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Edge Clique Partition and Cover Beyond Independence

Authors Fedor V. Fomin , Petr A. Golovach , Danil Sagunov , Kirill Simonov

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ACM Subject Classification
  • Mathematics of computing → Combinatorial algorithms
  • Theory of computation → Fixed parameter tractability
Keywords
  • edge clique partition
  • edge clique cover
  • independence number
  • parameterized complexity
  • above guarantee

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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-α parameterizations 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/α is fixed-parameter tractable, whereas ECC/α is NP-complete for all k ≥ 2, yet can be solved in polynomial time for k ∈ {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/α becomes 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 ω is typically small. For H-minor free graphs, we design a subexponential algorithm of running time f(H)^√k ⋅ n^𝒪(1).

Cite As Get BibTex

Fedor V. Fomin, Petr A. Golovach, Danil Sagunov, and Kirill Simonov. Edge Clique Partition and Cover Beyond Independence. In 33rd Annual European Symposium on Algorithms (ESA 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 351, pp. 43:1-43:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.ESA.2025.43

Author Details

Fedor V. Fomin
  • University of Bergen, Norway
Petr A. Golovach
  • University of Bergen, Norway
Danil Sagunov
  • Saint Petersburg State University, Russia
Kirill Simonov
  • University of Bergen, Norway

Funding

  • Fomin, Fedor V.: Supported by the Research Council of Norway under BWCA project (grant no. 314528).
  • Golovach, Petr A.: Supported by the Research Council of Norway under BWCA (grant no. 314528) and Extreme-Algorithms (grant no 355137) projects.

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