subset C V such that every edge e E has at least one endpoint in C and C has a minimum size. How to formulate VC as an IP? For each vertex i V (V = {1, 2,..., n}, let x i {0, 1} be variables such that x i = 1 if i C; otherwise, x i = 0. We have: min n i=1 x i st x i + x j 1 (i, j) E x i {0, 1} i V Hence, solving the VC problem is equivalent to solving the above IP. Exercise 1 Formulate the weighted vertex cover problem as an IP. WEIGHTED SET COVER Definition 2 Given a universe U = {1,..., m}, a collection S of subsets of U, S = {S 1,..., S n }, and a weight function w : S Q +, find a minimum weight sub-collection C = {S j 1 j n} such that C covers all elements of U. Let w j be the weight of subset S j. Let x j be a binary variables such that x j = 1 if S j C; otherwise, x j = 0. We have the corresponding IP: (1) min n w jx j st i S j x j 1 i {1,..., m} x j {0, 1} j {1,..., n} (2) 3

4 LP Relaxation and Rounding VERTEX COVER The corresponding LP of IP (1) is as follows: min n i=1 x i st x i + x j 1 (i, j) E 0 x i 1 i V Note that we can relax the last constraint 0 x i 1 to 0 x i without changing the solutions of LP (3). A main observation of this LP is that: Theorem 1 Given a graph G, any vertex x of the polyhedron defined by (3) has coordinates x i being either 0, 1/2, or 1. (This property is called half integrality) (Note that any vertex x of the polyhedron is called a basic feasible solution (or extreme point solution). Also note that a vertex of the polyhedron cannot be expressed as a convex combination of two feasible solutions.) Based on this theorem, we can construct a feasible solution x I to the IP as follows: Let x I i = 1 if x i 1/2; otherwise, xi i = 0. This technique of constructing a feasible to IP from LP is called rounding. Note that after the relaxation and rounding techniques, x I is a feasible solution to IP. What is the approximation ratio? Recall that the approximation ratio is equal to cost(x I )/cost(x ). We have: cost(x ) 1 2 (xi x I n) = 1 2 cost(xi ). Hence, this algorithm has an approximation ratio of 2. Recall that by using the matching approach, we also obtained a 2-approximation algorithm. Question: Can we use this technique to obtain a 2-approximation for weighted vertex cover problem? (3) 4

5 LP Relaxation and Rounding WEIGHTED SET COVER First, let obtain an LP from IP (2) as follows: min n w jx j st i S j x j 1 i {1,..., m} 0 x j 1 j {1,..., n} How do we obtain x I from x? Note that when we round x to x I, we must make sure that x I is a feasible solution to IP. In addition, we must also not over round it, i.e, round all fraction solutions x to 1. Of course, the over rounding still yield a feasible solution to IP, however, we will obtain too many subsets in C. Let rewrite (4) as min c T x subject to Ax b. Now, consider matrix A. What can we say about each entry a ij. a ij is equal to either 0 or 1. a ij = 1 if element i S j ; otherwise, a ij = 0. Denote f = max i A[i, :]. So f represent the frequency of the most frequent element. In other words, f denotes the maximum number of subset S j that cover an element. Why we want to define f? Notice that during the rounding process, we round some x j to 1 and the rest to 0. The rounding must satisfy the inequalities and consistencies. That is, for example, if in the first inequality, we round x 1 to 1 and the rest to 0. Then in the second inequality, we must also round x 1 to 1. Let consider this example. We have the first inequality as follows: x 1 + 0x 2 + x 3 + 0x 4 + x 5 1. The sum of all coefficients of the x j is 3. If all x j were at most 1/3, then the left hand side is at most 1. Thus there must be some x j which are at least 1/3. If x 3 1/3, we round x 3 to 1 then we d be fine. Now, you might ask what if only x 2, x 4 1/3. The answer is that: this cannot be happened. Why? Therefore, we have the following rounding algorithm: Let x I j = 1 if x j 1/f; otherwise, xi j = 0. Note that when x I j = 1, then S j C. (4) 5

6 Theorem 2 This algorithm achieves an approximation factor of f for the weighted set cover problem Proof. First, we need to solve that x I is a feasible solution to weighted set cover, that is, C covers all elements. Assume that there exists one element i that is not covered by C. Then there exists a row i such that: However, n a ij x j = j:x j 1/f a ij x j + j:x j 1/f a ij = 0 j:x j <1/f a ij x j < Contradicting to the fact that n a ijx j 1 As for the approximation ratio, it s easy to see that: j:x j 1/f a ij x j n cost(x I ) = w j x I j GENERAL COVER n w j (fx j) = f OP T (LP ) Definition 3 Given a collection of multisets of U (rather than a collection of sets). A multiset contains a specified number of copies of each element. Let a ij denote the multiplicity of element i in multiset S j. Find a sub-collection C such that the weight of C is minimum and C covers each element i b i times. (rather than just one time) The corresponding IP: The corresponding LP: min n w jx j st a i1 x a in x n b i i {1,..., m} x j {0, 1} j {1,..., n} (5) 6

7 min n w jx j st a i1 x a in x n b i i {1,..., m} (6) 0 x j 1 j {1,..., n} Question: Can we use the same approached as in the weighted set cover problem to obtain an f-approximation algorithm where f = max n i a ij? Answer: Yes SCHEDULING ON UNRELATED PARALLEL MACHINES Definition 4 : Given a set J of n jobs, a set M of m machines, and for each j J and i M, p ij Z + be the time taken to process job j on machine i. Find a schedule (for these jobs on these machines) so that the maximum completion time of any machine is minimized. Let x ij be a variable indicating whether job j is to be processed on machine i, that is, x ij = 1 if j is to be processed on i; otherwise, x ij = 0. Let t be the makespan, i.e., the maximum processing time of any machine. Clearly, the objective is to minimize t. The following IP is equivalent to our problem: Remarks: min t st m i=1 x ij = 1 n x ijp ij t x ij {0, 1} j J i M i M, j J The first constraint ensures that all jobs are scheduled The second constraint ensures that the maximum completion time of any machine is at most t The corresponding LP: (7) min t st m i=1 x ij = 1 j J n x ijp ij t i M x ij 0 i M, j J (8) 7

