Edge Clique Partition and Cover Beyond Independence

2025-06-26

AI Summary

This paper investigates the parameterized complexity of Edge Clique Cover (ECC) and Edge Clique Partition (ECP) with respect to the parameter (k = # ext{cliques} - alpha(G)), where (alpha(G)) is the independence number. Addressing the limitation of classical parameters on sparse graphs, it introduces and systematically analyzes the natural lower-bound-type parameter (alpha(G) + k). The results show that ECP parameterized by (alpha(G)) is fixed-parameter tractable (FPT), whereas ECC parameterized by (alpha(G)) is NP-complete for (k geq 2), revealing a fundamental complexity dichotomy. ECC is polynomial-time solvable for (k = 0,1) and FPT when parameterized by (k + omega(G)). Furthermore, a subexponential algorithm for ECP parameterized by (alpha(G)) is devised for (H)-minor-free graphs, running in time (f(H)^{sqrt{k}} cdot n^{O(1)}). These contributions significantly extend the parameterized complexity landscape of clique cover problems.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Machine Learning: Graph-based Machine Learning
• Data Mining & Knowledge Management: Graph Mining, Social Network Analysis & Community

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Security and Privacy: Applications of cryptography

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 α(G). Motivated by this observation, we investigate above αparameterizations of the edge clique cover and partition problems. Concretely, we introduce and study Edge Clique Cover Above Independent Set (ECC/α) and Edge Clique Partition Above Independent Set (ECP/α), where the goal is to cover or partition all edges of a graph using at most α(G) + k cliques, and k is the parameter. Our main results reveal a distinct complexity landscape for the two variants. We show that ECP/αis fixed-parameter tractable, whereas ECC/αis 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/αbecomes fixed-parameter tractable when parameterized by k + ω(G), where ω(G) is the size of a maximum clique of the graph G. This result is particularly relevant for sparse graphs, in which ωis typically small. For H-minor free graphs, we design a subexponential algorithm of running time f(H)^{sqrt{k}}n^{O(1)}.

Problem

• Study edge clique cover and partition problems above independence number.
• Investigate fixed-parameter tractability of ECP/α and ECC/α variants.
• Analyze complexity differences between edge cover and partition problems.

Innovation

• Parameterized edge clique cover above independent set
• Fixed-parameter tractable edge clique partition
• Sparse graph optimization with clique constraints