98 research outputs found
Hypergraph Acyclicity and Propositional Model Counting
We show that the propositional model counting problem #SAT for CNF- formulas
with hypergraphs that allow a disjoint branches decomposition can be solved in
polynomial time. We show that this class of hypergraphs is incomparable to
hypergraphs of bounded incidence cliquewidth which were the biggest class of
hypergraphs for which #SAT was known to be solvable in polynomial time so far.
Furthermore, we present a polynomial time algorithm that computes a disjoint
branches decomposition of a given hypergraph if it exists and rejects
otherwise. Finally, we show that some slight extensions of the class of
hypergraphs with disjoint branches decompositions lead to intractable #SAT,
leaving open how to generalize the counting result of this paper
Clique-width : harnessing the power of atoms.
Many NP-complete graph problems are polynomial-time solvable on graph classes of bounded clique-width. Several of these problems are polynomial-time solvable on a hereditary graph class G if they are so on the atoms (graphs with no clique cut-set) of G . Hence, we initiate a systematic study into boundedness of clique-width of atoms of hereditary graph classes. A graph G is H-free if H is not an induced subgraph of G, and it is (H1,H2) -free if it is both H1 -free and H2 -free. A class of H-free graphs has bounded clique-width if and only if its atoms have this property. This is no longer true for (H1,H2) -free graphs, as evidenced by one known example. We prove the existence of another such pair (H1,H2) and classify the boundedness of clique-width on (H1,H2) -free atoms for all but 18 cases
New Polynomial Cases of the Weighted Efficient Domination Problem
Let G be a finite undirected graph. A vertex dominates itself and all its
neighbors in G. A vertex set D is an efficient dominating set (e.d. for short)
of G if every vertex of G is dominated by exactly one vertex of D. The
Efficient Domination (ED) problem, which asks for the existence of an e.d. in
G, is known to be NP-complete even for very restricted graph classes.
In particular, the ED problem remains NP-complete for 2P3-free graphs and
thus for P7-free graphs. We show that the weighted version of the problem
(abbreviated WED) is solvable in polynomial time on various subclasses of
2P3-free and P7-free graphs, including (P2+P4)-free graphs, P5-free graphs and
other classes.
Furthermore, we show that a minimum weight e.d. consisting only of vertices
of degree at most 2 (if one exists) can be found in polynomial time. This
contrasts with our NP-completeness result for the ED problem on planar
bipartite graphs with maximum degree 3
Maximum Independent Sets in Subcubic Graphs: New Results
The maximum independent set problem is known to be NP-hard in the class of
subcubic graphs, i.e. graphs of vertex degree at most 3. We present a
polynomial-time solution in a subclass of subcubic graphs generalizing several
previously known results
Well-quasi-ordering versus clique-width : new results on bigenic classes.
Daligault, Rao and Thomassé conjectured that if a hereditary class of graphs is well-quasi-ordered by the induced subgraph relation then it has bounded clique-width. Lozin, Razgon and Zamaraev recently showed that this conjecture is not true for infinitely defined classes. For finitely defined classes the conjecture is still open. It is known to hold for classes of graphs defined by a single forbidden induced subgraph H, as such graphs are well-quasi-ordered and are of bounded clique-width if and only if H is an induced subgraph of P4P4. For bigenic classes of graphs i.e. ones defined by two forbidden induced subgraphs there are several open cases in both classifications. We reduce the number of open cases for well-quasi-orderability of such classes from 12 to 9. Our results agree with the conjecture and imply that there are only two remaining cases to verify for bigenic classes
Complexity of Coloring Graphs without Paths and Cycles
Let and denote a path on vertices and a cycle on
vertices, respectively. In this paper we study the -coloring problem for
-free graphs. Maffray and Morel, and Bruce, Hoang and Sawada,
have proved that 3-colorability of -free graphs has a finite forbidden
induced subgraphs characterization, while Hoang, Moore, Recoskie, Sawada, and
Vatshelle have shown that -colorability of -free graphs for
does not. These authors have also shown, aided by a computer search, that
4-colorability of -free graphs does have a finite forbidden induced
subgraph characterization. We prove that for any , the -colorability of
-free graphs has a finite forbidden induced subgraph
characterization. We provide the full lists of forbidden induced subgraphs for
and . As an application, we obtain certifying polynomial time
algorithms for 3-coloring and 4-coloring -free graphs. (Polynomial
time algorithms have been previously obtained by Golovach, Paulusma, and Song,
but those algorithms are not certifying); To complement these results we show
that in most other cases the -coloring problem for -free
graphs is NP-complete. Specifically, for we show that -coloring is
NP-complete for -free graphs when and ; for we show that -coloring is NP-complete for -free graphs
when , ; and additionally, for , we show that
-coloring is also NP-complete for -free graphs if and
. This is the first systematic study of the complexity of the
-coloring problem for -free graphs. We almost completely
classify the complexity for the cases when , and
identify the last three open cases
Tree decompositions with small cost
The f-cost of a tree decomposition ({Xi | i e I}, T = (I;F))
for a function f : N -> R+ is defined as EieI f(|Xi|). This measure
associates with the running time or memory use of some algorithms
that use the tree decomposition. In this paper we investigate the
problem to find tree decompositions of minimum f-cost.
A function f : N -> R+ is fast, if for every i e N: f(i+1) => 2*f(i).
We show that for fast functions f, every graph G has a tree decomposition
of minimum f-cost that corresponds to a minimal triangulation
of G; if f is not fast, this does not hold. We give polynomial time
algorithms for the problem, assuming f is a fast function, for graphs
that has a polynomial number of minimal separators, for graphs of
treewidth at most two, and for cographs, and show that the problem
is NP-hard for bipartite graphs and for cobipartite graphs.
We also discuss results for a weighted variant of the problem derived
of an application from probabilistic networks
Solving Problems on Graphs of High Rank-Width
A modulator of a graph G to a specified graph class H is a set of vertices
whose deletion puts G into H. The cardinality of a modulator to various
tractable graph classes has long been used as a structural parameter which can
be exploited to obtain FPT algorithms for a range of hard problems. Here we
investigate what happens when a graph contains a modulator which is large but
"well-structured" (in the sense of having bounded rank-width). Can such
modulators still be exploited to obtain efficient algorithms? And is it even
possible to find such modulators efficiently?
We first show that the parameters derived from such well-structured
modulators are strictly more general than the cardinality of modulators and
rank-width itself. Then, we develop an FPT algorithm for finding such
well-structured modulators to any graph class which can be characterized by a
finite set of forbidden induced subgraphs. We proceed by showing how
well-structured modulators can be used to obtain efficient parameterized
algorithms for Minimum Vertex Cover and Maximum Clique. Finally, we use
well-structured modulators to develop an algorithmic meta-theorem for deciding
problems expressible in Monadic Second Order (MSO) logic, and prove that this
result is tight in the sense that it cannot be generalized to LinEMSO problems.Comment: Accepted at WADS 201
Polynomial kernels for 3-leaf power graph modification problems
A graph G=(V,E) is a 3-leaf power iff there exists a tree T whose leaves are
V and such that (u,v) is an edge iff u and v are at distance at most 3 in T.
The 3-leaf power graph edge modification problems, i.e. edition (also known as
the closest 3-leaf power), completion and edge-deletion, are FTP when
parameterized by the size of the edge set modification. However polynomial
kernel was known for none of these three problems. For each of them, we provide
cubic kernels that can be computed in linear time for each of these problems.
We thereby answer an open problem first mentioned by Dom, Guo, Huffner and
Niedermeier (2005).Comment: Submitte
- …