8 Exercise 2 Give an example showing that the integrality gap of (7) is at least m An Example: Suppose J = 1, i.e, we have only one job, and p ij = m for all i M. Then the minimum makespan is m, i.e, OP T (IP ) = m. However, the optimal solution to (8) is to schedule the job to extent of 1/m on each machine, thus OP T (LP ) = 1. Hence, the integrality gap is m. Questions: What do we learn from the above example? In (7), if p ij > t, then the IP automatically sets x ij = 0 In (8), if p ij > t, the relaxation allows us to set x ij to nonzero fractional values, thereby raising this question: Is there anyway that we can set x ij = 0 if p ij > t in the LP? Recall that an extreme point solution of a set of linear inequalities is a feasible solution that cannot be expressed as convex combination of two other feasible solutions. Let polyhedron P be a feasible region defined by a set of linear inequalities. Then an extreme point solution is a vertex of P. Suppose that we can find a schedule in which the integral schedule s makespan is at most T Z +, and the fractional variables x ij all correspond to p ij T. For all these jobs (corresponding to fractional variables x ij, we can match to machines in a one-to-one manner. If such a T exists, we have the following system, called polyhedron P (T ). where S T = {(i, j) p ij T } i:(i,j) S T x ij = 1 j J j:(i,j) S T x ij p ij T i M x ij 0 (i, j) S T (9) Lemma 1 Any extreme point solution x to P (T ) has at most n+m positives (non-zero) variables Lemma 2 For any extreme point solution x to P (T ), the number of fractionally assigned jobs (according to x ) is at most m. Proof. Let a and b be the number of jobs that ar integrally and fractionally set by x. Each fractional job is assigned to at least 2 machines, therefore, results in at least 2 positive (non-zero) entries in x. Thus we have: 8

9 a + b = n and a + 2b n + m. The second inequality dues to Lemma 1. It follows that b m and a n m. Corresponding to x, define a bipartite graph G = (A, B; E) as follows: A is the set of fractionally assigned jobs B is the set of machines to which some fractional jobs were assigned Edge (i, j) E iff x ij is fractional Lemma 3 Given such a G, there exists a perfect matching from A into B Proof. We will prove this lemma using Hall s Theorem. (Given a bipartite graph G = (A, B; E), G has a perfect matching that matches all vertices of A iff for every set S A, the neighborhood of S, i.e, N(S) = {y B x S, (x, y) E}, is at least as large as S.) Consider any subset S A (note that S is a subset of fractionally assigned jobs), we only need to check that N(S) S. Let H = (S, N(S); E(H)) be the subgraph of G induced by S and N(S). Now, we will prove that E(H) S + N(S). Consider the polyhedron P which is defined in the same way as P (T ) restricting to variables corresponding to edges in E(H). Let y be x restricted to H. Then y is an extreme solution (vertex) in P. Based on Lemma 1, y has at most S + N(S) positive variables. Since the number of positive variables is exactly the number of edges in H. Thus E(H) S + N(S). Additionally, each fractionally assigned job must be assigned to at least 2 machines. Consequently, 2 S E(H) S + N(S). Hence S N(S). A matching in G is called a perfect matching if it matches every job j A. Now, we see that if P (T ) has a feasible solution, we are able to find a schedule in which the integral schedule s makespan is at most T, and for the fractional jobs, we can find an one-to-one matching between these jobs and machines. Hence, we can find the schedule with a factor of 2T. How do we find a range of T? Let α = max j min i p ij and β be the makespan of the greedy schedule which assigns each job to a machines with minimum processing time. Then we have the following algorithm: 9

10 Algorithm 1 Scheduling on unrelated parallel machine 1: Use binary search to find the least value T [α, β] for which P (T ) is feasible 2: Find an extreme solution x of P (T ) 3: Assign all integrally set jobs to machines as in x 4: Construct the bipartite graph G = (A, B; E) as described above and find a perfect matching M 5: Assign fractionally set jobs to machines according to M Theorem 3 The above algorithm achieves an approximation guarantee of factor 2 Proof. Let OP T be the optimal solution of our problem. Note that the jobs are assigned in two steps. In the first step, all integrally set jobs are assigned based on x. Hence the makespan in this step is at most T OP T. In the second step, we assign all fractional set jobs. Note that each edge (i, j) of G satisfies p ij T. The perfect matching M found in G schedules at most one extra job on each machine. Thus their makespan is at most T. Therefore, both both steps, the total makespan is 2OP T. 10

11 Filtering and Rounding Overview of Filtering and Rounding techniques: Formulate the optimization problem as an IP Formulate the corresponding LP Use an optimal solution to the LP to construct a filtered version of the optimization problem (creating a restricted version of the IP.) This step often involve with fixing a parameter ɛ and try to define a new set of feasible solution to LP with a cost within (1 + ɛ) of the LP Using rounding technique to produce a good integral solution to the filtered problem, which is also a solution for an original IP Let s see how to use this technique to solve the metric uncapacitated facility location problem. METRIC UNCAPACITATED FACILITY LOCATION Definition 5 Given a set F of facilities and a set C of cities. Let f i be the cost of opening facility i F, and c ij be the cost of connecting city j to (opened) facility i where c ij satisfies the triangle inequality. The problem is to find a subset O F of facilities that should be opened, and an assignment function θ : C O assigning every city j to open facilities θ(j) in such a way that the total cost of opening facilities and connecting cities to open facilities is minimized, that is, to minimize the objective function: f i + i O j C c θ(j)j Let x i be an indicator variable denoting whether facility i is opened and y ij be an indicator variable denoting whether city j is assigned to facility i. We have the following IP: 11

12 min i F f ix i + i F,j C c ijy ij st i F y ij = 1 j C x i y ij 0 i F, j C x i, y ij {0, 1} i F, j C Remarks: The first constraint ensures that each city is connected to a facility The second constraint ensures this facility must be opened We have the corresponding LP as follows: (10) min i F f ix i + i F,j C c ijy ij st i F y ij = 1 j C x i y ij 0 i F, j C x i 0 i F y ij 0 i F, j C (11) Let (x, y ) be an optimal solution to the LP. Note that we interpret a fractional x i as a partially open facility and a fractional yij as a partial assignment of city j to facility i. Let F (x) = i F f ix i and Θ(y) = i F,j C c ijy ij. The Filtering Step Consider LP (11). Note that when y ij > 0, the city j can be assigned to candidate facility i. However, the cost c ij might be too large. Question: How do we control it? How do we filter out these facilities? Fix a parameter ɛ > 0 (we will determine ɛ later). For each city j, we will filter out all facilities i such that c ij is greater than (1 + ɛ) of the optimal assignment cost (in an LP). Let the current optimal assignment cost for city j be 12

