11. APPROXIMATION ALGORITHMS


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1 Coping with NPcompleteness 11. APPROXIMATION ALGORITHMS load balancing center selection pricing method: vertex cover LP rounding: vertex cover generalized load balancing knapsack problem Q. Suppose I need to solve an NPhard problem. What should I do? A. Sacrifice one of three desired features. i. Solve arbitrary instances of the problem. ii. Solve problem to optimality. iii. Solve problem in polynomial time. ρapproximation algorithm. Guaranteed to run in polytime. Guaranteed to solve arbitrary instance of the problem Guaranteed to find solution within ratio ρ of true optimum. Lecture slides by Kevin Wayne Copyright 005 PearsonAddison Wesley Challenge. Need to prove a solution's value is close to optimum, without even knowing what optimum value is Last updated on /15/17 6:00 PM Load balancing 11. APPROXIMATION ALGORITHMS load balancing center selection pricing method: vertex cover LP rounding: vertex cover generalized load balancing knapsack problem Input. m identical machines; n jobs, job j has processing time t j. Job j must run contiguously on one machine. A machine can process at most one job at a time. Def. Let J(i) be the subset of jobs assigned to machine i. The load of machine i is L i = Σ j J(i) t j. Def. The makespan is the maximum load on any machine L = max i L i. Load balancing. Assign each job to a machine to minimize makespan. machine 1 a dmachine 1 f machine b c Machine e g 0 L1 L time
2 Load balancing on machines is NPhard Load balancing: list scheduling Claim. Load balancing is hard even if only machines. Pf. PARTITION P LOADBALANCE. NPcomplete by Exercise 8.6 Listscheduling algorithm. Consider n jobs in some fixed order. Assign job j to machine whose load is smallest so far. ListScheduling(m, n, t 1,t,,t n ) { e a b f c g d for i = 1 to m { L i 0 } J(i) load on machine i jobs assigned to machine i length of job f for j = 1 to n { i = argmin k L k J(i) J(i) {j} machine i has smallest load assign job j to machine i L i L i + t j update load of machine i machine 1 machine a dmachine 1 f b c Machine e g yes } } return J(1),, J(m) Implementation. O(n log m) using a priority queue. 0 L time 5 6 Load balancing: list scheduling analysis Believe it or not Theorem. [Graham 1966] Greedy algorithm is a approximation. First worstcase analysis of an approximation algorithm. Need to compare resulting solution with optimal makespan L*. Lemma 1. The optimal makespan L* max j t j. Pf. Some machine must process the most timeconsuming job. Lemma. The optimal makespan Pf. The total processing time is Σ j t j. L * m 1 j t j. One of m machines must do at least a 1 / m fraction of total work. L * m 1 j t j. 7 8
3 Load balancing: list scheduling analysis Load balancing: list scheduling analysis Theorem. Greedy algorithm is a approximation. Pf. Consider load L i of bottleneck machine i. Let j be last job scheduled on machine i. When job j assigned to machine i, i had smallest load. Its load before assignment is L i t j L i t j L k for all 1 k m. Theorem. Greedy algorithm is a approximation. Pf. Consider load L i of bottleneck machine i. Let j be last job scheduled on machine i. When job j assigned to machine i, i had smallest load. Its load before assignment is L i t j L i t j L k for all 1 k m. Sum inequalities over all k and divide by m: blue jobs scheduled before j L i t j 1 m k L k = 1 m k t k L * Lemma machine i j Now, L = L i = (L i t j )!# "# $ + t j % L* L* L *. Lemma 1 0 Li  tj L = Li time 9 10 Load balancing: list scheduling analysis Load balancing: list scheduling analysis Q. Is our analysis tight? A. Essentially yes. Q. Is our analysis tight? A. Essentially yes. Ex: m machines, m (m 1) jobs length 1 jobs, one job of length m. Ex: m machines, m (m 1) jobs length 1 jobs, one job of length m. list scheduling makespan = 19 = m  1 optimal makespan = 10 = m m = 10 machine idle machine 3 idle machine idle machine 5 idle machine 6 idle machine 7 idle machine 8 idle machine 9 idle machine 10 idle m =
