6 research outputs found
Triangle Packing in (Sparse) Tournaments: Approximation and Kernelization
Given a tournament T and a positive integer k, the C_3-Packing-T asks if there exists a least k (vertex-)disjoint directed 3-cycles in T. This is the dual problem in tournaments of the classical minimal feedback vertex set problem. Surprisingly C_3-Packing-T did not receive a lot of attention in the literature. We show that it does not admit a PTAS unless P=NP, even if we restrict the considered instances to sparse tournaments, that is tournaments with a feedback arc set (FAS) being a matching. Focusing on sparse tournaments we provide a (1+6/(c-1)) approximation algorithm for sparse tournaments having a linear representation where all the backward arcs have "length" at least c. Concerning kernelization, we show that C_3-Packing-T admits a kernel with O(m) vertices, where m is the size of a given feedback arc set. In particular, we derive a O(k) vertices kernel for C_3-Packing-T when restricted to sparse instances. On the negative size, we show that C_3-Packing-T does not admit a kernel of (total bit) size O(k^{2-epsilon}) unless NP is a subset of coNP / Poly. The existence of a kernel in O(k) vertices for C_3-Packing-T remains an open question
On Approximating Four Covering and Packing Problems
In this paper, we consider approximability issues of the following four
problems: triangle packing, full sibling reconstruction, maximum profit
coverage and 2-coverage. All of them are generalized or specialized versions of
set-cover and have applications in biology ranging from full-sibling
reconstructions in wild populations to biomolecular clusterings; however, as
this paper shows, their approximability properties differ considerably. Our
inapproximability constant for the triangle packing problem improves upon the
previous results; this is done by directly transforming the inapproximability
gap of Haastad for the problem of maximizing the number of satisfied equations
for a set of equations over GF(2) and is interesting in its own right. Our
approximability results on the full siblings reconstruction problems answers
questions originally posed by Berger-Wolf et al. and our results on the maximum
profit coverage problem provides almost matching upper and lower bounds on the
approximation ratio, answering a question posed by Hassin and Or.Comment: 25 page
Kernelization for Graph Packing Problems via Rainbow Matching
We introduce a new kernelization tool, called rainbow matching technique,
that is appropriate for the design of polynomial kernels for packing problems.
Our technique capitalizes on the powerful combinatorial results of [Graf,
Harris, Haxell, SODA 2021]. We apply the rainbow matching technique on two
(di)graph packing problems, namely the Triangle-Packing in Tournament problem
(\TPT), where we ask for a directed triangle packing in a tournament, and the
Induced 2-Path-Packing (\IPP) where we ask for a packing of induced paths
of length two in a graph. The existence of a sub-quadratic kernels for these
problems was proven for the first time in [Fomin, Le, Lokshtanov, Saurabh,
Thomass\'e, Zehavi. ACM Trans. Algorithms, 2019], where they gave a kernel of
vertices and vertices
respectively. In the same paper it was questioned whether these bounds can be
(optimally) improved to linear ones. Motivated by this question, we apply the
rainbow matching technique and prove that \TPT admits an (almost linear) kernel
of vertices and that \IPP admits
kernel of vertices
Approximation Algorithms for Effective Team Formation
This dissertation investigates the problem of creating multiple disjoint teams of maximum efficacy from a fixed set of workers. We identify three parameters which directly correlate to the team effectiveness â team expertise, team cohesion and team size â and propose efficient algorithms for optimizing each in various settings. We show that under standard assumptions the problems we explore are not optimally solvable in polynomial time, and thus we focus on developing efficient algorithms with guaranteed worst case approximation bounds. First, we investigate maximizing team expertise in a setting where each worker has different expertise for each job and each job may be completed only by teams of certain sizes. Second, we consider the problem of maximizing team cohesion when the set of workers form a social network with known pairwise compatibility. Third, we explore the problem from a game theoretic perspective in which multiple teams compete on a fixed number of workers and the true needs of each team are pri- vate. We present allocation algorithms that both incentivize teams to state their needs accurately and allocate workers effectively. Finally, we experimentally measure the correlation between team cohesiveness, team expertise and team efficacy on a social network graph of computer science research co-authorship
On the complexity of colouring antiprismatic graphs
International audienceA graph G is prismatic if for every triangle T of G, every vertex of G not in T has a unique neighbour in T. The complement of a prismatic graph is called \emph{antiprismatic}. The complexity of colouring antiprismatic graphs is still unknown. Equivalently, the complexity of the clique cover problem in prismatic graphs is not known. Chudnovsky and Seymour gave a full structural description of prismatic graphs. They showed that the class can be divided into two subclasses: the orientable prismatic graphs, and the non-orientable prismatic graphs. We give a polynomial time algorithm that solves the clique cover problem in every non-orientable prismatic graph. It relies on the the structural description and on later work of Javadi and Hajebi. We give a polynomial time algorithm which solves the vertex-disjoint triangles problem for every prismatic graph. It does not rely on the structural description
On the complexity of colouring antiprismatic graphs
International audienceA graph G is prismatic if for every triangle T of G, every vertex of G not in T has a unique neighbour in T. The complement of a prismatic graph is called \emph{antiprismatic}. The complexity of colouring antiprismatic graphs is still unknown. Equivalently, the complexity of the clique cover problem in prismatic graphs is not known. Chudnovsky and Seymour gave a full structural description of prismatic graphs. They showed that the class can be divided into two subclasses: the orientable prismatic graphs, and the non-orientable prismatic graphs. We give a polynomial time algorithm that solves the clique cover problem in every non-orientable prismatic graph. It relies on the the structural description and on later work of Javadi and Hajebi. We give a polynomial time algorithm which solves the vertex-disjoint triangles problem for every prismatic graph. It does not rely on the structural description