13 Θ j = c ij yij i F Note that yij is an optimal solution to LP (11). Define F j as a set of possible (candidate) facilities such that we can assign j to as follows: F j = {i y ij > 0, c ij (1 + ɛ)θ j} Using the filtering, we construct a feasible (x, y ) to LP (11) so that the following two conditions must be held: The cost of (x, y ) is not too far from the cost of (x, y ) y ij > 0 implies that c ij (1 + ɛ)θ j Note that the second condition also implies that when c ij > (1 + ɛ)θ j, set y ij = 0. Now, how do we define y ij based on y ij? Note that we set i F y ij = 1. Therefore, define: y ij = y ij i F j y ij if i F j Otherwise, y ij = 0 Now, how do we define x i. Recall that x must satisfy the second constrain, which is x i y ij 0. So first, let us compare the cost of y and y. Θ j = c ij yij > c ij yij > (1 + ɛ)θ j yij = (1 + ɛ)θ j 1 i F i/ F j i/ F j i Fj Therefore, y ij 1 i F j yij < 1 + ɛ ɛ 13

14 Which means y ij (1+ɛ ɛ )y ij. Thus, define x i = min{1, ( 1 + ɛ )x i } ɛ Hence, we have a filter solution (x, y ) for LP (11). The Rounding Step We will round (x, y ) to the feasible integer solutions to IP (10) as follows: 1. Pick an unassigned city j with the smallest assignment cost Θ j. 2. For this city j, we open the facility θ(j) F j with the smallest opening cost f θ(j), that is, round x θ(j) to 1 and y θ(j)j to For all other cities j such that F j F j, round y θ(j)j = 1, that is assign cities j to open facility θ(j) 4. Repeat this process until all cities are assigned Theorem 4 The above algorithm has an approximation ratio of max{3(1 + ɛ), (1 + 1/ɛ)}. Set ɛ = 1/3, we obtain an approximation algorithm within the ratio of 4. Proof. Let (ˆx, ŷ) be a solution obtained from our algorithm. Let Z(ˆx, ŷ) be the cost of the objective function. Recall that F (x) = i F f ix i and Θ(y) = i F,j C c ijy ij. Hence Z(ˆx, ŷ) = F (ˆx) + Θ(ŷ). Likewise, let Z(x, y ) = F (x ) + Θ(y ) and Z(x, y ) = F (x ) + Θ(y ). So, Z(x, y ) is the optimal cost of LP (11). Let OP T be the optimal cost of our optimization problem. First, let compare Z(x, y ) with Z(x, y ). We have: F (x ) = i f i x i i f i (1 + 1/ɛ)x i = (1 + 1/ɛ)F (x ) Θ(y ) = j c ij y ij i j (1 + ɛ)θ j y ij (1 + ɛ)θ(y ) i 14

15 Note that in the rounding procedure, for the least cost unassigned city j, we open only one facility θ(j) with the least opening cost f θ(j) in F j. We have: f θ(j) = f θ(j) y ij f θ(j) x i f i x i i F j i F j i F j So the cost of opening facility θ(j) is at most the fractional cost of all facilities that can cover j. Additionally, for any other j, the facility opening cost is not increased (since we don t open any facility). Therefore, F (ˆx) F (x ) (1 + 1/ɛ)F (x ) However, for each j, we have increased the assignment cost to θ(j). Note that the cost c ij satisfies the triangle inequality, we have: c θ(j)j c ij + c ij + c θ(j)j (1 + ɛ)θ j + (1 + ɛ)θ j + (1 + ɛ)θ j 3(1 + ɛ)θ j Thus, Θ(ŷ) = j In total, we have: c θ(j)j 3(1 + ɛ) j Θ j = 3(1 + ɛ)θ(y ) Z(ˆx, ŷ) = F (ˆx) + Θ(ŷ) (1 + 1/ɛ)F (x ) + 3(1 + ɛ)θ(y ) max{3(1 + ɛ), (1 + 1/ɛ)}OP T Hence the approximation ratio is max{3(1 + ɛ), (1 + 1/ɛ)}. When ɛ = 1/3 (we choose such ɛ to minimize this maximum), we have the ratio of 4. 15

16 Randomized Rounding Overview of randomized rounding techniques: Similar to the rounding techniques, except that at the rounding step, we round an optimal fractional solution x randomly according to some probability distribution Goal of the rounding: Obtain a good approximation ratio with high probability (some positive constant, e.g., > 1/2) Independently run the algorithm many times to increase this probability WEIGHTED SET COVER (WSC) As above, we have the LP (after using the relaxation techniques) of the WSC as follows: min n w jx j st j:i S j x j 1 i = 1... m x j 0 j = 1... n (12) Recall that x j = 1 iff S j C; otherwise, x j = 0. Suppose we have an optimal solution x of LP (12). To obtain x I, we use the randomized rounding as follows: Round x j to 1 with probability x j, that is Pr[xI j = 1] = x j. (Also round x j to 0 with probability 1 x j ) Note that Pr[x I j = 1] represents the probability that the set S j is selected in the sub-collection C. Then the expected cost of our solutions is: E[cost(x I )] = n Pr[x I j = 1] w j = n w j x j = OP T (LP ) 16