4 Load balancing: LPT rule Load balancing: LPT rule Longest processing time (LPT). Sort n jobs in descending order of processing time, and then run list scheduling algorithm. Observation. If at most m jobs, then listscheduling is optimal. Pf. Each job put on its own machine. Lemma 3. If there are more than m jobs, L* t m+1. LPTListScheduling(m, n, t 1,t,,t n ) { Sort jobs so that t 1 t for i = 1 to m { } L i 0 J(i) t n load on machine i jobs assigned to machine i Pf. Consider first m+1 jobs t 1,, t m+1. Since the t i 's are in descending order, each takes at least t m+1 time. There are m + 1 jobs and m machines, so by pigeonhole principle, at least one machine gets two jobs. for j = 1 to n { i = argmin k L k machine i has smallest load Theorem. LPT rule is a 3/approximation algorithm. Pf. Same basic approach as for list scheduling. } J(i) J(i) {j} L i L i + t j } return J(1),, J(m) assign job j to machine i update load of machine i 13 L i = (L i t j )!# "# $ + t j % 3 L *. L* 1 L* Lemma 3 ( by observation, can assume number of jobs > m ) 1 Load Balancing: LPT rule Q. Is our 3/ analysis tight? A. No. Theorem. [Graham 1969] LPT rule is a /3approximation. Pf. More sophisticated analysis of same algorithm. Q. Is Graham's /3 analysis tight? A. Essentially yes. 11. APPROXIMATION ALGORITHMS load balancing center selection pricing method: vertex cover LP rounding: vertex cover generalized load balancing knapsack problem Ex. m machines n = m + 1 jobs jobs of length m, m + 1,, m 1 and one more job of length m. Then, L / L* = (m 1) / (3m) 15
5 Center selection problem Center selection problem Input. Set of n sites s 1,, s n and an integer k > 0. Input. Set of n sites s 1,, s n and an integer k > 0. Center selection problem. Select set of k centers C so that maximum distance r(c) from a site to nearest center is minimized. Center selection problem. Select set of k centers C so that maximum distance r(c) from a site to nearest center is minimized. r(c) k = centers Notation. dist(x, y) = distance between sites x and y. dist(s i, C) = min c C dist(s i, c) = distance from s i to closest center. r(c) = max i dist(s i, C) = smallest covering radius. Goal. Find set of centers C that minimizes r(c), subject to C = k. center site Distance function properties. dist(x, x) = 0 [ identity ] dist(x, y) = dist(y, x) [ symmetry ] dist(x, y) dist(x, z) + dist(z, y) [ triangle inequality ] Center selection example Greedy algorithm: a false start Ex: each site is a point in the plane, a center can be any point in the plane, dist(x, y) = Euclidean distance. Remark: search can be infinite! Greedy algorithm. Put the first center at the best possible location for a single center, and then keep adding centers so as to reduce the covering radius each time by as much as possible. Remark: arbitrarily bad! k = centers k = centers r(c) greedy center 1 center site center site 19 0
6 Center selection: greedy algorithm Center selection: analysis of greedy algorithm Repeatedly choose next center to be site farthest from any existing center. GREEDYCENTERSELECTION (k, n, s 1, s,, s n) C. REPEAT k times Select a site s i with maximum distance dist(s i, C). C C s i. RETURN C. site farthest from any center Lemma. Let C* be an optimal set of centers. Then r(c) r(c*). Pf. [by contradiction] Assume r(c*) < ½ r(c). For each site c i C, consider ball of radius ½ r(c) around it. Exactly one c * i in each ball; let c i be the site paired with c i *. Consider any site s and its closest center c * i C*. dist(s, C) dist(s, c i ) dist(s, c i *) + dist(c i *, c i ) r(c*). Thus, r(c) r(c*). r(c*) since c i * is closest center