68 research outputs found
Efficient FPT algorithms for (strict) compatibility of unrooted phylogenetic trees
In phylogenetics, a central problem is to infer the evolutionary
relationships between a set of species ; these relationships are often
depicted via a phylogenetic tree -- a tree having its leaves univocally labeled
by elements of and without degree-2 nodes -- called the "species tree". One
common approach for reconstructing a species tree consists in first
constructing several phylogenetic trees from primary data (e.g. DNA sequences
originating from some species in ), and then constructing a single
phylogenetic tree maximizing the "concordance" with the input trees. The
so-obtained tree is our estimation of the species tree and, when the input
trees are defined on overlapping -- but not identical -- sets of labels, is
called "supertree". In this paper, we focus on two problems that are central
when combining phylogenetic trees into a supertree: the compatibility and the
strict compatibility problems for unrooted phylogenetic trees. These problems
are strongly related, respectively, to the notions of "containing as a minor"
and "containing as a topological minor" in the graph community. Both problems
are known to be fixed-parameter tractable in the number of input trees , by
using their expressibility in Monadic Second Order Logic and a reduction to
graphs of bounded treewidth. Motivated by the fact that the dependency on
of these algorithms is prohibitively large, we give the first explicit dynamic
programming algorithms for solving these problems, both running in time
, where is the total size of the input.Comment: 18 pages, 1 figur
Polynomial kernelization for removing induced claws and diamonds
A graph is called (claw,diamond)-free if it contains neither a claw (a
) nor a diamond (a with an edge removed) as an induced subgraph.
Equivalently, (claw,diamond)-free graphs can be characterized as line graphs of
triangle-free graphs, or as linear dominoes, i.e., graphs in which every vertex
is in at most two maximal cliques and every edge is in exactly one maximal
clique.
In this paper we consider the parameterized complexity of the
(claw,diamond)-free Edge Deletion problem, where given a graph and a
parameter , the question is whether one can remove at most edges from
to obtain a (claw,diamond)-free graph. Our main result is that this problem
admits a polynomial kernel. We complement this finding by proving that, even on
instances with maximum degree , the problem is NP-complete and cannot be
solved in time unless the Exponential Time
Hypothesis fai
Approximation Algorithms for the Capacitated Domination Problem
We consider the {\em Capacitated Domination} problem, which models a
service-requirement assignment scenario and is also a generalization of the
well-known {\em Dominating Set} problem. In this problem, given a graph with
three parameters defined on each vertex, namely cost, capacity, and demand, we
want to find an assignment of demands to vertices of least cost such that the
demand of each vertex is satisfied subject to the capacity constraint of each
vertex providing the service. In terms of polynomial time approximations, we
present logarithmic approximation algorithms with respect to different demand
assignment models for this problem on general graphs, which also establishes
the corresponding approximation results to the well-known approximations of the
traditional {\em Dominating Set} problem. Together with our previous work, this
closes the problem of generally approximating the optimal solution. On the
other hand, from the perspective of parameterization, we prove that this
problem is {\it W[1]}-hard when parameterized by a structure of the graph
called treewidth. Based on this hardness result, we present exact
fixed-parameter tractable algorithms when parameterized by treewidth and
maximum capacity of the vertices. This algorithm is further extended to obtain
pseudo-polynomial time approximation schemes for planar graphs
Approximate min-max relations on plane graphs
Let G be a plane graph, let τ(G) (resp. τ′(G)) be the minimum number of vertices (resp. edges) that meet all cycles of G, and let ν(G) (resp. ν′(G)) be the maximum number of vertex-disjoint (resp. edge-disjoint) cycles in G. In this note we show that τ(G)≤3 ν(G) and τ′(G)≤4 ν′(G)-1; our proofs are constructive, which yield polynomial-time algorithms for finding corresponding objects with the desired properties. © 2011 The Author(s).published_or_final_versionSpringer Open Choice, 28 May 201
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
Linear-time Algorithms for Eliminating Claws in Graphs
Since many NP-complete graph problems have been shown polynomial-time
solvable when restricted to claw-free graphs, we study the problem of
determining the distance of a given graph to a claw-free graph, considering
vertex elimination as measure. CLAW-FREE VERTEX DELETION (CFVD) consists of
determining the minimum number of vertices to be removed from a graph such that
the resulting graph is claw-free. Although CFVD is NP-complete in general and
recognizing claw-free graphs is still a challenge, where the current best
algorithm for a graph has the same running time of the best algorithm for
matrix multiplication, we present linear-time algorithms for CFVD on weighted
block graphs and weighted graphs with bounded treewidth. Furthermore, we show
that this problem can be solved in linear time by a simpler algorithm on
forests, and we determine the exact values for full -ary trees. On the other
hand, we show that CLAW-FREE VERTEX DELETION is NP-complete even when the input
graph is a split graph. We also show that the problem is hard to approximate
within any constant factor better than , assuming the Unique Games
Conjecture.Comment: 20 page
On Structural Parameterizations of the Edge Disjoint Paths Problem
In this paper we revisit the classical edge disjoint paths (EDP) problem, where one is given an undirected graph G and a set of terminal pairs P and asks whether G contains a set of pairwise edge-disjoint paths connecting every terminal pair in P. Our focus lies on structural parameterizations for the problem that allow for efficient (polynomial-time or FPT) algorithms. As our first result, we answer an open question stated in Fleszar et al. (Proceedings of the ESA, 2016), by showing that the problem can be solved in polynomial time if the input graph has a feedback vertex set of size one. We also show that EDP parameterized by the treewidth and the maximum degree of the input graph is fixed-parameter tractable. Having developed two novel algorithms for EDP using structural restrictions on the input graph, we then turn our attention towards the augmented graph, i.e., the graph obtained from the input graph after adding one edge between every terminal pair. In constrast to the input graph, where EDP is known to remain NP-hard even for treewidth two, a result by Zhou et al. (Algorithmica 26(1):3--30, 2000) shows that EDP can be solved in non-uniform polynomial time if the augmented graph has constant treewidth; we note that the possible improvement of this result to an FPT-algorithm has remained open since then. We show that this is highly unlikely by establishing the W[1]-hardness of the problem parameterized by the treewidth (and even feedback vertex set) of the augmented graph. Finally, we develop an FPT-algorithm for EDP by exploiting a novel structural parameter of the augmented graph
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