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 $X$; these relationships are often depicted via a phylogenetic tree -- a tree having its leaves univocally labeled by elements of $X$ 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 $X$), 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 $k$, 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 $k$ of these algorithms is prohibitively large, we give the first explicit dynamic programming algorithms for solving these problems, both running in time $2^{O(k^2)} \cdot n$, where $n$ 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 $K_{1,3}$) nor a diamond (a $K_4$ 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 $G$ and a parameter $k$, the question is whether one can remove at most $k$ edges from $G$ 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 $6$, the problem is NP-complete and cannot be solved in time $2^{o(k)}\cdot |V(G)|^{O(1)}$ 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 $G$ 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 $k$-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 $2$, 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