Δinequality ½ r(c) ½ r(c) ½ r(c) c i Property. Upon termination, all centers in C are pairwise at least r(c) apart. Pf. By construction of algorithm. C* site s c i * 1 Center selection Dominating set reduces to center selection Lemma. Let C* be an optimal set of centers. Then r(c) r (C*). Theorem. Greedy algorithm is a approximation for center selection problem. Remark. Greedy algorithm always places centers at sites, but is still within a factor of of best solution that is allowed to place centers anywhere. Question. Is there hope of a 3/approximation? /3? e.g., points in the plane Theorem. Unless P = NP, there no ρapproximation for center selection problem for any ρ <. Pf. We show how we could use a ( ε) approximation algorithm for CENTERSELECTION selection to solve DOMINATINGSET in polytime. Let G = (V, E), k be an instance of DOMINATINGSET. Construct instance G' of CENTERSELECTION with sites V and distances dist(u, v) = 1 if (u, v) E dist(u, v) = if (u, v) E Note that G' satisfies the triangle inequality. G has dominating set of size k iff there exists k centers C* with r(c*) = 1. Thus, if G has a dominating set of size k, a ( ε)approximation algorithm for CENTERSELECTION would find a solution C* with r(c*) = 1 since it cannot use any edge of distance. 3
7 Weighted vertex cover 11. APPROXIMATION ALGORITHMS load balancing center selection pricing method: vertex cover LP rounding: vertex cover generalized load balancing knapsack problem Definition. Given a graph G = (V, E), a vertex cover is a set S V such that each edge in E has at least one end in S. Weighted vertex cover. Given a graph G with vertex weights, find a vertex cover of minimum weight. 9 9 weight = + + weight = 11 6 Pricing method Pricing method Pricing method. Each edge must be covered by some vertex. Edge e = (i, j) pays price p e 0 to use both vertex i and j. Set prices and find vertex cover simultaneously. Fairness. Edges incident to vertex i should pay w i in total. WEIGHTEDVERTEXCOVER (G, w) for each vertex i : p e e=(i, j) w i S. FOREACH e E p e 0. 9 Fairness lemma. For any vertex cover S and any fair prices pe : e pe w(s). Pf. WHILE (there exists an edge (i, j) such that neither i nor j is tight) Select such an edge e = (i, j). Increase pe as much as possible until i or j tight. S set of all tight nodes. RETURN S. each edge e covered by at least one node in S sum fairness inequalities for each node in S 7 8
8 Pricing method example Pricing method: analysis Theorem. Pricing method is a approximation for WEIGHTEDVERTEXCOVER. Pf. Algorithm terminates since at least one new node becomes tight after each iteration of while loop. Let S = set of all tight nodes upon termination of algorithm. S is a vertex cover: if some edge (i, j) is uncovered, then neither i nor j is tight. But then while loop would not terminate. vertex weight price of edge ab Let S* be optimal vertex cover. We show w(s) w(s*). w(s) = w i = i S i S p e e=(i, j) i V p e e=(i, j) = p e w(s*). e E all nodes in S are tight S V, prices 0 each edge counted twice fairness lemma 9 30 Weighted vertex cover 11. APPROXIMATION ALGORITHMS Given a graph G = (V, E) with vertex weights w i 0, find a min weight subset of vertices S V such that every edge is incident to at least one vertex in S. load balancing center selection pricing method: vertex cover LP rounding: vertex cover generalized load balancing knapsack problem total weight = = 57 3