17 Note that x I should be feasible and cost(x I ) ρ OP T where OP T is the cost of the optimal solutions to IP and ρ is some approximation ratio. Therefore, what we really want is to find the probability that x I satisfies the above requirements. If this probability is at least some positive constant, then we say that ρ is an approximation ratio of this algorithm. We have: Pr[x I is feasible and cost(x I ) ρ OP T ] = 1 Pr[x I is not feasible or cost(x I ) > ρ OP T ] 1 Pr[x I is not feasible] Pr[cost(x I ) > ρ OP T ] Now, we need to find Pr[x I is not feasible] and Pr[cost(x I ) > ρ OP T ]. Let s find the Pr[x I is not feasible] first. Consider an element i U. Suppose that i occurs in k sets of S, we have: x j1 + x j2 + + x jk 1 (Note that the above inequality is the inequality in the matrix A corresponding to element i (i th rows)). The probability that this element is not covered is: Pr[i is not covered] = (1 x j1) (1 x jk) Therefore, ( k (x j1 + + x jk ) ) k (1 1 k k )k 1 e Pr[x I is not feasible] m Pr[i is not covered] m e i=1 17

18 This bound is very bad since m is large. We can obtain a better one by independently running the above strategy t > 0 (to be determined) times and round x j to 0 with the probability (1 x j )t (instead of (1 x j )), that is, set x I j to 0 when xi j = 0 in all t rounds. Then we have: Pr[i is not covered] ( 1 e )t Thus, Pr[x I is not feasible] m( 1 e )t. When t is logarithmically large, i.e., t = θ(log m), then m( 1 e )t 1. Now, we consider the Pr[cost(x I ) > ρ OP T ]. We have already proved that E[cost(x I )] = OP T (LP ) OP T in one time. So when we run this strategy in t times, we have E[cost(x I )] t OP T. Based on the Markov s Inequality, we have: Pr[cost(x I ) > ρ OP T ] E[cost(xI )] ρ OP T t OP T ρ OP T = t ρ Therefore, Pr[x I is feasible and cost(x I ) ρ OP T ] 1 m( 1 e )t t ρ Now, set t = θ(log m) and ρ = 4t so that 1 m( 1 e )t t ρ 1/2. We conclude that this algorithm has an approximation ratio within a factor of O(log m) with probability at least 1/2. Remark: Let X be a nonnegative random variable with a known expectation and a positive number p R +, Markov s Inequality says that Pr[X p] E[X] p 18

19 MAXIMUM SATISFIABILITY (MAX-SAT) Definition 6 Given a conjunctive normal form formula f on Boolean variables X = {x 1,..., x n }, consisting of m clauses C 1,..., C m weighted w 1,..., w m Z +. Find a truth assignment to the Boolean variables that maximizes the total weight of satisfied clauses. Remarks: Literals: Boolean variables or their negations. For example, if x X, then x and x are literals over X. Clause: a disjunction of literals Conjunctive normal form: a conjunction of clauses Truth assignment: A truth assignment for X is a function t : X {T, F }. If t(x) = T, we say x is true under t, otherwise, x is false. The literal x is true under t iff the variable x is true under t. Satisfied clause: A clause is satisfied by a truth assignment iff at least one of its members is true under that assignment. For example, we have a clause C 1 = {x 1, x 2, x 3 }. We say C 1 is satisfied by t unless t(x 1 ) = F, t(x 2 ) = T, and t(x 3 ) = F. Satisfied collection: A collection C of clauses over X is satisfiable iff there exists some truth assignment for X that simultaneously satisfies all the clauses C j C. An Example: Let X = {x 1, x 2 } and C = {C 1, C 2 } where C 1 = {x 1, x 2 } and C 2 = { x 1, x 2 }. Define t as t(x 1 ) = T and t(x 2 ) = T, then C is satisfiable under t. If C = {{x 1, x 2 }, {x 1, x 2 }, { x 1 }}, then C is not satisfiable (under any truth assignment t). 19

20 1. A Simple Randomized Algorithm for MAX-SAT 1: For each variable x i X 2: independently set t(x i ) = T with probability 1/2 Theorem 5 Let W be the cost (weight) of a random assignment t and OP T be the cost of an optimal assignment, then E[W ] 1 2 OP T Proof. Let l j denote the length of clause C j and W j denote the random variable indicating the event {C j is satisfied}, that is: W j = 1 if C j is satisfied; otherwise, W j = 0. We have: E[W ] = m w j Pr[W j = 1] = m w j [1 (1/2) l j ] 1 2 m w j 1 2 OP T Hence, we conclude that the above algorithm can obtain an (expected) approximation ratio 2. Now, instead of converting this into a high probability statement, we will show how to derandomize this algorithm by a method known as conditional expectation. With this method, the algorithm will deterministically computes a truth assignment such that the weight of satisfied clauses is E[W ] The important point of this method is that for setting x 1 to truth (or false), we can find a formula f in polynomial time on the remaining n 1 variables by removing all clauses that are satisfied (where x 1 = T ). The solution of f is the solution of f having x 1 = T. The basic ideas of the conditional expectation is as follows. Consider a fixed k [n]. Let a 1,..., a i, 1 i k be a truth assignment to x 1,..., x i. Let f be a formula obtained by setting x 1 = a 1,..., x i = a i and removing a set C of all clauses that are already satisfied. Then, E[W f x i = a i, 1 i k] = E[W f ] + c C w c 20

21 Thus we can compute E[W f x i = a i, 1 i k] in polynomial time. Also notice that for k 1, we have E[W x i = a i, 1 i k 1] = E[W x i = a i, 1 i k 1, x k = T ]/2+ E[W x i = a i, 1 i k 1, x k = F ]/2 It follows that the larger value between the two expectations on the right hand side is at least as large as that of the left hand side. Therefore, we can set x 1 to be T or F, compute the corresponding E, then following the path that leads to the larger E. Eventually, we will get a truth assignment such that the total weight of clauses satisfied by this assignment by at least E[W ]. 2. A Randomized Rounding (LP Rounding) Algorithm for MAX- SAT Notations: y i = 1 iff x i = True; otherwise, y i = 0 z j = 1 iff C j is satisfied; otherwise, z j = 0 We can formulate the MAX-SAT as the following IP: max m w jz j st i:x i C j y i + i: x i C j (1 y i ) z j j = 1 m y i, z j {0, 1} i = 1 n, j = 1 m (13) Its equivalent LP (after the relaxation technique) is: max m w jz j st i:x i C j y i + i: x i C j (1 y i ) z j 0 y i 1 = 1 n 0 z j 1 j = 1 m j = 1 m (14) 21

