Defective Coloring on Classes of Perfect Graphs

In Defective Coloring we are given a graph $G$ and two integers $\chi_d$, $\Delta^*$ and are asked if we can $\chi_d$-color $G$ so that the maximum degree induced by any color class is at most $\Delta^*$. We show that this natural generalization of Coloring is much harder on several basic graph classes. In particular, we show that it is NP-hard on split graphs, even when one of the two parameters $\chi_d$, $\Delta^*$ is set to the smallest possible fixed value that does not trivialize the problem ($\chi_d = 2$ or $\Delta^* = 1$). Together with a simple treewidth-based DP algorithm this completely determines the complexity of the problem also on chordal graphs. We then consider the case of cographs and show that, somewhat surprisingly, Defective Coloring turns out to be one of the few natural problems which are NP-hard on this class. We complement this negative result by showing that Defective Coloring is in P for cographs if either $\chi_d$ or $\Delta^*$ is fixed; that it is in P for trivially perfect graphs; and that it admits a sub-exponential time algorithm for cographs when both $\chi_d$ and $\Delta^*$ are unbounded

Parameterized (Approximate) Defective Coloring

In Defective Coloring we are given a graph G=(V,E) and two integers chi_d,Delta^* and are asked if we can partition V into chi_d color classes, so that each class induces a graph of maximum degree Delta^*. We investigate the complexity of this generalization of Coloring with respect to several well-studied graph parameters, and show that the problem is W-hard parameterized by treewidth, pathwidth, tree-depth, or feedback vertex set, if chi_d=2. As expected, this hardness can be extended to larger values of chi_d for most of these parameters, with one surprising exception: we show that the problem is FPT parameterized by feedback vertex set for any chi_d != 2, and hence 2-coloring is the only hard case for this parameter. In addition to the above, we give an ETH-based lower bound for treewidth and pathwidth, showing that no algorithm can solve the problem in n^{o(pw)}, essentially matching the complexity of an algorithm obtained with standard techniques. We complement these results by considering the problem\u27s approximability and show that, with respect to Delta^*, the problem admits an algorithm which for any epsilon>0 runs in time (tw/epsilon)^{O(tw)} and returns a solution with exactly the desired number of colors that approximates the optimal Delta^* within (1+epsilon). We also give a (tw)^{O(tw)} algorithm which achieves the desired Delta^* exactly while 2-approximating the minimum value of chi_d. We show that this is close to optimal, by establishing that no FPT algorithm can (under standard assumptions) achieve a better than 3/2-approximation to chi_d, even when an extra constant additive error is also allowed

Treewidth with a Quantifier Alternation Revisited

In this paper we take a closer look at the parameterized complexity of existsforall SAT, the prototypical complete problem of the class Sigma_2^p, the second level of the polynomial hierarchy. We provide a number of tight fine-grained bounds on the complexity of this problem and its variants with respect to the most important structural graph parameters. Specifically, we show the following lower bounds (assuming the ETH): - It is impossible to decide existsforall SAT in time less than double-exponential in the input formula\u27s treewidth. More strongly, we establish the same bound with respect to the formula\u27s primal vertex cover, a much more restrictive measure. This lower bound, which matches the performance of known algorithms, shows that the degeneration of the performance of treewidth-based algorithms to a tower of exponentials already begins in problems with one quantifier alternation. - For the more general existsforall CSP problem over a non-boolean domain of size B, there is no algorithm running in time 2^{B^{o(vc)}}, where vc is the input\u27s primal vertex cover. - existsforall SAT is already NP-hard even when the input formula has constant modular treewidth (or clique-width), indicating that dense graph parameters are less useful for problems in Sigma_2^p. - For the two weighted versions of existsforall SAT recently introduced by de Haan and Szeider, called exists_kforall SAT and existsforall_k SAT, we give tight upper and lower bounds parameterized by treewidth (or primal vertex cover) and the weight k. Interestingly, the complexity of these two problems turns out to be quite different: one is double-exponential in treewidth, while the other is double-exponential in k. We complement the above negative results by showing a double-exponential FPT algorithm for QBF parameterized by vertex cover, showing that for this parameter the complexity never goes beyond double-exponential, for any number of quantifier alternations

Graph modification for edge-coloured and signed graph homomorphism problems: parameterized and classical complexity

