30,494 research outputs found
The weighted stable matching problem
We study the stable matching problem in non-bipartite graphs with incomplete
but strict preference lists, where the edges have weights and the goal is to
compute a stable matching of minimum or maximum weight. This problem is known
to be NP-hard in general. Our contribution is two fold: a polyhedral
characterization and an approximation algorithm. Previously Chen et al. have
shown that the stable matching polytope is integral if and only if the subgraph
obtained after running phase one of Irving's algorithm is bipartite. We improve
upon this result by showing that there are instances where this subgraph might
not be bipartite but one can further eliminate some edges and arrive at a
bipartite subgraph. Our elimination procedure ensures that the set of stable
matchings remains the same, and thus the stable matching polytope of the final
subgraph contains the incidence vectors of all stable matchings of our original
graph. This allows us to characterize a larger class of instances for which the
weighted stable matching problem is polynomial-time solvable. We also show that
our edge elimination procedure is best possible, meaning that if the subgraph
we arrive at is not bipartite, then there is no bipartite subgraph that has the
same set of stable matchings as the original graph. We complement these results
with a -approximation algorithm for the minimum weight stable matching
problem for instances where each agent has at most two possible partners in any
stable matching. This is the first approximation result for any class of
instances with general weights.Comment: This is an extended version of a paper to appear at the The Fourth
International Workshop on Matching Under Preferences (MATCH-UP 2017
Fast Evaluation of Interlace Polynomials on Graphs of Bounded Treewidth
We consider the multivariate interlace polynomial introduced by Courcelle
(2008), which generalizes several interlace polynomials defined by Arratia,
Bollobas, and Sorkin (2004) and by Aigner and van der Holst (2004). We present
an algorithm to evaluate the multivariate interlace polynomial of a graph with
n vertices given a tree decomposition of the graph of width k. The best
previously known result (Courcelle 2008) employs a general logical framework
and leads to an algorithm with running time f(k)*n, where f(k) is doubly
exponential in k. Analyzing the GF(2)-rank of adjacency matrices in the context
of tree decompositions, we give a faster and more direct algorithm. Our
algorithm uses 2^{3k^2+O(k)}*n arithmetic operations and can be efficiently
implemented in parallel.Comment: v4: Minor error in Lemma 5.5 fixed, Section 6.6 added, minor
improvements. 44 pages, 14 figure
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
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