9 Weighted vertex cover: ILP formulation Weighted vertex cover: ILP formulation Given a graph G = (V, E) with vertex weights w i 0, find a min weight subset of vertices S V such that every edge is incident to at least one vertex in S. Integer linear programming formulation. Model inclusion of each vertex i using a 0/1 variable x i. x i = " # $ Vertex covers in 1 1 correspondence with 0/1 assignments: S = { i V : x i = 1}. 0 if vertex i is not in vertex cover 1 if vertex i is in vertex cover Objective function: minimize Σ i w i x i. Weighted vertex cover. Integer linear programming formulation. (ILP) i V w i x i x i + x j 1 (i, j) E x i {0, 1} i V Observation. If x* is optimal solution to (ILP), then S = { i V : x i * = 1} is a min weight vertex cover. For every edge (i, j), must take either vertex i or j (or both): x i + x j Integer linear programming Linear programming Given integers a ij, b i, and c j, find integers x j that satisfy: c T x Ax b x 0 x Observation. Vertex cover formulation proves that INTEGERPROGRAMMING is an NPhard search problem. n j=1 n j=1 c j x j a ij x j b i 1 i m x j 0 1 j n x j 1 j n Given integers a ij, b i, and c j, find real numbers x j that satisfy: c T x Ax b x 0 Linear. No x, xy, arccos(x), x(1 x), etc. Simplex algorithm. [Dantzig 197] Can solve LP in practice. Ellipsoid algorithm. [Khachian 1979] Can solve LP in polytime. Interior point algorithms. [Karmarkar 198, Renegar 1988, ] Can solve LP in polytime and in practice. n j=1 n j=1 c j x j a ij x j b i 1 i m x j 0 1 j n 35 36
10 LP feasible region Weighted vertex cover: LP relaxation LP geometry in D. Linear programming relaxation. x 1 = 0 (LP ) i V w i x i x i + x j 1 (i, j) E x i 0 i V Observation. Optimal value of (LP) is optimal value of (ILP). Pf. LP has fewer constraints. Note. LP is not equivalent to vertex cover. ½ ½ x 1 + x = 6 x = 0 x 1 + x = 6 Q. How can solving LP help us find a small vertex cover? A. Solve LP and round fractional values. ½ Weighted vertex cover: LP rounding algorithm Weighted vertex cover inapproximability Lemma. If x* is optimal solution to (LP), then S = { i V : x i * ½} is a vertex cover whose weight is at most twice the min possible weight. Theorem. [Dinur Safra 00] If P NP, then no ρapproximation for WEIGHTEDVERTEXCOVER for any ρ < (even if all weights are 1). Pf. [S is a vertex cover] Consider an edge (i, j) E. Since x i * + x j * 1, either x i * ½ or x j * ½ (or both) (i, j) covered. Pf. [S has desired cost] Let S* be optimal vertex cover. Then w i i S* * w i x i 1 w i i S LP is a relaxation xi* ½ i S Theorem. The rounding algorithm is a approximation algorithm. Pf. Lemma + fact that LP can be solved in polytime. On the Hardness of Approximating Minimum Vertex Cover Irit Dinur Samuel Safra May 6, 00 Abstract We prove the Minimum Vertex Cover problem to be NPhard to approximate to within a factor of , extending on previous PCP and hardness of approximation technique. To that end, one needs to develop a new proof framework, and borrow and extend ideas from several fields. Open research problem. Close the gap. 39 0
11 Generalized load balancing 11. APPROXIMATION ALGORITHMS load balancing center selection pricing method: vertex cover LP rounding: vertex cover generalized load balancing knapsack problem Input. Set of m machines M; set of n jobs J. Job j J must run contiguously on an authorized machine in M j M. Job j J has processing time t j. Each machine can process at most one job at a time. Def. Let J(i) be the subset of jobs assigned to machine i. The load of machine i is L i = Σ j J(i) t j. Def. The makespan is the maximum load on any machine = max i L i. Generalized load balancing. Assign each job to an authorized machine to minimize makespan. Generalized load balancing: integer linear program and relaxation Generalized load balancing: lower bounds ILP formulation. x ij = time machine i spends processing job j. (IP) min L s. t. x i j = t j for all j J i L for all i M x i j j x i j {0, t j } for all j J and i M j Lemma 1. The optimal makespan L* max j t j. Pf. Some machine must process the most timeconsuming job. Lemma. Let L be optimal value to the LP. Then, optimal makespan L* L. Pf. LP has fewer constraints than IP formulation. x i j = 0 for all j J and i M j LP relaxation. (LP) min L s. t. x i j = t j for all j J i L for all i M x i j j x i j 0 for all j J and i M j x i j = 0 for all j J and i M j 3
