A Note on Submodular Function Minimization with Covering Type Linear Constraints

In this paper, we consider the non-negative submodular function minimization problem with covering type linear constraints. Assume that there exist m linear constraints, and we denote by $$\varDelta _i$$Δi the number of non-zero coefficients in the ith constraints. Furthermore, we assume that $$\varDelta _1 \ge \varDelta _2 \ge \cdots \ge \varDelta _m$$Δ1≥Δ2≥⋯≥Δm. For this problem, Koufogiannakis and Young proposed a polynomial-time $$\varDelta _1$$Δ1-approximation algorithm. In this paper, we propose a new polynomial-time primal-dual approximation algorithm based on the approximation algorithm of Takazawa and Mizuno for the covering integer program with $$\{0,1\}$${0,1}-variables and the approximation algorithm of Iwata and Nagano for the submodular function minimization problem with set covering constraints. The approximation ratio of our algorithm is $$\max \{\varDelta _2, \min \{\varDelta _1, 1 + \varPi \}\}$$max{Δ2,min{Δ1,1+Π}}, where $$\varPi $$Π is the maximum size of a connected component of the input submodular function.