On Sparsification for Computing Treewidth

We investigate whether an n-vertex instance (G,k) of Treewidth, asking whether the graph G has treewidth at most k, can efficiently be made sparse without changing its answer. By giving a special form of OR-cross-composition, we prove that this is unlikely: if there is an e > 0 and a polynomial-time algorithm that reduces n-vertex Treewidth instances to equivalent instances, of an arbitrary problem, with O(n^{2-e}) bits, then NP is in coNP/poly and the polynomial hierarchy collapses to its third level. Our sparsification lower bound has implications for structural parameterizations of Treewidth: parameterizations by measures that do not exceed the vertex count, cannot have kernels with O(k^{2-e}) bits for any e > 0, unless NP is in coNP/poly. Motivated by the question of determining the optimal kernel size for Treewidth parameterized by vertex cover, we improve the O(k^3)-vertex kernel from Bodlaender et al. (STACS 2011) to a kernel with O(k^2) vertices. Our improved kernel is based on a novel form of treewidth-invariant set. We use the q-expansion lemma of Fomin et al. (STACS 2011) to find such sets efficiently in graphs whose vertex count is superquadratic in their vertex cover number.Comment: 21 pages. Full version of the extended abstract presented at IPEC 201

Kernel Bounds for Structural Parameterizations of Pathwidth

Assuming the AND-distillation conjecture, the Pathwidth problem of determining whether a given graph G has pathwidth at most k admits no polynomial kernelization with respect to k. The present work studies the existence of polynomial kernels for Pathwidth with respect to other, structural, parameters. Our main result is that, unless NP is in coNP/poly, Pathwidth admits no polynomial kernelization even when parameterized by the vertex deletion distance to a clique, by giving a cross-composition from Cutwidth. The cross-composition works also for Treewidth, improving over previous lower bounds by the present authors. For Pathwidth, our result rules out polynomial kernels with respect to the distance to various classes of polynomial-time solvable inputs, like interval or cluster graphs. This leads to the question whether there are nontrivial structural parameters for which Pathwidth does admit a polynomial kernelization. To answer this, we give a collection of graph reduction rules that are safe for Pathwidth. We analyze the success of these results and obtain polynomial kernelizations with respect to the following parameters: the size of a vertex cover of the graph, the vertex deletion distance to a graph where each connected component is a star, and the vertex deletion distance to a graph where each connected component has at most c vertices.Comment: This paper contains the proofs omitted from the extended abstract published in the proceedings of Algorithm Theory - SWAT 2012 - 13th Scandinavian Symposium and Workshops, Helsinki, Finland, July 4-6, 201

A Branch and Bound Algorithm for Exact, Upper, and Lower Bounds on Treewidth

In this paper, a branch and bound algorithm for computing the treewidth of a graph is presented. The method incorporates extensions of existing results, and uses new pruning and reduction rules, based upon roperties of the adopted branching strategy. We discuss how the algorithm can not only be used to obtain exact bounds for the treewidth, but also to obtain upper and/or lower bounds. Computational results of the algorithm are presented

A generic NP-hardness proof for a variant of Graph Coloring

In this note, a direct proof is given of the NP-completeness of a variant of GRAPH COLORING, i.e., a generic proof is given, similar to the proof of Cook of the NP-completeness of SATISFIABILITY. Then, transformations from this variant of GRAPH COLORING to INDEPENDENT SET and to SATISFIABILITY are given. These proofs could be useful in an educational setting, where basics of the theory of NP-completeness must be explained to students whose background in combinatorial optimisation and/or graph theory is stronger than their background in logic. In addition, I believe that the proof given here is slightly easier than older generic proofs of NP-completeness

A tourist guide through treewidth

A short overview is given of many recent results in algorithmic graph theory that deal with the notions treewidth, and pathwidth. We discuss algorithms that find tree-decompositions, algorithms that use tree-decompositions to solve hard problems efficiently, graph minor theory, and some applications. The paper contains an extensive bibliography

Some lower bound results for decentralized extrema-finding in rings of processors

AbstractWe consider the problem of finding the largest of a set of n uniquely numbered processors, arranged in a ring, by means of an asynchronous distributed algorithm without a central controller. Processors are identical, except for their unique number (identity). Using a technique of Frederickson and Lynch we show that arbitrary algorithms that solve this problem on rings where processors know the ring size cannot have a better worst-case number of messages than algorithms that use only comparisons between identities. We show a similar type of result for rings, where the ring size is not known. We use these results to answer a question, posed by Korach, Rotem, and Santoro in 1981 whether each extrema-finding algorithm that uses time n on a ring of n processors must use a quadratic number of messages; and to show a lower bound of 0.683 n log(n) on the worst-case number of messages for unidirectional rings with known ring size n. Also, we give a lower bound of 12n log(n) on the average number of messages for algorithms that use only comparisons on rings with known ring size n

A structural approach to kernels for ILPs: Treewidth and Total Unimodularity

Kernelization is a theoretical formalization of efficient preprocessing for NP-hard problems. Empirically, preprocessing is highly successful in practice, for example in state-of-the-art ILP-solvers like CPLEX. Motivated by this, previous work studied the existence of kernelizations for ILP related problems, e.g., for testing feasibility of Ax <= b. In contrast to the observed success of CPLEX, however, the results were largely negative. Intuitively, practical instances have far more useful structure than the worst-case instances used to prove these lower bounds. In the present paper, we study the effect that subsystems with (Gaifman graph of) bounded treewidth or totally unimodularity have on the kernelizability of the ILP feasibility problem. We show that, on the positive side, if these subsystems have a small number of variables on which they interact with the remaining instance, then we can efficiently replace them by smaller subsystems of size polynomial in the domain without changing feasibility. Thus, if large parts of an instance consist of such subsystems, then this yields a substantial size reduction. We complement this by proving that relaxations to the considered structures, e.g., larger boundaries of the subsystems, allow worst-case lower bounds against kernelization. Thus, these relaxed structures can be used to build instance families that cannot be efficiently reduced, by any approach.Comment: Extended abstract in the Proceedings of the 23rd European Symposium on Algorithms (ESA 2015

Computing the treewidth and the minimum fill-in with the modular decomposition

Using the notion of modular decomposition we extend the class of graphs on which both the TREEWIDTH and the MINIMUM-FILL- IN problems can be solved in polynomial time. We show that if C is a class of graphs which is modularly decomposable into graphs that have a polynomial number of minimal separators, or graphs formed by adding a matching between two cliques, then both the TREEWIDTH and the MINIMUM-FILL-IN problems on C can be solved in polyno- mial time. For the graphs that are modular decomposable into cycles we give algorithms, that use respectively O(n) and O(n³) time for TREEWIDTH and MINIMUM FILL-IN

An exact algorithm for graph coloring with polynomial memory

In this paper, we give an algorithm that computes the chromatic number of a graph in O(5.283n) time and polynomial memory