Maximizing Vertex-Disjoint and Edge-Disjoint Clique Packings in Bounded Degree Graphs

Conceitos essenciais

The authors study the problem of finding a maximum-cardinality set of r-cliques in an undirected graph of fixed maximum degree Δ, subject to the cliques being either vertex disjoint or edge disjoint. They provide a complete complexity classification for both the vertex-disjoint and edge-disjoint variants.

Resumo

The paper studies two related problems on clique packings in undirected graphs:

Vertex-Disjoint Kr-Packing Problem (VDKr): Find a maximum-cardinality set of r-cliques (Krs) in the graph where the cliques are pairwise vertex disjoint.
Edge-Disjoint Kr-Packing Problem (EDKr): Find a maximum-cardinality set of Krs in the graph where the cliques are pairwise edge disjoint.

The authors provide the following results:

If Δ < 3r/2 - 1, then VDKr and EDKr can be solved in linear time.
If Δ < 5r/3 - 1, then VDKr can be solved in polynomial time.
If r ≤ 5 and Δ ≤ 2r - 2, or if r ≥ 6 and Δ < 5r/3 - 1, then EDKr can be solved in polynomial time.
If Δ ≥ ⌈5r/3⌉ - 1, then VDKr is APX-hard.
If r ≥ 6 and Δ ≥ ⌈5r/3⌉ - 1, then EDKr is APX-hard.
If r = 4 and Δ > 2r - 2 = 6, then EDK4 is APX-hard.
The authors provide a complete complexity classification for both VDKr and EDKr in terms of the maximum degree Δ and the fixed clique size r.

Estatísticas

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Citações

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Principais Insights Extraídos De

Packing $K_r$s in bounded degree graphs

by Michael McKa... às arxiv.org 04-09-2024

https://arxiv.org/pdf/2209.03684.pdf

Perguntas Mais Profundas

What are some real-world applications of the vertex-disjoint and edge-disjoint clique packing problems studied in this paper

The vertex-disjoint and edge-disjoint clique packing problems studied in this paper have several real-world applications. One application is in network design, where the goal is to find a maximum-cardinality set of vertex-disjoint or edge-disjoint cliques in a network. This can help in optimizing network resources and improving network efficiency by identifying clusters of nodes that can communicate without interference. Another application is in bioinformatics, where clique packing can be used to analyze protein-protein interaction networks. By identifying sets of proteins that form cliques, researchers can gain insights into complex biological processes and pathways. Additionally, in social network analysis, clique packing can be used to identify groups of individuals with strong connections or common interests, aiding in targeted marketing or community detection efforts.

How might the algorithms and complexity results presented here be extended to more general packing problems, such as packing arbitrary subgraphs rather than just cliques

The algorithms and complexity results presented in this paper for vertex-disjoint and edge-disjoint clique packing can be extended to more general packing problems involving arbitrary subgraphs. One approach could be to generalize the problem to finding maximum-cardinality sets of arbitrary subgraphs that are either vertex-disjoint or edge-disjoint. By adapting the algorithms and complexity analysis to accommodate different types of subgraphs, researchers can address a wider range of optimization problems in graph theory. This extension could have applications in diverse fields such as computer networks, social network analysis, and computational biology.

Are there any connections between the clique packing problems studied here and other well-known graph optimization problems, such as finding maximum independent sets or maximum matchings

There are connections between the clique packing problems studied in this paper and other well-known graph optimization problems. For example, the vertex-disjoint clique packing problem is related to finding maximum independent sets in a graph. In the context of clique packing, a maximum independent set corresponds to a maximum-cardinality set of vertex-disjoint cliques. Similarly, the edge-disjoint clique packing problem can be connected to finding maximum matchings in a graph. In this case, a maximum matching corresponds to a maximum-cardinality set of edge-disjoint cliques. By drawing these connections, researchers can leverage existing algorithms and insights from related optimization problems to solve clique packing problems more efficiently.