An Algebraic Approach Towards the Fine-Grained Complexity of Graph Coloring Problems

International audienceIn this paper we are interested in the finegrained complexity of determining whether there is an homomorphism from an input graph G to a fixed graph H (the H-coloring problem). The starting point is that these problems can be viewed as constraint satisfaction problems (CSPs), and that (partial) polymorphisms of binary relations are of paramount importance in the study of complexity classes of such CSPs. Thus, we first investigate the expressivity of binary symmetric relations EH and their corresponding (partial) polymorphisms pPol(EH). For irreflexive graphs we observe that there is no pair of graphs H and H such that pPol(EH) ⊆ pPol(E H), unless E H = ∅ or H =H. More generally we show the existence of an nary relation R whose partial polymorphisms strictly subsume those of H and such that CSP(R) is N Pcomplete if and only if H contains an odd cycle of length at most n. Motivated by this we also describe the sets of total polymorphisms of every nontrivial clique, every odd cycle, as well as certain cores. We finish the paper with some noteworthy questions for future research

Component twin-width as a parameter for BINARY-CSP and its semiring generalizations

We investigate the fine-grained and the parameterized complexity of several generalizations of binary constraint satisfaction problems (BINARY-CSPs), that subsume variants of graph colouring problems. Our starting point is the observation that several algorithmic approaches that resulted in complexity upper bounds for these problems, share a common structure. We thus explore an algebraic approach relying on semirings that unifies different generalizations of BINARY-CSPs (such as the counting, the list, and the weighted versions), and that facilitates a general algorithmic approach to efficiently solving them. The latter is inspired by the (component) twin-width parameter introduced by Bonnet et al., which we generalize via edge-labelled graphs in order to formulate it to arbitrary binary constraints. We consider input instances with bounded component twin-width, as well as constraint templates of bounded component twin-width, and obtain an FPT algorithm as well as an improved, exponential-time algorithm, for broad classes of binary constraints. We illustrate the advantages of this framework by instantiating our general algorithmic approach on several classes of problems (e.g., the H-coloring problem and its variants), and showing that it improves the best complexity upper bounds in the literature for several well-known problems

On the parameterized complexity of non-hereditary relaxations of clique

We investigate the parameterized complexity of several problems formalizing cluster identification in graphs. In other words we ask whether a graph contains a large enough and sufficiently connected subgraph. We study here three relaxations of Clique: s-Club and s-Clique, in which the relaxation is focused on the distances in respectively the cluster and the original graph, and γ-Complete Subgraph in which the relaxation is made on the minimal degree in the cluster. As these three problems are known to be NP-hard, we study here their parameterized complexities. We prove that s-Club and s-Clique are NP-hard even restricted to graphs of degeneracy ≤ 3 whenever s ≥ 3, and to graphs of degeneracy ≤ 2 whenever s ≥ 5, which is a strictly stronger result than its W[1]-hardness parameterized by the degeneracy. We also obtain that these problems are solvable in polynomial time on graphs of degeneracy 1. Concerning γ-Complete Subgraph, we prove that it is W[1]-hard parameterized by both the degeneracy, which implies the W[1]-hardness parameterized by the number of vertices in the γ-complete-subgraph, and the number of elements outside the γ-complete subgraph

An Algebraic Approach Towards the Fine-Grained Complexity of Graph Coloring Problems

In this paper we are interested in the finegrained complexity of determining whether there is an homomorphism from an input graph G to a fixed graph H (the H-coloring problem). The starting point is the observation that these problems can be formulated in the language of CSPs, and that (partial) polymorphisms of binary relations are of paramount importance in the study of complexity classes of such CSPs. Thus, we first investigate the expressivity of binary symmetric relations EH and their corresponding (partial) polymorphisms pPol(EH). For irreflexive graphs we observe that there is no pair of graphs H and H such that pPol(EH) ⊆ pPol(E H), unless E H = ∅ or H =H. More generally we show the existence of an nary relation R whose partial polymorphisms strictly subsume those of H and such that CSP(R) is N Pcomplete if and only if H contains an odd cycle of length at most n. Motivated by this we also describe the sets of total polymorphisms of every nontrivial clique, every odd cycle, as well as certain cores. We finish the paper with some noteworthy questions for future research

On the parameterized complexity of non-hereditary relaxations of clique

We investigate the parameterized complexity of several problems formalizing cluster identification in graphs. In other words we ask whether a graph contains a large enough and sufficiently connected subgraph. We study here three relaxations of Clique: s-Club and s-Clique, in which the relaxation is focused on the distances in respectively the cluster and the original graph, and γ-Complete Subgraph in which the relaxation is made on the minimal degree in the cluster. As these three problems are known to be NP-hard, we study here their parameterized complexities. We prove that s-Club and s-Clique are NP-hard even restricted to graphs of degeneracy ≤ 3 whenever s ≥ 3, and to graphs of degeneracy ≤ 2 whenever s ≥ 5, which is a strictly stronger result than its W[1]-hardness parameterized by the degeneracy. We also obtain that these problems are solvable in polynomial time on graphs of degeneracy 1. Concerning γ-Complete Subgraph, we prove that it is W[1]-hard parameterized by both the degeneracy, which implies the W[1]-hardness parameterized by the number of vertices in the γ-complete-subgraph, and the number of elements outside the γ-complete subgraph