515 research outputs found
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
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