A Linear-Time and Linear-Space Algorithm for the Minimum Vertex Cover Problem on Giant Graphs

In this paper, we develop the message passing based lineartime and linear-space MVC algorithm (MVC-MPL) for solving the minimum vertex cover (MVC) problem. MVC-MPL is based on heuristics derived from a theoretical analysis of message passing algorithms in the context of belief propagation. We show that MVC-MPL produces smaller vertex covers than other linear-time and linear-space algorithms. Introduction Given an undirected graph G = 〈V,E〉, a vertex cover (VC) of G is defined as a set of vertices S ⊆ V such that every edge in E has at least one of its endpoint vertices in S. A minimum vertex cover (MVC) of G is a vertex cover of minimum cardinality. The MVC problem is to find an MVC for a given graph. It is an NP-hard problem (Karp 1972) that has been used across a wide range of application domains, such as crew scheduling, VLSI design, nurse rostering and industrial machine assignments (Cai et al. 2013). Many heuristics have been developed to tackle the MVC problem and its generalizations (Cai et al. 2013; Xu, Kumar, and Koenig 2016). However, none of these algorithms are linear-time and linear-space in the number of vertices and edges. (Henceforth, we use the term “linear” to refer to being linear-time and linear-space in the number of vertices and edges.) Therefore, they are difficult to apply to giant graphs with billions of vertices and edges. Belief propagation (BP) is a well-known technique used for solving queries such as probability marginalization and maximum-a-posteriori estimation in probabilistic models (Yedidia, Freeman, and Weiss 2003). Message passing is a class of techniques that generalize BP. It, too, has been applied to problems across various fields, including the Ising model in statistical physics and error correcting codes in information science (Yedidia, Freeman, and Weiss 2003). In this paper, we develop the message passing based linear MVC algorithm (MVC-MPL) that solves the problem heuristically using the theory of message passing. Empirically, we show that MVC-MPL produces smaller VCs than other linear MVC algorithms. ∗The research at the University of Southern California was supported by NSF under grant numbers 1409987 and 1319966. Copyright c © 2017, Association for the Advancement of Artificial Intelligence (www.aaai.org). All rights reserved. Background A Linear Factor-2 Approximation Algorithm A well-known linear factor-2 approximation algorithm (MVC-2) for solving the MVC problem works as follows (Vazirani 2003). In each iteration, the algorithm arbitrarily selects an uncovered edge, then marks it as well as the edges incident on its two endpoint vertices as being covered, and adds its endpoint vertices into the vertex cover V C. The algorithm then proceeds to the next iteration until all edges are marked as being covered. This is the only existing well-known linear MVC algorithm that is known to the authors. The Warning Propagation Algorithm Thewarning propagation algorithm is a special message passing algorithm where messages can only take one of two values, namely 0 or 1 (Weigt and Zhou 2006). (Weigt and Zhou 2006) developed an algorithm that uses warning propagation to solve the MVC problem and analyzed it theoretically. In their algorithm, messages are passed between adjacent vertices. A message of 1 from vi ∈ V to vj ∈ V indicates that vi is not in the MVC and thus it “warns” vj to be included in the MVC. Their theoretical analysis mainly focused on Erdős-Rényi (ER) random graphs, in which each edge is generated with a constant probability (Erdős and Rényi 1959). They show that, on an infinitely large ER random graph, a message has a probability of W (c)/c to be equal to 1, where c is the average degree of vertices andW is the Lambert-W function.