Complexity and Approximation Results for the Connected Vertex Cover Problem in Graphs and Hypergraphs Bruno Escoffier, Laurent Gourvès, Jérôme Monnot
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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. -
The Travelling Salesman Problem in Bounded Degree Graphs*
The Travelling Salesman Problem in Bounded Degree Graphs? Andreas Bj¨orklund1, Thore Husfeldt1, Petteri Kaski2, and Mikko Koivisto2 1 Lund University, Department of Computer Science, P.O.Box 118, SE-22100 Lund, Sweden, e-mail: [email protected], [email protected] 2 Helsinki Institute for Information Technology HIIT, Department of Computer Science, University of Helsinki, P.O.Box 68, FI-00014 University of Helsinki, Finland, e-mail: [email protected], Abstract. We show that the travelling salesman problem in bounded- degree graphs can be solved in time O((2 − )n), where > 0 depends only on the degree bound but not on the number of cities, n. The algo- rithm is a variant of the classical dynamic programming solution due to Bellman, and, independently, Held and Karp. In the case of bounded in- teger weights on the edges, we also present a polynomial-space algorithm with running time O((2 − )n) on bounded-degree graphs. 1 Introduction There is no faster algorithm known for the travelling salesman problem than the classical dynamic programming solution from the early 1960s, discovered by Bellman [2, 3], and, independently, Held and Karp [9]. It runs in time within a polynomial factor of 2n, where n is the number of cities. Despite the half a cen- tury of algorithmic development that has followed, it remains an open problem whether the travelling salesman problem can be solved in time O(1.999n) [15]. In this paper we provide such an upper bound for graphs with bounded maximum vertex degree. For this restricted graph class, previous attemps have succeeded to prove such bounds when the degree bound, ∆, is three or four. -
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. -
On Snarks That Are Far from Being 3-Edge Colorable
On snarks that are far from being 3-edge colorable Jonas H¨agglund Department of Mathematics and Mathematical Statistics Ume˚aUniversity SE-901 87 Ume˚a,Sweden [email protected] Submitted: Jun 4, 2013; Accepted: Mar 26, 2016; Published: Apr 15, 2016 Mathematics Subject Classifications: 05C38, 05C70 Abstract In this note we construct two infinite snark families which have high oddness and low circumference compared to the number of vertices. Using this construction, we also give a counterexample to a suggested strengthening of Fulkerson's conjecture by showing that the Petersen graph is not the only cyclically 4-edge connected cubic graph which require at least five perfect matchings to cover its edges. Furthermore the counterexample presented has the interesting property that no 2-factor can be part of a cycle double cover. 1 Introduction A cubic graph is said to be colorable if it has a 3-edge coloring and uncolorable otherwise. A snark is an uncolorable cubic cyclically 4-edge connected graph. It it well known that an edge minimal counterexample (if such exists) to some classical conjectures in graph theory, such as the cycle double cover conjecture [15, 18], Tutte's 5-flow conjecture [19] and Fulkerson's conjecture [4], must reside in this family of graphs. There are various ways of measuring how far a snark is from being colorable. One such measure which was introduced by Huck and Kochol [8] is the oddness. The oddness of a bridgeless cubic graph G is defined as the minimum number of odd components in any 2-factor in G and is denoted by o(G). -
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. -
Switching 3-Edge-Colorings of Cubic Graphs Arxiv:2105.01363V1 [Math.CO] 4 May 2021
Switching 3-Edge-Colorings of Cubic Graphs Jan Goedgebeur Department of Computer Science KU Leuven campus Kulak 8500 Kortrijk, Belgium and Department of Applied Mathematics, Computer Science and Statistics Ghent University 9000 Ghent, Belgium [email protected] Patric R. J. Osterg˚ard¨ Department of Communications and Networking Aalto University School of Electrical Engineering P.O. Box 15400, 00076 Aalto, Finland [email protected] In loving memory of Johan Robaey Abstract The chromatic index of a cubic graph is either 3 or 4. Edge- Kempe switching, which can be used to transform edge-colorings, is here considered for 3-edge-colorings of cubic graphs. Computational results for edge-Kempe switching of cubic graphs up to order 30 and bipartite cubic graphs up to order 36 are tabulated. Families of cubic graphs of orders 4n + 2 and 4n + 4 with 2n edge-Kempe equivalence classes are presented; it is conjectured that there are no cubic graphs arXiv:2105.01363v1 [math.CO] 4 May 2021 with more edge-Kempe equivalence classes. New families of nonplanar bipartite cubic graphs with exactly one edge-Kempe equivalence class are also obtained. Edge-Kempe switching is further connected to cycle switching of Steiner triple systems, for which an improvement of the established classification algorithm is presented. Keywords: chromatic index, cubic graph, edge-coloring, edge-Kempe switch- ing, one-factorization, Steiner triple system. 1 1 Introduction We consider simple finite undirected graphs without loops. For such a graph G = (V; E), the number of vertices jV j is the order of G and the number of edges jEj is the size of G. -
