Parameterized Complexity of Equitable Coloring

A graph on $n$ vertices is equitably $k$-colorable if it is $k$-colorable and every color is used either $\left\lfloor n/k \right\rfloor$ or $\left\lceil n/k \right\rceil$ times. Such a problem appears to be considerably harder than vertex coloring, being $\mathsf{NP\text{-}Complete}$ even for cographs and interval graphs. In this work, we prove that it is $\mathsf{W[1]\text{-}Hard}$ for block graphs and for disjoint union of split graphs when parameterized by the number of colors; and $\mathsf{W[1]\text{-}Hard}$ for $K_{1,4}$-free interval graphs when parameterized by treewidth, number of colors and maximum degree, generalizing a result by Fellows et al. (2014) through a much simpler reduction. Using a previous result due to Dominique de Werra (1985), we establish a dichotomy for the complexity of equitable coloring of chordal graphs based on the size of the largest induced star. Finally, we show that \textsc{equitable coloring} is $\mathsf{FPT}$ when parameterized by the treewidth of the complement graph

Complexity of Grundy coloring and its variants

The Grundy number of a graph is the maximum number of colors used by the greedy coloring algorithm over all vertex orderings. In this paper, we study the computational complexity of GRUNDY COLORING, the problem of determining whether a given graph has Grundy number at least $k$. We also study the variants WEAK GRUNDY COLORING (where the coloring is not necessarily proper) and CONNECTED GRUNDY COLORING (where at each step of the greedy coloring algorithm, the subgraph induced by the colored vertices must be connected). We show that GRUNDY COLORING can be solved in time $O^*(2.443^n)$ and WEAK GRUNDY COLORING in time $O^*(2.716^n)$ on graphs of order $n$. While GRUNDY COLORING and WEAK GRUNDY COLORING are known to be solvable in time $O^*(2^{O(wk)})$ for graphs of treewidth $w$ (where $k$ is the number of colors), we prove that under the Exponential Time Hypothesis (ETH), they cannot be solved in time $O^*(2^{o(w\log w)})$. We also describe an $O^*(2^{2^{O(k)}})$ algorithm for WEAK GRUNDY COLORING, which is therefore \fpt for the parameter $k$. Moreover, under the ETH, we prove that such a running time is essentially optimal (this lower bound also holds for GRUNDY COLORING). Although we do not know whether GRUNDY COLORING is in \fpt, we show that this is the case for graphs belonging to a number of standard graph classes including chordal graphs, claw-free graphs, and graphs excluding a fixed minor. We also describe a quasi-polynomial time algorithm for GRUNDY COLORING and WEAK GRUNDY COLORING on apex-minor graphs. In stark contrast with the two other problems, we show that CONNECTED GRUNDY COLORING is \np-complete already for $k=7$ colors.Comment: 24 pages, 7 figures. This version contains some new results and improvements. A short paper based on version v2 appeared in COCOON'1

Improved FPT algorithms for weighted independent set in bull-free graphs

Very recently, Thomass\'e, Trotignon and Vuskovic [WG 2014] have given an FPT algorithm for Weighted Independent Set in bull-free graphs parameterized by the weight of the solution, running in time $2^{O(k^5)} \cdot n^9$. In this article we improve this running time to $2^{O(k^2)} \cdot n^7$. As a byproduct, we also improve the previous Turing-kernel for this problem from $O(k^5)$ to $O(k^2)$. Furthermore, for the subclass of bull-free graphs without holes of length at most $2p-1$ for $p \geq 3$, we speed up the running time to $2^{O(k \cdot k^{\frac{1}{p-1}})} \cdot n^7$. As $p$ grows, this running time is asymptotically tight in terms of $k$, since we prove that for each integer $p \geq 3$, Weighted Independent Set cannot be solved in time $2^{o(k)} \cdot n^{O(1)}$ in the class of $\{bull,C_4,\ldots,C_{2p-1}\}$-free graphs unless the ETH fails.Comment: 15 page

On local search and LP and SDP relaxations for k-Set Packing

Set packing is a fundamental problem that generalises some well-known combinatorial optimization problems and knows a lot of applications. It is equivalent to hypergraph matching and it is strongly related to the maximum independent set problem. In this thesis we study the k-set packing problem where given a universe U and a collection C of subsets over U, each of cardinality k, one needs to find the maximum collection of mutually disjoint subsets. Local search techniques have proved to be successful in the search for approximation algorithms, both for the unweighted and the weighted version of the problem where every subset in C is associated with a weight and the objective is to maximise the sum of the weights. We make a survey of these approaches and give some background and intuition behind them. In particular, we simplify the algebraic proof of the main lemma for the currently best weighted approximation algorithm of Berman ([Ber00]) into a proof that reveals more intuition on what is really happening behind the math. The main result is a new bound of k/3 + 1 + epsilon on the integrality gap for a polynomially sized LP relaxation for k-set packing by Chan and Lau ([CL10]) and the natural SDP relaxation [NOTE: see page iii]. We provide detailed proofs of lemmas needed to prove this new bound and treat some background on related topics like semidefinite programming and the Lovasz Theta function. Finally we have an extended discussion in which we suggest some possibilities for future research. We discuss how the current results from the weighted approximation algorithms and the LP and SDP relaxations might be improved, the strong relation between set packing and the independent set problem and the difference between the weighted and the unweighted version of the problem.Comment: There is a mistake in the following line of Theorem 17: "As an induced subgraph of H with more edges than vertices constitutes an improving set". Therefore, the proofs of Theorem 17, and hence Theorems 19, 23 and 24, are false. It is still open whether these theorems are tru

Hitting forbidden minors: Approximation and Kernelization

We study a general class of problems called F-deletion problems. In an F-deletion problem, we are asked whether a subset of at most $k$ vertices can be deleted from a graph $G$ such that the resulting graph does not contain as a minor any graph from the family F of forbidden minors. We obtain a number of algorithmic results on the F-deletion problem when F contains a planar graph. We give (1) a linear vertex kernel on graphs excluding $t$-claw $K_{1,t}$, the star with $t$ leves, as an induced subgraph, where $t$ is a fixed integer. (2) an approximation algorithm achieving an approximation ratio of $O(\log^{3/2} OPT)$, where $OPT$ is the size of an optimal solution on general undirected graphs. Finally, we obtain polynomial kernels for the case when F contains graph $\theta_c$ as a minor for a fixed integer $c$. The graph $\theta_c$ consists of two vertices connected by $c$ parallel edges. Even though this may appear to be a very restricted class of problems it already encompasses well-studied problems such as {\sc Vertex Cover}, {\sc Feedback Vertex Set} and Diamond Hitting Set. The generic kernelization algorithm is based on a non-trivial application of protrusion techniques, previously used only for problems on topological graph classes