Statistical Mechanics of the Minimum Dominating Set Problem

Abstract

The minimum dominating set (MDS) problem has wide applications in network science and related fields. It aims at constructing a node set of smallest size such that any node of the network is either in this set or is adjacent to at least one node of this set. Although this optimization problem is generally very difficult, we show it can be exactly solved by a generalized leaf-removal (GLR) process if the network contains no core. We present a percolation theory to describe the GLR process on random networks, and solve a spin glass model by mean field method to estimate the MDS size. We also implement a message-passing algorithm and a local heuristic algorithm that combines GLR with greedy node-removal to obtain near-optimal solutions for single random networks. Our algorithms also perform well on real-world network instances.

Generalization of core percolation on complex networks

“…Moreover it was shown that the formation of the core is related to controllability robustness [13,14] and some combinatorial optimization problems such as maximum matching and minimum vertex cover [9,12,15]. Also a generalized leaf removal process, which is applicable in minimum dominating set problem, has been introduced in [16]. Using a time-dependent analysis, people have studied the core percolation related to this generalized leaf-removal algorithm.…”

Two faces of greedy leaf removal procedure on graphs

“…For w (c,t) , we follow a similar calculation in the context of a generalized GLR procedure for the minimum dominating set problem [49,50]. For the GLR procedure here, we have…”

Minimal dominating set problem studied by simulated annealing and cavity method: analytics and population dynamics

“…(3)for the given real number x c (0 < x c < 1), can we judge if there exist a solution of size not exceeding x c N (in there, N represent the node number of the given graph)? Previously we have tried to answer the question one of minimal dominating set problem through cavity method and get good results [1,2]. Now the following work inspired by [3], in this work we mainly answer the last question of minimal dominating set problem through cavity method too.…”

“…MDS problem is a nondeterministic polynomialcomplete(NP-complete) optimization problem [4], So finding exact solution is extremely difficult task in general.Even we hard to find the approximate MDS solution of a given graph. There are some heuristic algorithms [5-7, 9, 14, 15] and statistical physics algorithm [1,2] to solve the MDS problem, but the only small part of this work related to solution space structure, bounds (threshold value x c ) and size of MDS problem. In this work we use one step replica symmetry breaking theory of statistical physics to study the solution space of minimal dominating set problem.…”