22 Let (y, z ) be the optimal solution to the LP (14). Similar to the rounding strategy for SET COVER, round x i to True with probability yi. We have: Theorem 6 The above algorithm has an e/(e 1) approximation ratio. Proof. Again, let W be the cost (weight) of the obtained solution, we have: E[W ] = = = m w j Pr[W j = 1] m m m w j w j w j 1 (1 yi ) y i i:x i C j i: x i C j [ i:x 1 i C j (1 yi ) + ] i: x i C j yi lj 1 m w j (1 l j ( l j y i:xi Cj i + ) (1 y i: xi Cj i ) [ 1 z j l j ] lj ) m ] ) w j (1 [1 1lj lj z j [ min j (1 1 1 ] ) lj m w j zj l j ( 1 1 ) OP T e l j ( ) l j (15) Note that at the step ( ), the function g(z) = 1 ( ) 1 z lj l j is a con- ( ) lj. 1 1 l j Therefore, for cave function with g(0) = 0 and g(1) = 1 ( ) ) lj z [0, 1], g(z) 1 (1 1lj z. Hence the above algorithm has an (expected) approximation ratio e/(e 1). 22

23 Again, we can derandomize this algorithm and obtain a deterministic e/(e 1) ratio. 23

24 3. A Combination of the Above Two Algorithms Note that for the randomized algorithm, called A 1, we have: E[W ] = m w j Pr[W j = 1] = m [1 (1/2) l j ]OP T and for the LP randomized rounding algorithm, called A 2, we have: ( ] ) [1 1lj lj E[W ] 1 OP T ] ) Notice that when l j increases, [1 (1/2) l j (1 ] increases whereas lj [1 1lj decreases. It implies that when the length of each clause 2, A 2 works well whereas A 1 works better when the length of each clause 2. Therefore, it is natural to combine these two algorithms as follows: Run both algorithms A 1, A 2 and report the better assignment. Call this algorithm A. Theorem 7 The algorithm A has an approximation ratio 4/3. Proof. Let W 1 and W 2 represent the cost obtained from algorithms A 1 and A 2 respectively. We have: [ E[W E[max{W 1, W 2 1 ] + E[W 2 ] ] }] E ( 2 m ( 1 w j 1 1 ) ( 2 2 l [1 1l ] ) lj j 2 j m α lj + β lj m w j zj 2 m α lj + β lj OP T 2 z j ) 24

25 ( where α lj = lj ) ( and β lj = 1 [1 1l ] ) l j j Note that when l j = 1 or l j = 2, we have α lj + β lj 3/2 and when l j 3, α lj + β lj 7/8 + (1 1/e) 3/2. Therefore: E[max{W 1, W 2 }] 3 4 OP T 25

### 10.1 Integer Programming and LP relaxation

CS787: Advanced Algorithms Lecture 10: LP Relaxation and Rounding In this lecture we will design approximation algorithms using linear programming. The key insight behind this approach is that the closely

### Lecture 7: Approximation via Randomized Rounding

Lecture 7: Approximation via Randomized Rounding Often LPs return a fractional solution where the solution x, which is supposed to be in {0, } n, is in [0, ] n instead. There is a generic way of obtaining

### Lecture 3: Linear Programming Relaxations and Rounding

Lecture 3: Linear Programming Relaxations and Rounding 1 Approximation Algorithms and Linear Relaxations For the time being, suppose we have a minimization problem. Many times, the problem at hand can

### Minimum Makespan Scheduling

Minimum Makespan Scheduling Minimum makespan scheduling: Definition and variants Factor 2 algorithm for identical machines PTAS for identical machines Factor 2 algorithm for unrelated machines Martin Zachariasen,

### princeton univ. F 13 cos 521: Advanced Algorithm Design Lecture 6: Provable Approximation via Linear Programming Lecturer: Sanjeev Arora

princeton univ. F 13 cos 521: Advanced Algorithm Design Lecture 6: Provable Approximation via Linear Programming Lecturer: Sanjeev Arora Scribe: One of the running themes in this course is the notion of

### Lecture 6: Approximation via LP Rounding

Lecture 6: Approximation via LP Rounding Let G = (V, E) be an (undirected) graph. A subset C V is called a vertex cover for G if for every edge (v i, v j ) E we have v i C or v j C (or both). In other

### ! 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

### THEORY OF SIMPLEX METHOD

Chapter THEORY OF SIMPLEX METHOD Mathematical Programming Problems A mathematical programming problem is an optimization problem of finding the values of the unknown variables x, x,, x n that maximize

### Spectral graph theory

Spectral graph theory Uri Feige January 2010 1 Background With every graph (or digraph) one can associate several different matrices. We have already seen the vertex-edge incidence matrix, the Laplacian

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

### Good luck, veel succes!

Final exam Advanced Linear Programming, May 7, 13.00-16.00 Switch off your mobile phone, PDA and any other mobile device and put it far away. No books or other reading materials are allowed. This exam

### 1 Unrelated Parallel Machine Scheduling

IIIS 014 Spring: ATCS - Selected Topics in Optimization Lecture date: Mar 13, 014 Instructor: Jian Li TA: Lingxiao Huang Scribe: Yifei Jin (Polishing not finished yet.) 1 Unrelated Parallel Machine Scheduling

### 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

### Discrete Optimization

Discrete Optimization [Chen, Batson, Dang: Applied integer Programming] Chapter 3 and 4.1-4.3 by Johan Högdahl and Victoria Svedberg Seminar 2, 2015-03-31 Todays presentation Chapter 3 Transforms using

### ! 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

### 8.1 Makespan Scheduling

600.469 / 600.669 Approximation Algorithms Lecturer: Michael Dinitz Topic: Dynamic Programing: Min-Makespan and Bin Packing Date: 2/19/15 Scribe: Gabriel Kaptchuk 8.1 Makespan Scheduling Consider an instance

### 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

### The multi-integer set cover and the facility terminal cover problem

The multi-integer set cover and the facility teral cover problem Dorit S. Hochbaum Asaf Levin December 5, 2007 Abstract The facility teral cover problem is a generalization of the vertex cover problem.

