NP-Complete Graph Problems
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Vertex Cover Might Be Hard to Approximate to Within 2 Ε − Subhash Khot ∗ Oded Regev †
Vertex Cover Might be Hard to Approximate to within 2 ε − Subhash Khot ∗ Oded Regev † Abstract Based on a conjecture regarding the power of unique 2-prover-1-round games presented in [Khot02], we show that vertex cover is hard to approximate within any constant factor better than 2. We actually show a stronger result, namely, based on the same conjecture, vertex cover on k-uniform hypergraphs is hard to approximate within any constant factor better than k. 1 Introduction Minimum vertex cover is the problem of finding the smallest set of vertices that touches all the edges in a given graph. This is one of the most fundamental NP-complete problems. A simple 2- approximation algorithm exists for this problem: construct a maximal matching by greedily adding edges and then let the vertex cover contain both endpoints of each edge in the matching. It can be seen that the resulting set of vertices indeed touches all the edges and that its size is at most twice the size of the minimum vertex cover. However, despite considerable efforts, state of the art techniques can only achieve an approximation ratio of 2 o(1) [16, 21]. − Given this state of affairs, one might strongly suspect that vertex cover is NP-hard to approxi- mate within 2 ε for any ε> 0. This is one of the major open questions in the field of approximation − algorithms. In [18], H˚astad showed that approximating vertex cover within constant factors less 7 than 6 is NP-hard. This factor was recently improved by Dinur and Safra [10] to 1.36. -
3.1 Matchings and Factors: Matchings and Covers
1 3.1 Matchings and Factors: Matchings and Covers This copyrighted material is taken from Introduction to Graph Theory, 2nd Ed., by Doug West; and is not for further distribution beyond this course. These slides will be stored in a limited-access location on an IIT server and are not for distribution or use beyond Math 454/553. 2 Matchings 3.1.1 Definition A matching in a graph G is a set of non-loop edges with no shared endpoints. The vertices incident to the edges of a matching M are saturated by M (M-saturated); the others are unsaturated (M-unsaturated). A perfect matching in a graph is a matching that saturates every vertex. perfect matching M-unsaturated M-saturated M Contains copyrighted material from Introduction to Graph Theory by Doug West, 2nd Ed. Not for distribution beyond IIT’s Math 454/553. 3 Perfect Matchings in Complete Bipartite Graphs a 1 The perfect matchings in a complete b 2 X,Y-bigraph with |X|=|Y| exactly c 3 correspond to the bijections d 4 f: X -> Y e 5 Therefore Kn,n has n! perfect f 6 matchings. g 7 Kn,n The complete graph Kn has a perfect matching iff… Contains copyrighted material from Introduction to Graph Theory by Doug West, 2nd Ed. Not for distribution beyond IIT’s Math 454/553. 4 Perfect Matchings in Complete Graphs The complete graph Kn has a perfect matching iff n is even. So instead of Kn consider K2n. We count the perfect matchings in K2n by: (1) Selecting a vertex v (e.g., with the highest label) one choice u v (2) Selecting a vertex u to match to v K2n-2 2n-1 choices (3) Selecting a perfect matching on the rest of the vertices. -
Approximation Algorithms
Lecture 21 Approximation Algorithms 21.1 Overview Suppose we are given an NP-complete problem to solve. Even though (assuming P = NP) we 6 can’t hope for a polynomial-time algorithm that always gets the best solution, can we develop polynomial-time algorithms that always produce a “pretty good” solution? In this lecture we consider such approximation algorithms, for several important problems. Specific topics in this lecture include: 2-approximation for vertex cover via greedy matchings. • 2-approximation for vertex cover via LP rounding. • Greedy O(log n) approximation for set-cover. • Approximation algorithms for MAX-SAT. • 21.2 Introduction Suppose we are given a problem for which (perhaps because it is NP-complete) we can’t hope for a fast algorithm that always gets the best solution. Can we hope for a fast algorithm that guarantees to get at least a “pretty good” solution? E.g., can we guarantee to find