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On the Complexity of Multipacking

Authors Sandip Das , Sk Samim Islam , Daniel Lokshtanov

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LIPIcs.ESA.2026.152.pdf
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Subject Classification

ACM Subject Classification
  • Theory of computation → Design and analysis of algorithms
  • Mathematics of computing → Discrete mathematics
Keywords
  • Multipacking
  • Complexity
  • Exact exponential algorithms

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Abstract

A multipacking in an undirected graph G = (V, E) is a set M ⊆ V such that for every vertex v ∈ V and for every integer r ≥ 1, the ball of radius r around v contains at most r vertices of M, that is, there are at most r vertices in M at a distance at most r from v in G. The Multipacking problem asks whether a graph contains a multipacking of size at least k.
For more than a decade, it remained an open question whether the Multipacking problem is NP-complete or solvable in polynomial time, although the problem is known to be polynomial-time solvable for certain graph classes (e.g., strongly chordal graphs, grids, etc). Foucaud, Gras, Perez, and Sikora [Foucaud et al., 2021] [Algorithmica 2021] made a step towards solving the open question by showing that the Multipacking problem is NP-complete for directed graphs and W[1]-hard when parameterized by the solution size. 
In this paper, we prove that the Multipacking problem is NP-complete on undirected graphs, which answers the open question. Moreover, the problem is W[2]-hard on undirected graphs when parameterized by the solution size. Furthermore, we show that the problem is NP-complete and W[2]-hard (parameterized by solution size) on chordal, bipartite, and claw-free graphs, and remains NP-complete on regular and CONV graphs (intersection graphs of convex sets in the plane). Additionally, the problem is NP-complete and W[2]-hard (parameterized by the solution size) on chordal ∩ 1/2-hyperbolic graphs, which is a superclass of strongly chordal graphs on which the problem is polynomial-time solvable. On the positive side, we present an exact exponential-time algorithm for the Multipacking problem on general graphs that breaks the 2ⁿ barrier, with running time O^*(1.58ⁿ), where n is the number of vertices.

Cite As Get BibTex

Sandip Das, Sk Samim Islam, and Daniel Lokshtanov. On the Complexity of Multipacking. In 34th Annual European Symposium on Algorithms (ESA 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 388, pp. 152:1-152:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.ESA.2026.152

Author Details

Sandip Das
  • Indian Statistical Institute, Kolkata, India
Sk Samim Islam
  • Indian Statistical Institute, Kolkata, India
Daniel Lokshtanov
  • University of California, Santa Barbara, CA, USA

References

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