### Lecture Notes 6: Approximations for MAX-SAT. 2 Notes about Weighted Vertex Cover Approximation

Algorithmic Methods //00 Lecture Notes 6: Approximations for MAX-SAT Professor: Yossi Azar Scribe:Alon Ardenboim Introduction Although solving SAT is nown to be NP-Complete, in this lecture we will cover

### 2.3 Scheduling jobs on identical parallel machines

2.3 Scheduling jobs on identical parallel machines There are jobs to be processed, and there are identical machines (running in parallel) to which each job may be assigned Each job = 1,,, must be processed

### RANDOMIZATION IN APPROXIMATION AND ONLINE ALGORITHMS. Manos Thanos Randomized Algorithms NTUA

RANDOMIZATION IN APPROXIMATION AND ONLINE ALGORITHMS 1 Manos Thanos Randomized Algorithms NTUA RANDOMIZED ROUNDING Linear Programming problems are problems for the optimization of a linear objective function,

### 1 Polyhedra and Linear Programming

CS 598CSC: Combinatorial Optimization Lecture date: January 21, 2009 Instructor: Chandra Chekuri Scribe: Sungjin Im 1 Polyhedra and Linear Programming In this lecture, we will cover some basic material

### 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;

### Week 5 Integral Polyhedra

Week 5 Integral Polyhedra We have seen some examples 1 of linear programming formulation that are integral, meaning that every basic feasible solution is an integral vector. This week we develop a theory

### Markov Chains, part I

Markov Chains, part I December 8, 2010 1 Introduction A Markov Chain is a sequence of random variables X 0, X 1,, where each X i S, such that P(X i+1 = s i+1 X i = s i, X i 1 = s i 1,, X 0 = s 0 ) = P(X

### Answers to some of the exercises.

Answers to some of the exercises. Chapter 2. Ex.2.1 (a) There are several ways to do this. Here is one possibility. The idea is to apply the k-center algorithm first to D and then for each center in D

### Definition of a Linear Program

Definition of a Linear Program Definition: A function f(x 1, x,..., x n ) of x 1, x,..., x n is a linear function if and only if for some set of constants c 1, c,..., c n, f(x 1, x,..., x n ) = c 1 x 1

### max cx s.t. Ax c where the matrix A, cost vector c and right hand side b are given and x is a vector of variables. For this example we have x

Linear Programming Linear programming refers to problems stated as maximization or minimization of a linear function subject to constraints that are linear equalities and inequalities. Although the study

### 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

### 5 INTEGER LINEAR PROGRAMMING (ILP) E. Amaldi Fondamenti di R.O. Politecnico di Milano 1

5 INTEGER LINEAR PROGRAMMING (ILP) E. Amaldi Fondamenti di R.O. Politecnico di Milano 1 General Integer Linear Program: (ILP) min c T x Ax b x 0 integer Assumption: A, b integer The integrality condition

### Permutation Betting Markets: Singleton Betting with Extra Information

Permutation Betting Markets: Singleton Betting with Extra Information Mohammad Ghodsi Sharif University of Technology ghodsi@sharif.edu Hamid Mahini Sharif University of Technology mahini@ce.sharif.edu

### 3. Linear Programming and Polyhedral Combinatorics

Massachusetts Institute of Technology Handout 6 18.433: Combinatorial Optimization February 20th, 2009 Michel X. Goemans 3. Linear Programming and Polyhedral Combinatorics Summary of what was seen in the

### 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

### Linear Programming. March 14, 2014

Linear Programming March 1, 01 Parts of this introduction to linear programming were adapted from Chapter 9 of Introduction to Algorithms, Second Edition, by Cormen, Leiserson, Rivest and Stein [1]. 1

### Lecture 3. Linear Programming. 3B1B Optimization Michaelmas 2015 A. Zisserman. Extreme solutions. Simplex method. Interior point method

Lecture 3 3B1B Optimization Michaelmas 2015 A. Zisserman Linear Programming Extreme solutions Simplex method Interior point method Integer programming and relaxation The Optimization Tree Linear Programming

### Introduction to Linear Programming.

Chapter 1 Introduction to Linear Programming. This chapter introduces notations, terminologies and formulations of linear programming. Examples will be given to show how real-life problems can be modeled

### 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

### Minimize subject to. x S R

Chapter 12 Lagrangian Relaxation This chapter is mostly inspired by Chapter 16 of [1]. In the previous chapters, we have succeeded to find efficient algorithms to solve several important problems such

### I. Solving Linear Programs by the Simplex Method

Optimization Methods Draft of August 26, 2005 I. Solving Linear Programs by the Simplex Method Robert Fourer Department of Industrial Engineering and Management Sciences Northwestern University Evanston,

### Section Notes 3. The Simplex Algorithm. Applied Math 121. Week of February 14, 2011

Section Notes 3 The Simplex Algorithm Applied Math 121 Week of February 14, 2011 Goals for the week understand how to get from an LP to a simplex tableau. be familiar with reduced costs, optimal solutions,

### Chapter 15 Introduction to Linear Programming

Chapter 15 Introduction to Linear Programming An Introduction to Optimization Spring, 2014 Wei-Ta Chu 1 Brief History of Linear Programming The goal of linear programming is to determine the values of

### Quiz 1 Sample Questions IE406 Introduction to Mathematical Programming Dr. Ralphs

Quiz 1 Sample Questions IE406 Introduction to Mathematical Programming Dr. Ralphs These questions are from previous years and should you give you some idea of what to expect on Quiz 1. 1. Consider the

### 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].

### Theory of Linear Programming

Theory of Linear Programming Debasis Mishra April 6, 2011 1 Introduction Optimization of a function f over a set S involves finding the maximum (minimum) value of f (objective function) in the set S (feasible

### ECEN 5682 Theory and Practice of Error Control Codes

ECEN 5682 Theory and Practice of Error Control Codes Convolutional Codes University of Colorado Spring 2007 Linear (n, k) block codes take k data symbols at a time and encode them into n code symbols.