a solution that’s within 10% of optimal? If not that, then how about within a factor of 2 of optimal? Or, anything non-trivial? As seen in the last two lectures, the class of NP-complete problems are all equivalent in the sense that a polynomial-time algorithm to solve any one of them would imply a polynomial-time algorithm to solve all of them (and, moreover, to solve any problem in NP). However, the difficulty of getting a good approximation to these problems varies quite a bit. In this lecture we will examine several important NP-complete problems and look at to what extent we can guarantee to get approximately optimal solutions, and by what algorithms. -
A New Spectral Bound on the Clique Number of Graphs
A New Spectral Bound on the Clique Number of Graphs Samuel Rota Bul`o and Marcello Pelillo Dipartimento di Informatica - University of Venice - Italy {srotabul,pelillo}@dsi.unive.it Abstract. Many computer vision and patter recognition problems are intimately related to the maximum clique problem. Due to the intractabil- ity of this problem, besides the development of heuristics, a research di- rection consists in trying to find good bounds on the clique number of graphs. This paper introduces a new spectral upper bound on the clique number of graphs, which is obtained by exploiting an invariance of a continuous characterization of the clique number of graphs introduced by Motzkin and Straus. Experimental results on random graphs show the superiority of our bounds over the standard literature. 1 Introduction Many problems in computer vision and pattern recognition can be formulated in terms of finding a completely connected subgraph (i.e. a clique) of a given graph, having largest cardinality. This is called the maximum clique problem (MCP). One popular approach to object recognition, for example, involves matching an input scene against a stored model, each being abstracted in terms of a relational structure [1,2,3,4], and this problem, in turn, can be conveniently transformed into the equivalent problem of finding a maximum clique of the corresponding association graph. This idea was pioneered by Ambler et. al. [5] and was later developed by Bolles and Cain [6] as part of their local-feature-focus method. Now, it has become a standard technique in computer vision, and has been employing in such diverse applications as stereo correspondence [7], point pattern matching [8], image sequence analysis [9]. -
Complexity and Approximation Results for the Connected Vertex Cover Problem in Graphs and Hypergraphs Bruno Escoffier, Laurent Gourvès, Jérôme Monnot
Complexity and approximation results for the connected vertex cover problem in graphs and hypergraphs Bruno Escoffier, Laurent Gourvès, Jérôme Monnot To cite this version: Bruno Escoffier, Laurent Gourvès, Jérôme Monnot. Complexity and approximation results forthe connected vertex cover problem in graphs and hypergraphs. 2007. hal-00178912 HAL Id: hal-00178912 https://hal.archives-ouvertes.fr/hal-00178912 Preprint submitted on 12 Oct 2007 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Laboratoire d'Analyse et Modélisation de Systèmes pour l'Aide à la Décision CNRS UMR 7024 CAHIER DU LAMSADE 262 Juillet 2007 Complexity and approximation results for the connected vertex cover problem in graphs and hypergraphs Bruno Escoffier, Laurent Gourvès, Jérôme Monnot Complexity and approximation results for the connected vertex cover problem in graphs and hypergraphs Bruno Escoffier∗ Laurent Gourv`es∗ J´erˆome Monnot∗ Abstract We study a variation of the vertex cover problem where it is required that the graph induced by the vertex cover is connected. We prove that this problem is polynomial in chordal graphs, has a PTAS in planar graphs, is APX-hard in bipartite graphs and is 5/3-approximable in any class of graphs where the vertex cover problem is polynomial (in particular in bipartite graphs). -
Approximation Algorithms Instructor: Richard Peng Nov 21, 2016
