An Extension of the Nemhauser–Trotter Theorem to Generalized Vertex Cover with Applications

Abstract

Abstract. The Nemhauser-Trotter theorem provides an algorithm which is frequently used as a subroutine in approximation algorithms for the classical Vertex Cover problem. In this paper we present an extension of this theorem so it fits a more general variant of Vertex Cover, namely, the Generalized Vertex Cover problem, where edges are allowed not to be covered at a certain predetermined penalty. We show that many applications of the original Nemhauser-Trotter theorem can be applied using our extension to Generalized Vertex Cover. These applications include a (2 − 2/d)-approximation algorithm for graphs of bounded degree d, a polynomial-time approximation scheme (PTAS) for planar graphs, a (2 − lg lg n/2 lg n)-approximation algorithm for general graphs, and a 2k kernel for the parameterized Generalized Vertex Cover problem. Key words.approximation algorithms, generalized vertex cover, local ratio technique, Nemhauser-Trotter theorem AMS subject classifications. 68W25, 05C85, 68W40, 90C27 DOI. 10.1137/090773313 1. Introduction. Given a graph G = (V, E) with vertex weights, the classical Vertex Cover problem asks to find a minimum weight subset of vertices S ⊆ V that covers all edges in G, i.e., a subset S with S ∩ e = ∅ for all e ∈ E. The Vertex Cover problem is one of the most well studied problems in theoretical computer science and discrete mathematics in general, dating back to König's classical early 1930s result [20] and probably even prior to that. In 1972, Karp listed the decision version of Vertex Cover in his famous list of initial 21 NP-complete problems [18].One of the most well known results about Vertex Cover is the half-integrality of the LP-relaxation of the standard integer programming formulation of Vertex Cover (see, e.g., [22]). This result directly implies a 2-approximation algorithm for vertex cover (as observed by Hochbaum [14]). In 1975, only three years after the publication of Karp's famous NP-complete list, Nemhauser and Trotter published their seminal paper [23] in which they present a reduction that reduces the problem of finding a vertex cover in an arbitrary graph G to that of finding a vertex cover in a subgraph of G whose total weight is not much more than the weight of any of its vertex covers. This reduction is based on the half-integrality of Vertex Cover, and it adds additional structure to the Vertex Cover problem in general. Indeed, after applying the Nemhauser-Trotter reduction, one can use the total weight of the graph as a yardstick for analyzing approximate solutions rather than use the weight of the optimal solution of which there is rarely any knowledge. Below is a precise statement of the Nemhauser-Trotter theorem:

The generalized vertex cover problem and some variations

“…Bipartite graph is a subclass of perfect graphs on which the MWVCP and MWISP can be solved in polynomial time [33]. Houchbaum [16] and Bar-Yehuda et al [7] showed that GVC1-HB solvable in polynomial time on bipartite graphs. Hochbaum and Pathria [15] showed that GVC1-HP is solvable in polynomial time on bipartite graphs.…”

“…Proof. If the first set of conditions are satisfied, then from Theorem 2.1, GVC reduces to GVC1-HB, which is solvable in polynomial time on bipartite graphs [7]. If the second set of conditions are satisfied, then GVC reduces to GVC2-HP, which is solvable in polynomial time on bipartite graphs [15].…”

“…Houchbaum [16] provided an integer linear programming formulation of GVC1-HB and showed that the corresponding linear programming relaxation admits half integrality property. Bar-Yehuda et al [7] provided an extension of a well known theorem by Nemhauser and Trotter [24] for the vertex cover problem (independent set problem) to GVC1-HB, and presented a (2 − 2/d)-approximation algorithm on graphs with maximum degree of a node is bounded above by d. They also presented a polynomial time approximation scheme (PTAS) for GVC1-HB on planner graphs, and a (2 − log log n/2 log n)-approximation algorithm for a general graph. In the same paper they showed that GVC1-HB is NP-hard on complete graphs but solvable in polynomial time on bipartite graphs.…”

“…Another special case of GVC where q 1 ij = q 2 ij = 0 for all (i, j) ∈ E was considered by Houchbaum [16] and Bar-Yehuda et al [7]. This problem is also known as the generalized vertex cover problem in the literature and for definiteness we denote this problem by GVC1.…”

“…This problem is also known as the generalized vertex cover problem in the literature and for definiteness we denote this problem by GVC1. In fact, Houchbaum [16] and Bar-Yehuda et al [7] studied primarily a special case of GVC1 where c i ≥ 0 for all i ∈ V , q 0 ij ≥ 0 for all (i, j) ∈ E. We refer to this version of GVC1 as GVC1-HB. Houchbaum [16] provided an integer linear programming formulation of GVC1-HB and showed that the corresponding linear programming relaxation admits half integrality property.…”