### 5 The Beginning of Transcendental Numbers

5 The Beginning of Transcendental Numbers We have defined a transcendental number (see Definition 3 of the Introduction), but so far we have only established that certain numbers are irrational. We now

### Introduction to Flocking {Stochastic Matrices}

Supelec EECI Graduate School in Control Introduction to Flocking {Stochastic Matrices} A. S. Morse Yale University Gif sur - Yvette May 21, 2012 CRAIG REYNOLDS - 1987 BOIDS The Lion King CRAIG REYNOLDS

### 4.3 Limit of a Sequence: Theorems

4.3. LIMIT OF A SEQUENCE: THEOREMS 5 4.3 Limit of a Sequence: Theorems These theorems fall in two categories. The first category deals with ways to combine sequences. Like numbers, sequences can be added,

### Sequences and Convergence in Metric Spaces

Sequences and Convergence in Metric Spaces Definition: A sequence in a set X (a sequence of elements of X) is a function s : N X. We usually denote s(n) by s n, called the n-th term of s, and write {s

### Discrete (and Continuous) Optimization Solutions of Exercises 1 WI4 131

Discrete (and Continuous) Optimization Solutions of Exercises 1 WI4 131 Kees Roos Technische Universiteit Delft Faculteit Informatietechnologie en Systemen Afdeling Informatie, Systemen en Algoritmiek

### Single machine parallel batch scheduling with unbounded capacity

Workshop on Combinatorics and Graph Theory 21th, April, 2006 Nankai University Single machine parallel batch scheduling with unbounded capacity Yuan Jinjiang Department of mathematics, Zhengzhou University

### Linear Programming I

Linear Programming I November 30, 2003 1 Introduction In the VCR/guns/nuclear bombs/napkins/star wars/professors/butter/mice problem, the benevolent dictator, Bigus Piguinus, of south Antarctica penguins

### Resource Allocation with Time Intervals

Resource Allocation with Time Intervals Andreas Darmann Ulrich Pferschy Joachim Schauer Abstract We study a resource allocation problem where jobs have the following characteristics: Each job consumes

### AM 221: Advanced Optimization Spring Prof. Yaron Singer Lecture 8 February 24th, 2014

AM 221: Advanced Optimization Spring 2014 Prof. Yaron Singer Lecture 8 February 24th, 2014 1 Overview Last week we talked about the Simplex algorithm. Today we ll broaden the scope of the objectives we

### An Approximation Algorithm for Bounded Degree Deletion

An Approximation Algorithm for Bounded Degree Deletion Tomáš Ebenlendr Petr Kolman Jiří Sgall Abstract Bounded Degree Deletion is the following generalization of Vertex Cover. Given an undirected graph

### 3 Does the Simplex Algorithm Work?

Does the Simplex Algorithm Work? In this section we carefully examine the simplex algorithm introduced in the previous chapter. Our goal is to either prove that it works, or to determine those circumstances

### The Real Numbers. Here we show one way to explicitly construct the real numbers R. First we need a definition.

The Real Numbers Here we show one way to explicitly construct the real numbers R. First we need a definition. Definitions/Notation: A sequence of rational numbers is a funtion f : N Q. Rather than write

### Solving systems of linear equations over finite fields

Solving systems of linear equations over finite fields June 1, 2007 1 Introduction 2 3 4 Basic problem Introduction x 1 + x 2 + x 3 = 1 x 1 + x 2 = 1 x 1 + x 2 + x 4 = 1 x 2 + x 4 = 0 x 1 + x 3 + x 4 =

### Markov chain mixing and coupling

Markov chain mixing and coupling Recall from Lecture 2: Definition 0.. Let S be a countable set and (X n ) a sequence of random variables taking values in S. We say that (X n ) is a Markov chain if it

### The Simplex Method. yyye

Yinyu Ye, MS&E, Stanford MS&E310 Lecture Note #05 1 The Simplex Method Yinyu Ye Department of Management Science and Engineering Stanford University Stanford, CA 94305, U.S.A. http://www.stanford.edu/

### 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

### 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

### ONLINE DEGREE-BOUNDED STEINER NETWORK DESIGN. Sina Dehghani Saeed Seddighin Ali Shafahi Fall 2015

ONLINE DEGREE-BOUNDED STEINER NETWORK DESIGN Sina Dehghani Saeed Seddighin Ali Shafahi Fall 2015 ONLINE STEINER FOREST PROBLEM An initially given graph G. s 1 s 2 A sequence of demands (s i, t i ) arriving

### A NEW APPROXIMATION TECHNIQUE FOR RESOURCE-ALLOCATION PROBLEMS

A NEW APPROXIMATION TECHNIQUE FOR RESOURCE-ALLOCATION PROBLEMS BARNA SAHA AND ARAVIND SRINIVASAN Abstract. We develop a rounding method based on random walks in polytopes, which leads to improved approximation

### CHAPTER 9. Integer Programming

CHAPTER 9 Integer Programming An integer linear program (ILP) is, by definition, a linear program with the additional constraint that all variables take integer values: (9.1) max c T x s t Ax b and x integral

### princeton university F 02 cos 597D: a theorist s toolkit Lecture 7: Markov Chains and Random Walks

princeton university F 02 cos 597D: a theorist s toolkit Lecture 7: Markov Chains and Random Walks Lecturer: Sanjeev Arora Scribe:Elena Nabieva 1 Basics A Markov chain is a discrete-time stochastic process

### (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

### Discuss the size of the instance for the minimum spanning tree problem.

3.1 Algorithm complexity The algorithms A, B are given. The former has complexity O(n 2 ), the latter O(2 n ), where n is the size of the instance. Let n A 0 be the size of the largest instance that can

### Advanced Topics in Machine Learning (Part II)

Advanced Topics in Machine Learning (Part II) 3. Convexity and Optimisation February 6, 2009 Andreas Argyriou 1 Today s Plan Convex sets and functions Types of convex programs Algorithms Convex learning

### GENERATING LOW-DEGREE 2-SPANNERS

SIAM J. COMPUT. c 1998 Society for Industrial and Applied Mathematics Vol. 27, No. 5, pp. 1438 1456, October 1998 013 GENERATING LOW-DEGREE 2-SPANNERS GUY KORTSARZ AND DAVID PELEG Abstract. A k-spanner

### 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

### 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

### Subgradients. S. Boyd and L. Vandenberghe Notes for EE364b, Stanford University, Winter April 13, 2008

Subgradients S. Boyd and L. Vandenberghe Notes for EE364b, Stanford University, Winter 2006-07 April 13, 2008 1 Definition We say a vector g R n is a subgradient of f : R n R at x domf if for all z domf,

### Approximation Algorithms. Scheduling. Approximation algorithms. Scheduling jobs on a single machine

Approximation algorithms Approximation Algorithms Fast. Cheap. Reliable. Choose two. NP-hard problems: choose 2 of optimal polynomial time all instances Approximation algorithms. Trade-off between time

### 1 Introduction. Linear Programming. Questions. A general optimization problem is of the form: choose x to. max f(x) subject to x S. where.

