1,096 research outputs found

    The world of hereditary graph classes viewed through Truemper configurations

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    In 1982 Truemper gave a theorem that characterizes graphs whose edges can be labeled so that all chordless cycles have prescribed parities. The characterization states that this can be done for a graph G if and only if it can be done for all induced subgraphs of G that are of few speci c types, that we will call Truemper con gurations. Truemper was originally motivated by the problem of obtaining a co-NP characterization of bipartite graphs that are signable to be balanced (i.e. bipartite graphs whose node-node incidence matrices are balanceable matrices). The con gurations that Truemper identi ed in his theorem ended up playing a key role in understanding the structure of several seemingly diverse classes of objects, such as regular matroids, balanceable matrices and perfect graphs. In this survey we view all these classes, and more, through the excluded Truemper con gurations, focusing on the algorithmic consequences, trying to understand what structurally enables e cient recognition and optimization algorithms

    Cliques, colouring and satisfiability : from structure to algorithms

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    We examine the implications of various structural restrictions on the computational complexity of three central problems of theoretical computer science (colourability, independent set and satisfiability), and their relatives. All problems we study are generally NP-hard and they remain NP-hard under various restrictions. Finding the greatest possible restrictions under which a problem is computationally difficult is important for a number of reasons. Firstly, this can make it easier to establish the NP-hardness of new problems by allowing easier transformations. Secondly, this can help clarify the boundary between tractable and intractable instances of the problem. Typically an NP-hard graph problem admits an infinite sequence of narrowing families of graphs for which the problem remains NP-hard. We obtain a number of such results; each of these implies necessary conditions for polynomial-time solvability of the respective problem in restricted graph classes. We also identify a number of classes for which these conditions are sufficient and describe explicit algorithms that solve the problem in polynomial time in those classes. For the satisfiability problem we use the language of graph theory to discover the very first boundary property, i.e. a property that separates tractable and intractable instances of the problem. Whether this property is unique remains a big open problem

    Efficient domination and polarity

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    The thesis considers the following graph problems: Efficient (Edge) Domination seeks for an independent vertex (edge) subset D such that all other vertices (edges) have exactly one neighbor in D. Polarity asks for a vertex subset that induces a complete multipartite graph and that contains a vertex of every induced P_3. Monopolarity is the special case of Polarity where the wanted vertex subset has to be independent. These problems are NP-complete in general, but efficiently solvable on various graph classes. The thesis sharpens known NP-completeness results and presents new solvable cases

    Clique Generalizations and Related Problems

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    A large number of real-world problems can be model as optimization problems in graphs. The clique model was introduced to aid the study of network structure for social interaction. Each vertex represented an actor and the edges represented the relations between them. Nevertheless, the model has been shown to be restrictive for modeling real-world problems, since it leaves out subgraphs that do not have all pos- sible edges. As a consequence, clique generalizations were introduced to overcome the disadvantages of the clique model. In this thesis, I present three computationally dif- ficult combinatorial optimization problems related to clique generalization problems: co-2-plexes and k-cores. A k-core is a subgraph with minimum degree greater than or equal to k. In this work, I discuss the minimal k-core problem and the minimum k-core problem. I present a backtracking algorithm to find all minimal k-cores of a given undirected graph and its applications to the study of associative memory. The proposed method is a modification of the Bron and Kerbosch algorithm for finding all cliques of an undirected graph. In addition, I study the polyhedral structure of the k-core polytope. The minimum k-core problem is modeled as a binary integer program and relaxed as a linear program. Since the relaxation yields to a non-integral solution, cuts must be added in order to improve the solution. I show that edge and cycle transversals of the graph give valid inequalities for the convex hull of k-cores. A set of pairwise non-adjacent vertices defines a stable set. A stable set is the complement of a clique. A co-2-plex is a subgraph with degree less than or equal to one, and it is a stable set relaxation. I introduce a study of the maximum weighted co-2-plex (MWC2P) problem for {claw, bull}-free graphs and present two polynomial time algorithms to solve it. One of the algorithms transforms the original graph to solve an instance of the maximum weighted stable set problem utilizing Minty’s algorithm. The second algorithm is an extension of Minty’s algorithm and solves the problem in the original graph. All the algorithms discussed in this thesis were implemented and tested. Numerical results are provided for each one of them

