Multipacking on graphs and Euclidean metric space

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 number of G is the maximum cardinality of a multipacking of G and is denoted by mp(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. Whereas 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 [57] [Algorithmica 2021] made a step towards solving the open question by showing that the MULTIPACKING problem is NP-COMPLETE for directed graphs and it is W[1]-HARD when parameterized by the solution size. We study the hardness of the MULTIPACKING problem and we prove that the MULTIPACKING problem is NP-COMPLETE for undirected graphs, which answers the open question. Moreover, it is W[2]-HARD for undirected graphs when parameterized by the solution size. Furthermore, we have shown that the problem is NP-COMPLETE and W[2]-HARD (when parameterized by the solution size) even for various subclasses: chordal, bipartite, and claw-free graphs. Whereas, it is NP-COMPLETE for regular, and CONV graphs. Additionally, the problem is NP-COMPLETE and W[2]-HARD (when parameterized by the solution size) for chordal ∩ 12 -hyperbolic graphs, which is a superclass of strongly chordal graphs where the problem is polynomial-time solvable. Moreover, we provide approximation algorithms for the MULTIPACKING problem for cactus (an unbounded hyperbolic graph-class), chordal (a bounded hyperbolic graph-class) and δ-hyperbolic graphs. For a graph G = (V,E) with diameter diam(G), a function f : V → {0, 1, 2, ..., diam(G)} is called a broadcast on G. For each vertex u ∈ V , if there exists a vertex v in G (possibly, u = v) such that f(v) > 0 and d(u, v) ≤ f(v), then f is called a dominating broadcast on G. The cost of the dominating broadcast f is the quantity P v∈V f(v). The minimum cost of a dominating broadcast is the broadcast domination number of G, denoted by γb(G). We study the relationship between mp(G) and γb(G) for cactus, chordal and δ-hyperbolic graphs. An r-multipacking in an undirected graph G = (V,E) is a setM ⊆ V such that for every vertex v ∈ V and for every integer s, 1 ≤ s ≤ r, the ball of radius s around v contains at most s vertices of M, that is, there are at most s vertices in M at a distance at most s from v in G. The r-MULTIPACKING problem asks whether a graph contains an r-multipacking of size at least k. It is known that 1-MULTIPACKING problem is NP-COMPLETE for planar bipartite graphs of maximum degree 3, and chordal graphs. We study the hardness of the r-MULTIPACKING problem, for r ≥ 2. We prove that, for r ≥ 2, the r-MULTIPACKING problem is NP-COMPLETE even for planar bipartite graphs with bounded degree. Furthermore, we have shown that the problem is NP-COMPLETE for bounded diameter chordal graphs and bounded diameter bipartite graphs. Further, we study some variants of MULTIPACKING problem for geometric point sets with respect to their Euclidean distances. We show that, for a point set in R2, a maximum 1-multipacking can be computed in polynomial time but computing a maximum 2-multipacking is NP-HARD. Further, we provide approximation and parameterized solutions to the 2-multipacking problem. Next, we study the MINIMUM DOMINATING BROADCAST problem for geometric point set in Rd where the pairwise distance between the points are measured in Euclidean metric. We present a polynomial time algorithm for solving the MINIMUM DOMINATING BROADCAST problem on a point set in Rd. We provide bounds of the broadcast domination number using kissing number. Further, we prove tight upper and lower bounds of the broadcast domination number of point sets in R2.