We study the complexity of graph modification problems with respect to homomorphism-based colouring properties of edge-coloured graphs. A homomorphism from edge-coloured graph $G$ to edge-coloured graph $H$ is a vertex-mapping from $G$ to $H$ that preserves adjacencies and edge-colours. We consider the property of having a homomorphism to a fixed edge-coloured graph $H$. The question we are interested in is: given an edge-coloured graph $G$, can we perform $k$ graph operations so that the resulting graph admits a homomorphism to $H$? The operations we consider are vertex-deletion, edge-deletion and switching (an operation that permutes the colours of the edges incident to a given vertex). Switching plays an important role in the theory of signed graphs, that are 2-edge-coloured graphs whose colours are the signs $+$ and $-$. We denote the corresponding problems (parameterized by $k$) by VD-$H$-COLOURING, ED-$H$-COLOURING and SW-$H$-COLOURING. These problems generalise $H$-COLOURING (to decide if an input graph admits a homomorphism to a fixed target $H$). Our main focus is when $H$ is an edge-coloured graph with at most two vertices, a case that is already interesting as it includes problems such as VERTEX COVER, ODD CYCLE RANSVERSAL and EDGE BIPARTIZATION. For such a graph $H$, we give a P/NP-c complexity dichotomy for VD-$H$-COLOURING, ED-$H$-COLOURING and SW-$H$-COLOURING. We then address their parameterized complexity. We show that VD-$H$-COLOURING and ED-$H$-COLOURING for all such $H$ are FPT. In contrast, already for some $H$ of order 3, unless P=NP, none of the three problems is in XP, since 3-COLOURING is NP-c. We show that SW-$H$-COLOURING is different: there are three 2-edge-coloured graphs $H$ of order 2 for which SW-$H$-COLOURING is W-hard, and assuming the ETH, admits no algorithm in time $f(k)n^{o(k)}$. For the other cases, SW-$H$-COLOURING is FPT.Comment: 17 pages, 9 figures, 2 table

Complexity of planar signed graph homomorphisms to cycles

We study homomorphism problems of signed graphs. A signed graph is an undirected graph where each edge is given a sign, positive or negative. An important concept for signed graphs is the operation of switching at a vertex, which is to change the sign of each incident edge. A homomorphism of a graph is a vertex-mapping that preserves the adjacencies; in the case of signed graphs, we also preserve the edge-signs. Special homomorphisms of signed graphs, called s-homomorphisms, have been studied. In an s-homomorphism, we allow, before the mapping, to perform any number of switchings on the source signed graph. This concept has been extensively studied, and a full complexity classification (polynomial or NP-complete) for s-homomorphism to a fixed target signed graph has recently been obtained. Such a dichotomy is not known when we restrict the input graph to be planar (not even for non-signed graph homomorphisms). We show that deciding whether a (non-signed) planar graph admits a homomorphism to the square $C_t^2$ of a cycle with $t\ge 6$, or to the circular clique $K_{4t/(2t-1)}$ with $t\ge2$, are NP-complete problems. We use these results to show that deciding whether a planar signed graph admits an s-homomorphism to an unbalanced even cycle is NP-complete. (A cycle is unbalanced if it has an odd number of negative edges). We deduce a complete complexity dichotomy for the planar s-homomorphism problem with any signed cycle as a target. We also study further restrictions involving the maximum degree and the girth of the input signed graph. We prove that planar s-homomorphism problems to signed cycles remain NP-complete even for inputs of maximum degree~$3$ (except for the case of unbalanced $4$-cycles, for which we show this for maximum degree~$4$). We also show that for a given integer $g$, the problem for signed bipartite planar inputs of girth $g$ is either trivial or NP-complete.Comment: 17 pages, 10 figure

Complexity and Approximability of Parameterized MAX-CSPs

International audienceWe study the optimization version of constraint satisfaction problems (Max-CSPs) in the framework of parameterized complexity; the goal is to compute the maximum fraction of constraints that can be satisfied simultaneously. In standard CSPs, we want to decide whether this fraction equals one. The parameters we investigate are structural measures, such as the treewidth or the clique-width of the variable-constraint incidence graph of the CSP instance.We consider Max-CSPs with the constraint types AND, OR, PARITY, and MAJORITY, and with various parameters k, and we attempt to fully classify them into the following three cases: 1. The exact optimum can be computed in FPT time. 2. It is W[1]-hard to compute the exact optimum, but there is a randomized FPT approximation scheme (FPTAS), which computes a (1−ϵ)-approximation in time f(k,ϵ)⋅poly(n). 3. There is no FPTAS unless FPT=W[1].For the corresponding standard CSPs, we establish FPT vs. W[1]-hardness results