141 research outputs found
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
Bounded Search Tree Algorithms for Parameterized Cograph Deletion: Efficient Branching Rules by Exploiting Structures of Special Graph Classes
Many fixed-parameter tractable algorithms using a bounded search tree have
been repeatedly improved, often by describing a larger number of branching
rules involving an increasingly complex case analysis. We introduce a novel and
general search strategy that branches on the forbidden subgraphs of a graph
class relaxation. By using the class of -sparse graphs as the relaxed
graph class, we obtain efficient bounded search tree algorithms for several
parameterized deletion problems. We give the first non-trivial bounded search
tree algorithms for the cograph edge-deletion problem and the trivially perfect
edge-deletion problems. For the cograph vertex deletion problem, a refined
analysis of the runtime of our simple bounded search algorithm gives a faster
exponential factor than those algorithms designed with the help of complicated
case distinctions and non-trivial running time analysis [21] and computer-aided
branching rules [11].Comment: 23 pages. Accepted in Discrete Mathematics, Algorithms and
Applications (DMAA
The PACE 2017 Parameterized Algorithms and Computational Experiments Challenge: The Second Iteration
In this article, the Program Committee of the Second Parameterized Algorithms and Computational Experiments challenge (PACE 2017) reports on the second iteration of the PACE challenge. Track A featured the Treewidth problem and Track B the Minimum Fill-In problem. Over 44 participants on 17 teams from 11 countries submitted their implementations to the competition
Cooperative Games with Bounded Dependency Degree
Cooperative games provide a framework to study cooperation among
self-interested agents. They offer a number of solution concepts describing how
the outcome of the cooperation should be shared among the players.
Unfortunately, computational problems associated with many of these solution
concepts tend to be intractable---NP-hard or worse. In this paper, we
incorporate complexity measures recently proposed by Feige and Izsak (2013),
called dependency degree and supermodular degree, into the complexity analysis
of cooperative games. We show that many computational problems for cooperative
games become tractable for games whose dependency degree or supermodular degree
are bounded. In particular, we prove that simple games admit efficient
algorithms for various solution concepts when the supermodular degree is small;
further, we show that computing the Shapley value is always in FPT with respect
to the dependency degree. Finally, we note that, while determining the
dependency among players is computationally hard, there are efficient
algorithms for special classes of games.Comment: 10 pages, full version of accepted AAAI-18 pape
The Use of a Pruned Modular Decomposition for Maximum Matching Algorithms on Some Graph Classes
We address the following general question: given a graph class C on which we can solve Maximum Matching in (quasi) linear time, does the same hold true for the class of graphs that can be modularly decomposed into C? As a way to answer this question for distance-hereditary graphs and some other superclasses of cographs, we study the combined effect of modular decomposition with a pruning process over the quotient subgraphs. We remove sequentially from all such subgraphs their so-called one-vertex extensions (i.e., pendant, anti-pendant, twin, universal and isolated vertices). Doing so, we obtain a "pruned modular decomposition", that can be computed in quasi linear time. Our main result is that if all the pruned quotient subgraphs have bounded order then a maximum matching can be computed in linear time. The latter result strictly extends a recent framework in (Coudert et al., SODA\u2718). Our work is the first to explain why the existence of some nice ordering over the modules of a graph, instead of just over its vertices, can help to speed up the computation of maximum matchings on some graph classes
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