MATHEMATICAL ENGINEERING TECHNICAL REPORTS The Complexity of Maximizing the Difference of Two Matroid Rank Functions

In the context of discrete DC programming, Maehara and Murota (Mathematical Programming, Series A, 2014) posed the problem of determining the complexity of minimizing the difference of two M-convex set functions. In this paper, we show the NP-hardness of this minimization problem by proving a stronger result: maximizing the difference of two matroid rank functions is NP-hard. Maehara and Murota [3] established a theoretical framework of difference of discrete convex functions and studied the problem of minimizing the difference of two discrete convex functions (discrete DC programs). The computational complexities of several types of discrete DC programs were revealed in their paper, but determining the complexity of minimizing the difference of two M♮-convex set functions was posed as an open question (see Section 4.3 and Table 1 in [3]).1 In this paper, we show the NP-hardness of this minimization problem. As we will describe later, a special case of this problem is to maximize the difference of two matroid rank functions. We first show the NP-hardness of this special case. Theorem 1. The following problem is NP-hard: for two matroids M1 and M2 on a common ground set E with rank functions f1 and f2, maximize X⊆E f1(X)− f2(X). (1) Proof. Let B1 be the base family of M1. We show that Problem (1) is equivalent to the following problem: maximize |X| − f2(X) subject to X ∈ B1. (2) Claim 2. Problems (1) and (2) are equivalent. ∗This work is supported by JST, ERATO, Kawarabayashi Large Graph Project, and by KAKENHI Grant Number 24106002, 24700004. Although the term “M-convex functions on {0, 1}” is used in [3], they are equivalent to “M-convex set functions” by identifying a subset of the ground set and its characteristic vector.