Approximation algorithms for the submodular edge cover problem with submodular penalties

Highlights

• In this paper, we consider the submodular edge cover problem with submodular penalties.

• We give a 2Δ-approximation algorithm for the above problem, where Δ is the maximal degree of the graph G.

• We transform the submodular edge cover problem into a submodular set cover problem and show that there exists a $\mathrm{\Delta }+1$-approximation algorithm.

Abstract

In this paper, we consider the submodular edge cover problem with submodular penalties. In this problem, we are given an undirected graph $G=(V,E)$ with vertex set V and edge set E. Assume the covering cost function $c:2^E\to {\mathbb{R}}_{+}$ and the penalty function $p:2^V\to {\mathbb{R}}_{+}$ are both submodular with p non-decreasing, $c(\varnothing )=0$ and $p(\varnothing )=0$. The goal of the submodular edge cover problem with submodular penalties is to select an edge subset to cover some vertices and penalize the vertex subset containing uncovered vertices such that the total cost of covering and penalty is minimized. For this problem, we first give a 2Δ-approximation algorithm by using a primal-dual technique, where Δ is the maximal degree of the graph G. Then we transform this problem into a submodular set cover problem, and by applying a known result for the submodular set cover problem we conclude that there is an approximation algorithm with an approximation ratio $\mathrm{\Delta }+1$.

Introduction

This paper is focus on the submodular edge cover problem with submodular penalties (SEC-SP problem). In this problem, we are given an undirected graph $G=(V,E)$ with vertex set V and edge set E. Assume $c:2^E\to {\mathbb{R}}_{+}$ and $p:2^V\to {\mathbb{R}}_{+}$ are both submodular functions with p non-decreasing, $c(\varnothing )=0$ and $p(\varnothing )=0$. Here $c(D)$ means the covering cost for each edge subset D and $p(Q)$ means the penalty cost for each vertex subset Q. The goal of the submodular edge cover problem with submodular penalties is to select an edge subset to cover some vertices and penalize the vertex subset containing uncovered vertices such that the total cost of covering and penalty is minimized.

By a submodular function we mean a set function $f:2^U\to {\mathbb{R}}_{+}$ with the property that for any pair of subsets $X,Y$ of U,

$$f(X\cup Y)+f(X\cap Y)\le f(X)+f(Y),$$

where U is a nonempty finite set [2]. A submodular function is said to be modular if the equality holds in (1). A submodular function f is non-decreasing if for two subsets $X_1,X_2$ of U with $X_1\subseteq X_2$, we always have $f(X_1)\le f(X_2)$. Submodular functions play an important role in the field of optimization and there has been a lot of work on the submodular function optimization [4], [7], [8].

The submodular edge cover (SEC) problem is one type of submodular function optimization problem and is proved to be NP-hard [11]. In this problem, we are given an undirected graph $G=(V,E)$ and a submodular function $c:2^E\to {\mathbb{R}}_{+}$. The objective is to select an edge subset to cover the vertex set V such that the covering cost is minimized. Edge cover problem has many applications in optimization field and is often adapted into the minimum spanning tree problem, the edge coloring problem etc. Iwata and Nagano [11] considered the submodular edge cover problem and pointed out this problem can not be approximated within a factor of $o(|V|/{\ln }^2|V|)$. Partly based on the work of Iwata and Nagano, Kamiyama [12] studied the submodular function minimization problem with covering type linear constraints and proposed an improved primal-dual approximation algorithm.

When we consider approximating a constrained optimization problem permitting some violations, the penalty method is utilized by adding a term to the objective function. There has been a great amount of work applying this penalty idea [3], [5], [9], [10]. In particular, Xu et al. [16] studied the submodular vertex cover problems with linear/submodular penalties using primal-dual technique and gave approximation algorithms with ratios 2 and 4, respectively. Kamiyama [13] proved that there exists a combinatorial 3-approximation algorithm for the submodular vertex cover problem with submodular penalties. Plesník [15] proved that arbitrary profit partial edge cover problem is NP-hard and Parekh [14] suggested a polynomial-time algorithm for the prize-collecting edge cover problem.

The SEC-SP problem contains the SEC problem as a special case by assuming that the penalty is infinity. The NP-hardness of the special case implies the NP-hardness of the SEC-SP problem itself. So it is interesting to consider approximation algorithms for it. In this paper, we first present a 2Δ-approximation algorithm for the SEC-SP problem by using a primal-dual technique. Then we transform the SEC-SP problem to a submodular set cover problem, where Δ is the maximal degree of the graph G. By applying a known result for the submodular set cover problem, we show that there exists an approximation algorithm with $\mathrm{\Delta }+1$ ratio for the SEC-SP problem.

The paper is organized as follows: In Section 2 we formulate the primal and dual programs, present a primal-dual algorithm for the SEC-SP problem, and analyze the approximation ratio for this algorithm. In Section 3, we transform the SEC-SP problem to a submodular set cover problem and apply a known result for the submodular set cover problem to the SEC-SP problem to show that there exists an approximation algorithm with an improved approximation ratio.