Introduction to Approximation Algorithms PDF
Preview Introduction to Approximation Algorithms
Introduction toApproxi- mation Algorithms Marcin Sydow Introduction to Approximation Algorithms Introduction Exponential Algorithms LocalSearch Marcin Sydow Combinatorial Appr. Algs. VertexCover SetCover SteinerTree TSP LinearPro- gramming Rounding LP-Duality Primal-Dual Projectco-(cid:28)nancedbyEuropeanUnionwithintheframeworkofEuropeanSocialFund Schema DualFitting Selected Bibliography Introduction toApproxi- Easy introduction: mation Algorithms T.H.Cormen et al. (cid:16)Introduction to Algorithms(cid:17) 3rd Marcin edition, MIT Press 2009, chapters 34,35 Sydow Specialised textbooks: Introduction Exponential V.Vazirani (cid:16)Approximation Algorithms(cid:17), Springer 2003, Algorithms LocalSearch chapters 1,2,3,12 Combinatorial Appr. Algs. D.Williamson, D.Shmoys (cid:16)The Design of Approximation VertexCover SetCover Algorithms(cid:17), Cambridge University Press, 2011 SteinerTree TSP (Ed. by T.Gonzalez) (cid:16)Handbook of Approximation LinearPro- gramming Algorithms and Metaheurisitcs(cid:17), Chapman & Hall/CRC, Rounding 2007 LP-Duality (Ed. by D.Hochbaum) (cid:16)Approximation Algorithms for Primal-Dual NP-hard Problems(cid:17), PWS 1997 Schema DualFitting NP-hard Introduction toApproxi- mation Algorithms Marcin Sydow Multiple ways to deal with NP-hard problems: Introduction considering special cases Exponential Algorithms fast heuristics (local search, genetic algorithms, etc.) LocalSearch Combinatorial fast, exponential algorithms Appr. Algs. VertexCover randomised algorithms SetCover SteinerTree TSP approximation algorithms LinearPro- gramming Rounding LP-Duality Primal-Dual Schema DualFitting NP-optimisation problem (a bit more formally) Introduction NP-optimisation problem Π consists of: toApproxi- mation Algorithms set of valid instances, DΠ, recognisable in polynomial time Marcin (assume: all the numbers are rational, and encoded in Sydow binary, |I| denotes the size of encoded instance I, in bits). Introduction each instance I ∈ D has a set of feasible solutions, Π Exponential Algorithms S (I) (cid:54)= ∅. Each feasible solution s ∈ S (I) is of length LocalSearch Π Π Combinatorial bounded by polynomial of |I|. Moreover, there is a Appr. Algs. polynomial algorithm that given a pair (I,s) decides VertexCover SetCover whether s ∈ S (I) SteinerTree Π TSP there is a polynomially computable objective function obj LinearPro- Π gramming which assigns a nonnegative rational number to each pair Rounding (I,s) (an instance and its feasible solution). LP-Duality Π is speci(cid:28)ed to be either minimisation or maximisation Primal-Dual Schema problem DualFitting NP-optimisation problem, cont. Introduction toApproxi- mation Optimal solution of an instance of a minimisation Algorithms (maximisation) problem is a feasible solution which achieves the Marcin Sydow minimum (maximum) possible value of the objective function (called also (cid:16)cost(cid:17) for minimisation or (cid:16)pro(cid:28)t(cid:17) for Introduction maximisation). Exponential Algorithms LocalSearch Combinatorial OPTΠ(I) denotes optimum objective function value for an Appr. Algs. instance I VertexCover SetCover SteinerTree TSP Decision version of an NP-optimisation problem I: a pair (I,B), LinearPro- where B ∈ Q and the decision problem is stated as (cid:16)does there gramming exist a feasible solution to I of cost ≤ B, for minimisation Rounding problem I(cid:17) (or, analogously (cid:16)of pro(cid:28)t ≥ B(cid:17), for a maximisation LP-Duality Primal-Dual problem) Schema DualFitting Extending the de(cid:28)nition of NP-hardness for optimisation problems Introduction toApproxi- mation Algorithms Marcin Sydow Decision version can be (cid:16)reduced(cid:17) to optimisation version. (i.e. Introduction Exponential polynomial algorithm for optimisation version can obviously Algorithms LocalSearch solve the decision version) Combinatorial Appr. Algs. VertexCover NP-optimisation problem can be called NP-hard if its decision SetCover SteinerTree version is NP-hard. TSP LinearPro- gramming Rounding LP-Duality Primal-Dual Schema DualFitting Example: Vertex Cover Introduction toApproxi- mation Algorithms Given a graph G = (V,E), (cid:28)nd a subset of its vertices V(cid:48) ⊆ V Marcin that: Sydow (cid:16)covers(cid:17) all edges, i.e. each edge e ∈ E is incident with at Introduction Exponential least one vertex from V(cid:48) (feasibility constraint) Algorithms LocalSearch |V(cid:48)| is minimum possible (cost function to be minimised) Combinatorial Appr. Algs. VertexCover VC is NP-complete (e.g. reduction from 3-SAT via Independent SetCover SteinerTree Set) TSP LinearPro- gramming Solving NP-hard problems on special cases may be easy. Rounding E.g. VC on cycles. LP-Duality Primal-Dual Schema DualFitting Fast Exponential Algorithms Introduction toApproxi- mation Algorithms Marcin Sydow Introduction Exponential Algorithms LocalSearch Combinatorial Appr. Algs. VertexCover SetCover SteinerTree TSP LinearPro- gramming Rounding LP-Duality Primal-Dual Schema DualFitting Finding small VC for (cid:28)xed k Introduction toApproxi- mation Algorithms Marcin If k is (cid:28)xed (e.g. k=3 or 10) then VC is has an algorithm that Sydow is polynomial of n: Introduction try all k-subsets of V (there are nk such) and check each (in Exponential (n−k)! ALolgcoarlitShemarsch time O(kn)) if it forms a VC. (total: O(knk+1) - a polynomial Combinatorial of n) Appr. Algs. VertexCover However, such a polynomial-time algorithm is infeasible even for SetCover SteinerTree moderate values of k and n (e.g. n= 1000, and k=10). TSP Interestingly, for small k there is a exponential-time algoritm LinearPro- gramming for VC that is more e(cid:30)cient Rounding LP-Duality Primal-Dual Schema DualFitting Two observations Introduction toApproxi- mation Algorithms Marcin If G has a vertex cover of size at most k, then G has at most Sydow k(n−1) edges. Introduction Exponential Lemma Algorithms LocalSearch Let G = (V,E) be a graph and (u,v) ∈ E. G has a vertex Combinatorial Appr. Algs. cover of size at most k if at least one of the graphs G \{u} or VSeetrtCexovCeorver G \{v} has a vertex cover of size at most k-1. SteinerTree TSP The above two observations lead directly to a recursive LinearPro- algorithm for VC that is feasible for small values of k. gramming Rounding LP-Duality Primal-Dual Schema DualFitting
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