Computational approaches for zero forcing and related problems

Abstract

In this paper, we propose computational approaches for the zero forcing problem, the connected zero forcing problem, and the problem of forcing a graph within a specified number of timesteps.Our approaches are based on a combination of integer programming models and combinatorial algorithms, and include formulations for zero forcing as a dynamic process, and as a set-covering problem. We explore several solution strategies for these models, test them on various types of graphs, and show that they are competitive with the state-of-the-art algorithm for zero forcing.Our proposed algorithms for connected zero forcing and for controlling the number of zero forcing timesteps are the first general-purpose computational methods for these problems, and are superior to brute force computation.Connected zero forcing is a variant of zero forcing in which the initially colored set of vertices induces a connected subgraph. The connected zero forcing number of a graph is the cardinality of the smallest connected set of initially colored vertices which forces the entire graph to be colored (i.e., the smallest connected zero forcing set). Applications and various structural and computational aspects of connected zero forcing have been investigated in [15,16,17]; in particular, it can be used for modeling the spread of ideas or diseases originating from a single connected source in a network, or for power network monitoring accounting for the cost of supporting infrastructure. Other variants of zero forcing, such as positive semidefinite zero forcing [11,33,44,57], fractional zero forcing, signed zero forcing [41], and k-forcing [5,48] have also been studied. These are typically obtained by modifying the zero forcing color change rule, or adding certain restrictions to a zero forcing set. The number of timesteps in the zero forcing process after which a graph becomes colored is also a problem of interest (see, e.g., [14,21,28,43,57]). Connected variants of other graph problems -such as connected domination and connected power domination [25,31,36,40] -have been extensively studied as well.A closely related problem to zero forcing is power domination, where given a set S of initially colored vertices, the zero forcing color change rule is applied to N [S] instead of to S. Integer programming formulations for power domination and its variants have been explored in [1,18].The power domination problem is derived from the phase measurement unit (PMU) placement problem in electrical engineering, which has also been studied extensively; see, e.g., [47,49] and the bibliographies therein for various integer programming models and combinatorial algorithms for the PMU placement problem. Another closely related problem to zero forcing is the target set selection problem, where given a set S of initially colored vertices and a threshold function θ : V (G) → Z, all uncolored vertices v that have at least θ(v) colored neighbors become colored. Thus, the zero forcing problem constrains the infectors, but the target set selection problem constrai...

The zero forcing polynomial of a graph

“…Some of these results, as well as analogous results about a connected variant of zero forcing, have been reported in[10].…”

“…A fort of a graph G, as defined in [30], is a non-empty set F ⊂ V such that no vertex outside F is adjacent to exactly one vertex in F . Let F (G) be the set of all forts of G. In [11], it was shown that Z(G) is equal to the optimum of the following integer program:…”

Connected power domination in graphs

“…This disparity has been attributed to the non-locality of the connected domination problem, since exact algorithms are often unable to capture global properties like connectivity [29]. In some contrast, computational experiments in [13] have shown that algorithms for connected zero forcing are slightly faster than algorithms for zero forcing. Thus, in this aspect, power domination seems to behave more like domination than like zero forcing.…”

“…Requiring a power dominating set to be connected is motivated by the application in monitoring electrical networks: the data from PMUs is relayed by high-speed communication infrastructure to processing stations which collect and manage this data; thus, in addition to minimizing the production costs of the PMUs, an electric power company may seek to place all PMUs in a compact, connected region in the network in order to reduce the number of processing stations and related infrastructure required to collect the data.Connected power domination was explored from a computational perspective in [27] (although the problem called "connected power domination" in [27] is slightly different from the one considered here; see Section 6 for details). The connected variants of other graph problems, including connected zero forcing [13,14,15], connected domination [20,23,29,47], and connected vertex cover [19,36], have also been extensively studied. Imposing connectivity often fundamentally changes the nature of a problem, including its complexity, structural properties, and applications.…”

“…Connected power domination was explored from a computational perspective in [27] (although the problem called "connected power domination" in [27] is slightly different from the one considered here; see Section 6 for details). The connected variants of other graph problems, including connected zero forcing [13,14,15], connected domination [20,23,29,47], and connected vertex cover [19,36], have also been extensively studied. Imposing connectivity often fundamentally changes the nature of a problem, including its complexity, structural properties, and applications.…”