    Algorithms for the Maximum Independent Set Problem

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    This thesis focuses mainly on the Maximum Independent Set (MIS) problem. Some related graph theoretical combinatorial problems are also considered. As these problems are generally NP-hard, we study their complexity in hereditary graph classes, i.e. graph classes defined by a set F of forbidden induced subgraphs. We revise the literature about the issue, for example complexity results, applications, and techniques tackling the problem. Through considering some general approach, we exhibit several cases where the problem admits a polynomial-time solution. More specifically, we present polynomial-time algorithms for the MIS problem in: + some subclasses of S2;j;kS_{2;j;k}-free graphs (thus generalizing the classical result for S1;2;kS_{1;2;k}-free graphs); + some subclasses of treektree_{k}-free graphs (thus generalizing the classical results for subclasses of P5-free graphs); + some subclasses of P7P_{7}-free graphs and S2;2;2S_{2;2;2}-free graphs; and various subclasses of graphs of bounded maximum degree, for example subcubic graphs. Our algorithms are based on various approaches. In particular, we characterize augmenting graphs in a subclass of S2;k;kS_{2;k;k}-free graphs and a subclass of S2;2;5S_{2;2;5}-free graphs. These characterizations are partly based on extensions of the concept of redundant set [125]. We also propose methods finding augmenting chains, an extension of the method in [99], and finding augmenting trees, an extension of the methods in [125]. We apply the augmenting vertex technique, originally used for P5P_{5}-free graphs or banner-free graphs, for some more general graph classes. We consider a general graph theoretical combinatorial problem, the so-called Maximum -Set problem. Two special cases of this problem, the so-called Maximum F-(Strongly) Independent Subgraph and Maximum F-Induced Subgraph, where F is a connected graph set, are considered. The complexity of the Maximum F-(Strongly) Independent Subgraph problem is revised and the NP-hardness of the Maximum F-Induced Subgraph problem is proved. We also extend the augmenting approach to apply it for the general Maximum Π -Set problem. We revise on classical graph transformations and give two unified views based on pseudo-boolean functions and αff-redundant vertex. We also make extensive uses of α-redundant vertices, originally mainly used for P5P_{5}-free graphs, to give polynomial solutions for some subclasses of S2;2;2S_{2;2;2}-free graphs and treektree_{k}-free graphs. We consider some classical sequential greedy heuristic methods. We also combine classical algorithms with αff-redundant vertices to have new strategies of choosing the next vertex in greedy methods. Some aspects of the algorithms, for example forbidden induced subgraph sets and worst case results, are also considered. Finally, we restrict our attention on graphs of bounded maximum degree and subcubic graphs. Then by using some techniques, for example ff-redundant vertex, clique separator, and arguments based on distance, we general these results for some subclasses of Si;j;kS_{i;j;k}-free subcubic graphs

    On local search and LP and SDP relaxations for k-Set Packing

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    Set packing is a fundamental problem that generalises some well-known combinatorial optimization problems and knows a lot of applications. It is equivalent to hypergraph matching and it is strongly related to the maximum independent set problem. In this thesis we study the k-set packing problem where given a universe U and a collection C of subsets over U, each of cardinality k, one needs to find the maximum collection of mutually disjoint subsets. Local search techniques have proved to be successful in the search for approximation algorithms, both for the unweighted and the weighted version of the problem where every subset in C is associated with a weight and the objective is to maximise the sum of the weights. We make a survey of these approaches and give some background and intuition behind them. In particular, we simplify the algebraic proof of the main lemma for the currently best weighted approximation algorithm of Berman ([Ber00]) into a proof that reveals more intuition on what is really happening behind the math. The main result is a new bound of k/3 + 1 + epsilon on the integrality gap for a polynomially sized LP relaxation for k-set packing by Chan and Lau ([CL10]) and the natural SDP relaxation [NOTE: see page iii]. We provide detailed proofs of lemmas needed to prove this new bound and treat some background on related topics like semidefinite programming and the Lovasz Theta function. Finally we have an extended discussion in which we suggest some possibilities for future research. We discuss how the current results from the weighted approximation algorithms and the LP and SDP relaxations might be improved, the strong relation between set packing and the independent set problem and the difference between the weighted and the unweighted version of the problem.Comment: There is a mistake in the following line of Theorem 17: "As an induced subgraph of H with more edges than vertices constitutes an improving set". Therefore, the proofs of Theorem 17, and hence Theorems 19, 23 and 24, are false. It is still open whether these theorems are tru
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