12 Generalized load balancing: structure of LP solution Generalized load balancing: rounding Lemma 3. Let x be solution to LP. Let G(x) be the graph with an edge between machine i and job j if x ij > 0. Then G(x) is acyclic. Pf. (deferred) can transform x into another LP solution where G(x) is acyclic if LP solver doesn't return such an x Rounded solution. Find LP solution x where G(x) is a forest. Root forest G(x) at some arbitrary machine node r. If job j is a leaf node, assign j to its parent machine i. If job j is not a leaf node, assign j to any one of its children. Lemma. Rounded solution only assigns jobs to authorized machines. Pf. If job j is assigned to machine i, then x ij > 0. LP solution can only assign positive value to authorized machines. x ij > 0 G(x) acyclic job G(x) cyclic job machine machine 5 6 Generalized load balancing: analysis Generalized load balancing: analysis Lemma 5. If job j is a leaf node and machine i = parent(j), then x ij = t j. Pf. Since i is a leaf, x ij = 0 for all j parent(i). LP constraint guarantees Σ i x ij = t j. Theorem. Rounded solution is a approximation. Pf. Let J(i) be the jobs assigned to machine i. By LEMMA 6, the load L i on machine i has two components: Lemma 6. At most one nonleaf job is assigned to a machine. Pf. The only possible nonleaf job assigned to machine i is parent(i). leaf nodes: t j j J(i) j is a leaf = x ij x ij L L * Lemma 1 Lemma 5 j J(i) j is a leaf j J LP Lemma (LP is a relaxation) optimal value of LP parent: t parent(i) L * job machine Thus, the overall load L i L*. 7 8
13 Generalized load balancing: flow formulation Generalized load balancing: structure of solution Flow formulation of LP. x i j i x i j j = t j for all j J L for all i M x i j 0 for all j J and i M j x i j = 0 for all j J and i M j Lemma 3. Let (x, L) be solution to LP. Let G(x) be the graph with an edge from machine i to job j if x ij > 0. We can find another solution (x', L) such that G(x') is acyclic. Pf. Let C be a cycle in G(x). Augment flow along the cycle C. flow conservation maintained At least one edge from C is removed (and none are added). Repeat until G(x') is acyclic. 3 3 augment flow along cycle C Observation. Solution to feasible flow problem with value L are in 1to1 correspondence with LP solutions of value L. 1 3 G(x) 5 1 G(x') Conclusions Running time. The bottleneck operation in our approximation is solving one LP with m n + 1 variables. Remark. Can solve LP using flow techniques on a graph with m + n + 1 nodes: given L, find feasible flow if it exists. Binary search to find L*. Extensions: unrelated parallel machines. [Lenstra Shmoys Tardos 1990] Job j takes t ij time if processed on machine i. approximation algorithm via LP rounding. If P NP, then no no ρapproximation exists for any ρ < 3/. 11. APPROXIMATION ALGORITHMS load balancing center selection pricing method: vertex cover LP rounding: vertex cover generalized load balancing knapsack problem Mathematical Programming 6 (1990) NorthHolland APPROXIMATION ALGORITHMS FOR SCHEDULING UNRELATED PARALLEL MACHINES Jan Karel LENSTRA Eindhoven University of Technology, Eindhoven, The Netherlands, and Centre for Mathematics and Computer Science, Amsterdam, The Netherlands David B. SHMOYS and l~va TARDOS Cornell University, Ithaca, NY, USA 51