CS 3510 Design & Analysis of Algorithms Section B, Lecture #34 Approximation Algorithms Instructor: Richard Peng Nov 21, 2016 • DISCLAIMER: These notes are not necessarily an accurate representation of what I said during the class. They are mostly what I intend to say, and have not been carefully edited. Last week we formalized ways of showing that a problem is hard, specifically the definition of NP , NP -completeness, and NP -hardness. We now discuss ways of saying something useful about these hard problems. Specifi- cally, we show that the answer produced by our algorithm is within a certain fraction of the optimum. For a maximization problem, suppose now that we have an algorithm A for our prob- lem which, given an instance I, returns a solution with value A(I). The approximation ratio of algorithm A is dened to be OP T (I) max : I A(I) This is not restricted to NP-complete problems: we can apply this to matching to get a 2-approximation. Lemma 0.1. The maximal matching has size at least 1=2 of the optimum. Proof. Let the maximum matching be M ∗. Consider the process of taking a maximal matching: an edge in M ∗ can't be chosen only after one (or both) of its endpoints are picked. Each edge we add to the maximal matching only has 2 endpoints, so it takes at least M ∗ 2 edges to make all edges in M ∗ ineligible. This ratio is indeed close to tight: consider a graph that has many length 3 paths, and the maximal matching keeps on taking the middle edges. -
What Is the Set Cover Problem
18.434 Seminar in Theoretical Computer Science 1 of 5 Tamara Stern 2.9.06 Set Cover Problem (Chapter 2.1, 12) What is the set cover problem? Idea: “You must select a minimum number [of any size set] of these sets so that the sets you have picked contain all the elements that are contained in any of the sets in the input (wikipedia).” Additionally, you want to minimize the cost of the sets. Input: Ground elements, or Universe U= { u1, u 2 ,..., un } Subsets SSSU1, 2 ,..., k ⊆ Costs c1, c 2 ,..., ck Goal: Find a set I ⊆ {1,2,...,m} that minimizes ∑ci , such that U SUi = . i∈ I i∈ I (note: in the un-weighted Set Cover Problem, c j = 1 for all j) Why is it useful? It was one of Karp’s NP-complete problems, shown to be so in 1972. Other applications: edge covering, vertex cover Interesting example: IBM finds computer viruses (wikipedia) elements- 5000 known viruses sets- 9000 substrings of 20 or more consecutive bytes from viruses, not found in ‘good’ code A set cover of 180 was found. It suffices to search for these 180 substrings to verify the existence of known computer viruses. Another example: Consider General Motors needs to buy a certain amount of varied supplies and there are suppliers that offer various deals for different combinations of materials (Supplier A: 2 tons of steel + 500 tiles for $x; Supplier B: 1 ton of steel + 2000 tiles for $y; etc.). You could use set covering to find the best way to get all the materials while minimizing cost. -
1 Bipartite Matching and Vertex Covers
ORF 523 Lecture 6 Princeton University Instructor: A.A. Ahmadi Scribe: G. Hall Any typos should be emailed to a a [email protected]. In this lecture, we will cover an application of LP strong duality to combinatorial optimiza- tion: • Bipartite matching • Vertex covers • K¨onig'stheorem • Totally unimodular matrices and integral polytopes. 1 Bipartite matching and vertex covers Recall that a bipartite graph G = (V; E) is a graph whose vertices can be divided into two disjoint sets such that every edge connects one node in one set to a node in the other. Definition 1 (Matching, vertex cover). A matching is a disjoint subset of edges, i.e., a subset of edges that do not share a common vertex. A vertex cover is a subset of the nodes that together touch all the edges. (a) An example of a bipartite (b) An example of a matching (c) An example of a vertex graph (dotted lines) cover (grey nodes) Figure 1: Bipartite graphs, matchings, and vertex covers 1 Lemma 1. The cardinality of any matching is less than or equal to the cardinality of any vertex cover. This is easy to see: consider any matching. Any vertex cover must have nodes that at least touch the edges in the matching. Moreover, a single node can at most cover one edge in the matching because the edges are disjoint. As it will become clear shortly, this property can also be seen as an immediate consequence of weak duality in linear programming. Theorem 1 (K¨onig). If G is bipartite, the cardinality of the maximum matching is equal to the cardinality of the minimum vertex cover. -