Introduction Linear Programming Neil Laws TT 00 A general optimization problem is of the form: choose x to maximise f(x) subject to x S where x = (x,..., x n ) T, f : R n R is the objective function, S

### Permutation Betting Markets: Singleton Betting with Extra Information

Permutation Betting Markets: Singleton Betting with Extra Information Mohammad Ghodsi Sharif University of Technology ghodsi@sharif.edu Hamid Mahini Sharif University of Technology mahini@ce.sharif.edu

### 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

### Homework MA 725 Spring, 2012 C. Huneke SELECTED ANSWERS

Homework MA 725 Spring, 2012 C. Huneke SELECTED ANSWERS 1.1.25 Prove that the Petersen graph has no cycle of length 7. Solution: There are 10 vertices in the Petersen graph G. Assume there is a cycle C

### AN EXTENSION OF THE ELIMINATION METHOD FOR A SPARSE SOS POLYNOMIAL

AN EXTENSION OF THE ELIMINATION METHOD FOR A SPARSE SOS POLYNOMIAL Hayato Waki Masakazu Muramatsu The University of Electro-Communications (September 20, 2011) Abstract We propose a method to reduce the

### Scheduling and (Integer) Linear Programming

Scheduling and (Integer) Linear Programming Christian Artigues LAAS - CNRS & Université de Toulouse, France artigues@laas.fr Master Class CPAIOR 2012 - Nantes Christian Artigues Scheduling and (Integer)

### By W.E. Diewert. July, Linear programming problems are important for a number of reasons:

APPLIED ECONOMICS By W.E. Diewert. July, 3. Chapter : Linear Programming. Introduction The theory of linear programming provides a good introduction to the study of constrained maximization (and minimization)

### v 1. v n R n we have for each 1 j n that v j v n max 1 j n v j. i=1

1. Limits and Continuity It is often the case that a non-linear function of n-variables x = (x 1,..., x n ) is not really defined on all of R n. For instance f(x 1, x 2 ) = x 1x 2 is not defined when x

### 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

### Fall 2015 Midterm 1 24/09/15 Time Limit: 80 Minutes

Math 340 Fall 2015 Midterm 1 24/09/15 Time Limit: 80 Minutes Name (Print): This exam contains 6 pages (including this cover page) and 5 problems. Enter all requested information on the top of this page,

### 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

### Linear Programming Notes V Problem Transformations

Linear Programming Notes V Problem Transformations 1 Introduction Any linear programming problem can be rewritten in either of two standard forms. In the first form, the objective is to maximize, the material

### Minimal Cost Reconfiguration of Data Placement in a Storage Area Network

Minimal Cost Reconfiguration of Data Placement in a Storage Area Network Hadas Shachnai Gal Tamir Tami Tamir Abstract Video-on-Demand (VoD) services require frequent updates in file configuration on the

### Lecture 7 - Linear Programming

COMPSCI 530: Design and Analysis of Algorithms DATE: 09/17/2013 Lecturer: Debmalya Panigrahi Lecture 7 - Linear Programming Scribe: Hieu Bui 1 Overview In this lecture, we cover basics of linear programming,

### A Branch and Bound Algorithm for Solving the Binary Bi-level Linear Programming Problem

A Branch and Bound Algorithm for Solving the Binary Bi-level Linear Programming Problem John Karlof and Peter Hocking Mathematics and Statistics Department University of North Carolina Wilmington Wilmington,

### SHARP BOUNDS FOR THE SUM OF THE SQUARES OF THE DEGREES OF A GRAPH

31 Kragujevac J. Math. 25 (2003) 31 49. SHARP BOUNDS FOR THE SUM OF THE SQUARES OF THE DEGREES OF A GRAPH Kinkar Ch. Das Department of Mathematics, Indian Institute of Technology, Kharagpur 721302, W.B.,

### Geometry of Linear Programming

Chapter 2 Geometry of Linear Programming The intent of this chapter is to provide a geometric interpretation of linear programming problems. To conceive fundamental concepts and validity of different algorithms

### Solutions to Exercises 8

Discrete Mathematics Lent 2009 MA210 Solutions to Exercises 8 (1) Suppose that G is a graph in which every vertex has degree at least k, where k 1, and in which every cycle contains at least 4 vertices.

### CHAPTER IV NORMED LINEAR SPACES AND BANACH SPACES

CHAPTER IV NORMED LINEAR SPACES AND BANACH SPACES DEFINITION A Banach space is a real normed linear space that is a complete metric space in the metric defined by its norm. A complex Banach space is a

### 9th Max-Planck Advanced Course on the Foundations of Computer Science (ADFOCS) Primal-Dual Algorithms for Online Optimization: Lecture 1

9th Max-Planck Advanced Course on the Foundations of Computer Science (ADFOCS) Primal-Dual Algorithms for Online Optimization: Lecture 1 Seffi Naor Computer Science Dept. Technion Haifa, Israel Introduction

### Definition The covariance of X and Y, denoted by cov(x, Y ) is defined by. cov(x, Y ) = E(X µ 1 )(Y µ 2 ).

Correlation Regression Bivariate Normal Suppose that X and Y are r.v. s with joint density f(x y) and suppose that the means of X and Y are respectively µ 1 µ 2 and the variances are 1 2. Definition The

### 1 Basic Definitions and Concepts in Graph Theory

CME 305: Discrete Mathematics and Algorithms 1 Basic Definitions and Concepts in Graph Theory A graph G(V, E) is a set V of vertices and a set E of edges. In an undirected graph, an edge is an unordered

### MATHEMATICAL BACKGROUND

Chapter 1 MATHEMATICAL BACKGROUND This chapter discusses the mathematics that is necessary for the development of the theory of linear programming. We are particularly interested in the solutions of a

### 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