Edge Clique Partition and Cover Beyond Independence

TL;DR

This paper explores the complexity of edge clique cover and partition problems parameterized above the independence number, revealing fixed-parameter tractability for partition but NP-completeness for cover, with implications for sparse graphs.

Contribution

It introduces and analyzes above  parameterizations for edge clique cover and partition, establishing fixed-parameter tractability for partition and NP-completeness for cover, along with algorithms for sparse graphs.

Findings

ECP/ is fixed-parameter tractable.

ECC/ is NP-complete for k , polynomial for k
.

ECC/ is fixed-parameter tractable when parameterized by k + (G).

Abstract

Covering and partitioning the edges of a graph into cliques are classical problems at the intersection of combinatorial optimization and graph theory, having been studied through a range of algorithmic and complexity-theoretic lenses. Despite the well-known fixed-parameter tractability of these problems when parameterized by the total number of cliques, such a parameterization often fails to be meaningful for sparse graphs. In many real-world instances, on the other hand, the minimum number of cliques in an edge cover or partition can be very close to the size of a maximum independent set \alpha(G). Motivated by this observation, we investigate above \alpha parameterizations of the edge clique cover and partition problems. Concretely, we introduce and study Edge Clique Cover Above Independent Set (ECC/\alpha) and Edge Clique Partition Above Independent Set (ECP/\alpha), where the goal is to cover or partition all edges of a graph using at most \alpha(G) + k cliques, and k is the parameter. Our main results reveal a distinct complexity landscape for the two variants. We show that ECP/\alpha is fixed-parameter tractable, whereas ECC/\alpha is NP-complete for all k \geq 2, yet can be solved in polynomial time for k \in {0,1}. These findings highlight intriguing differences between the two problems when viewed through the lens of parameterization above a natural lower bound. Finally, we demonstrate that ECC/\alpha becomes fixed-parameter tractable when parameterized by k + \omega(G), where \omega(G) is the size of a maximum clique of the graph G. This result is particularly relevant for sparse graphs, in which \omega is typically small. For H-minor free graphs, we design a subexponential algorithm of running time f(H)^{\sqrt{k}}n^{O(1)}.