Bounded Treewidth
Algorithms for Graphs of (Locally) Bounded Treewidth by MohammadTaghi Hajiaghayi A thesis presented to the University of Waterloo in ful¯llment of the thesis requirement for the degree of Master of Mathematics in Computer Science Waterloo, Ontario, September 2001 c MohammadTaghi Hajiaghayi, 2001 ° I hereby declare that I am the sole author of this thesis. I authorize the University of Waterloo to lend this thesis to other institutions or individuals for the purpose of scholarly research. MohammadTaghi Hajiaghayi I further authorize the University of Waterloo to reproduce this thesis by photocopying or other means, in total or in part, at the request of other institutions or individuals for the purpose of scholarly research. MohammadTaghi Hajiaghayi ii The University of Waterloo requires the signatures of all persons using or photocopying this thesis. Please sign below, and give address and date. iii Abstract Many real-life problems can be modeled by graph-theoretic problems. These graph problems are usually NP-hard and hence there is no e±cient algorithm for solving them, unless P= NP. One way to overcome this hardness is to solve the problems when restricted to special graphs. Trees are one kind of graph for which several NP-complete problems can be solved in polynomial time. Graphs of bounded treewidth, which generalize trees, show good algorithmic properties similar to those of trees. Using ideas developed for tree algorithms, Arnborg and Proskurowski introduced a general dynamic programming approach which solves many problems such as dominating set, vertex cover and independent set. Others used this approach to solve other NP-hard problems. -
Snarks and Flow-Critical Graphs 1
Snarks and Flow-Critical Graphs 1 CˆandidaNunes da Silva a Lissa Pesci a Cl´audioL. Lucchesi b a DComp – CCTS – ufscar – Sorocaba, sp, Brazil b Faculty of Computing – facom-ufms – Campo Grande, ms, Brazil Abstract It is well-known that a 2-edge-connected cubic graph has a 3-edge-colouring if and only if it has a 4-flow. Snarks are usually regarded to be, in some sense, the minimal cubic graphs without a 3-edge-colouring. We defined the notion of 4-flow-critical graphs as an alternative concept towards minimal graphs. It turns out that every snark has a 4-flow-critical snark as a minor. We verify, surprisingly, that less than 5% of the snarks with up to 28 vertices are 4-flow-critical. On the other hand, there are infinitely many 4-flow-critical snarks, as every flower-snark is 4-flow-critical. These observations give some insight into a new research approach regarding Tutte’s Flow Conjectures. Keywords: nowhere-zero k-flows, Tutte’s Flow Conjectures, 3-edge-colouring, flow-critical graphs. 1 Nowhere-Zero Flows Let k > 1 be an integer, let G be a graph, let D be an orientation of G and let ϕ be a weight function that associates to each edge of G a positive integer in the set {1, 2, . , k − 1}. The pair (D, ϕ) is a (nowhere-zero) k-flow of G 1 Support by fapesp, capes and cnpq if every vertex v of G is balanced, i. e., the sum of the weights of all edges leaving v equals the sum of the weights of all edges entering v. -
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. -
1 Vertex Cover on Hypergraphs
15-859FF: Coping with Intractability CMU, Fall 2019 Lecture #3: Reductions for Problems in FPT September 11, 2019 Lecturer: Venkatesan Guruswami Scribe: Anish Sevekari 1 Vertex Cover on Hypergraphs We define a d-regular hypergraph H = (V; F) where F is a family of subsets of V of size d. The vertex cover problem on hypergraphs is to find the smallest set S such that every edge X 2 F intersects S at least once. We parameterize the vertex cover problem by size of the optimum solution. Then the parameterized version of the problem is given a hypergraph H and an integer k, does H have a vertex cover of size k? We will now give a dkpoly(n) algorithm that solves the vertex cover problem and a O(d2d!kd) kernel for d-regular Hypergraph Vertex Cover problem. 1.1 An algorithm for Hypergraph Vertex Cover The algorithm is a simple generalization of the 2kpoly(n) algorithm for vertex cover over graphs. We pick any edge, then branch on the vertex on this edge which is in the vertex cover. Algorithm 1 VertexCover(H = (V; F); k) 1: if k = 0 and jFj > 0 then 2: return ? 3: end if 4: Pick any edge e 2 F. 5: for v 2 e do 6: H0 = (V n fvg; F n ff 2 F : v 2 fg) 7: if VertexCover(H0; k) 6=? then 8: return fvg [ VertexCover(H0; k − 1) 9: end if 10: end for 11: return ? Note that the recursion depth of this algorithm is k, and we branch on at most d points inside each call.