Approximation algorithms for the submodular edge cover problem with submodular penalties
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 with vertex set V and edge set E. Assume and are both submodular functions with p non-decreasing, and . Here means the covering cost for each edge subset D and 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 with the property that for any pair of subsets of U, 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 of U with , we always have . 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 and a submodular function . 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 . 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 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.
Section snippets
A primal-dual algorithm for the SEC-SP problem
In this section, we first formulate the SEC-SP problem as a linear integer program. Then we relax it to a linear program and give the dual of the linear program. In order to use the primal-dual technique to design an approximation algorithm and control the dual ascending process in polynomial time, we consider a slightly weaker version of the dual program. Then by implementing the primal-dual framework, we present an approximation algorithm with 2Δ ratio for the SEC-SP problem, where Δ is the
An improved algorithm for the SEC-SP problem
In this section, we prove that the SEC-SP problem can be transformed to the submodular set cover (SSC) problem and by applying a result for the SSC problem, we show that there exists an approximation algorithm with better approximation ratio for the SEC-SP problem.
In the SSC problem, we are given a ground set S, a collection of subsets of S, and a submodular function with . A subset I of is called a set cover if for every s in S, there exists an integer such
Conclusion
Both covering and coloring [1] are important research topics in combinatorial optimization with applications in networking. Xu et al. [16] studied submodular objectives for vertex cover problem. We successfully introduced submodularity to edge cover problem. So far, it seems a challenge task to bring the submodularity to the coloring problem.
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgements
The authors are grateful to Professor Dingzhu Du and Professor Weili Wu for the valuable advice they offered during the authors' study of approximation algorithms. This work was supported by the NSF of China (No. 11971146), the NSF of Hebei Province of China (No. A2019205089, No. A2019205092), Hebei Province Foundation for Returnees (CL201714) and Overseas Expertise Introduction Program of Hebei Auspices (2530-5008).
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