14 Polynomialtime approximation scheme Knapsack problem PTAS. (1 + ε)approximation algorithm for any constant ε > 0. Load balancing. [Hochbaum Shmoys 1987] Euclidean TSP. [Arora, Mitchell 1996] Consequence. PTAS produces arbitrarily high quality solution, but trades off accuracy for time. Knapsack problem. Given n objects and a knapsack. Item i has value v i > 0 and weighs w i > 0. Knapsack has weight limit W. Goal: fill knapsack so as to maximize total value. Ex: { 3, } has value 0. we assume wi W for each i This section. PTAS for knapsack problem via rounding and scaling. item value weight original instance (W = 11) 53 5 Knapsack is NPcomplete Knapsack problem: dynamic programming I KNAPSACK. Given a set X, weights w i 0, values v i 0, a weight limit W, and a target value V, is there a subset S X such that: SUBSETSUM. Given a set X, values u i 0, and an integer U, is there a subset S X whose elements sum to exactly U? Theorem. SUBSETSUM P KNAPSACK. W w i i S v i i S V Pf. Given instance (u1,, un, U) of SUBSETSUM, create KNAPSACK instance: v i = w i = u i V = W = U u i i S u i i S U U Def. OPT(i, w) = max value subset of items 1,..., i with weight limit w. Case 1. OPT does not select item i. OPT selects best of 1,, i 1 using up to weight limit w. Case. OPT selects item i. New weight limit = w w i. OPT selects best of 1,, i 1 using up to weight limit w w i. # 0 if i = 0 % OPT(i, w) = $ OPT(i 1, w) if w i > w % & max{ OPT(i 1, w), v i + OPT(i 1, w w i )} otherwise Theorem. Computes the optimal value in O(n W) time. Not polynomial in input size. Polynomial in input size if weights are small integers
15 Knapsack problem: dynamic programming II Knapsack problem: dynamic programming II Def. OPT(i, v) = min weight of a knapsack for which we can obtain a solution of value v using a subset of items 1,..., i. Note. Optimal value is the largest value v such that OPT(n, v) W. Case 1. OPT does not select item i. OPT selects best of 1,, i 1 that achieves value v. Case. OPT selects item i. Consumes weight w i, need to achieve value v v i. OPT selects best of 1,, i 1 that achieves value v v i. Theorem. Dynamic programming algorithm II computes the optimal value in O(n vmax) time, where vmax is the maximum of any value. Pf. The optimal value V* n v max. There is one subproblem for each item and for each value v V*. It takes O(1) time per subproblem. Remark 1. Not polynomial in input size! Remark. Polynomial time if values are small integers. OPT(i, v) = 0 v 0 i =0 v>0 min {OPT(i 1,v), w i + OPT(i 1,v v i )} Knapsack problem: polynomialtime approximation scheme Knapsack problem: polynomialtime approximation scheme Intuition for approximation algorithm. Round all values up to lie in smaller range. Run dynamic programming algorithm II on rounded/scaled instance. Return optimal items in rounded instance. item value weight item value weight Round up all values: 0 < ε 1 = precision parameter. v max = largest value in original instance. θ = scaling factor = ε v max / n. v i = Observation. Optimal solutions to problem with v are equivalent to optimal solutions to problem with v ˆ. Intuition. v close to v so optimal solution using v is nearly optimal; v ˆ small and integral so dynamic programming algorithm II is fast. v i, ˆv i = v i original instance (W = 11) rounded instance (W = 11) 59 60
16 Knapsack problem: polynomialtime approximation scheme Knapsack problem: polynomialtime approximation scheme Theorem. If S is solution found by rounding algorithm and S* is any other feasible solution, then Pf. Let S* be any feasible solution satisfying weight constraint. v i v i (1 + ) always round up v i v i subset containing only the item of largest value Theorem. For any ε > 0, the rounding algorithm computes a feasible solution whose value is within a (1 + ε) factor of the optimum in O(n 3 / ε) time. Pf. We have already proved the accuracy bound. Dynamic program II running time is O(n v ˆ max ), where i i S S v i (v i + ) solve rounded instance optimally never round up by more than θ choosing S* = { max } v v i + 1 v ˆv max = v max = n = v i + n S n v i + 1 v θ = ε v max / n v i + 1 v thus v v i (1 + ) i S v i v max Σ i S v i 61 6
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