On the approximability of the maximum feasible subsystem problem with 0/1-coefficients

Abstract

Given a system of constraints i ≤ a T i x ≤ u i , where a i ∈ {0, 1} n , and i , u i ∈ R + , for i = 1, . . . , m, we consider the problem Mrfs of finding the largest subsystem for which there exists a feasible solution x ≥ 0. We present approximation algorithms and inapproximability results for this problem, and study some important special cases. Our main contributions are :1. In the general case, where a i ∈ {0, 1} n , a sharp separation in the approximability between the case when L = max{ 1 , · · · , m } is bounded above by a polynomial in n and m, and the case when it is not.2. In the case where A is an interval matrix, a sharp separation in approximability between the case where we allow a violation of the upper bounds by at most a (1 + ) factor, for any fixed > 0 and the case where no violations are allowed.Along the way, we prove that the induced matching problem on bipartite graphs is inapproximable beyond a factor of Ω(n 1 3 − ), for any > 0 unless NP=ZPP. Finally, we also show applications of Mrfs to some recently studied pricing problems.

Independent Set, Induced Matching, and Pricing: Connections and Tight (Subexponential Time) Approximation Hardnesses

“…It is, for example, studied as the "risk-free" marriage problem in [52] and is a subtask of finding a strong edge coloring. This problem and its variations also have connections to various problems such as maximum feasible subsystem [25,16], maximum expanding sequence [11], storylines extraction [38] and network scheduling, gathering and testing (e.g., [27,52,36,42,7]). In general graphs, the problem was shown to be NP-complete in [52,13] and was later shown in [20] to be hard to approximate to within a factor of n 1−ǫ and d 1−ǫ unless P = NP, where n is the number of vertices and d is the maximum degree of a graph.…”

On the approximability of the maximum interval constrained coloring problem

“…The main technique that we use is to decompose the problem into simpler instances using Dilworth's Theorem [4], and then show how we can solve these simpler instances in polynomial time. A similar technique was used earlier [5] to obtain an O( |Opt| log n)-approximation algorithm for the following Maximum Feasible Subsystem problem: Given an infeasible linear system l ≤ Ax ≤ u, x ≥ 0, where A is a consecutive 1's matrix, the problem is to find the largest subsystem for which there is a feasible solution satisfying the nonnegativity constraints x ≥ 0. However, the same decomposition does not work for the Mfc problem.…”

“…For a variable i, we let m i denote the number of clauses that the variable appears in. The reduction is inspired by the APXhardness proof for the maximum feasible subsystem on interval matrices [5]. We construct gadgets for each variable and each clause.…”

Complexity and Approximability of Parameterized MAX-CSPs

“…MAJORITY and THRESHOLD constraints are of course some of the most natural and well-studied predicates in many contexts: for example, MAX-CSP for such constraints contains the complexity of finding an assignment that satisfies as many inequalities as possible in a 0-1 Integer Linear Program whose coefficients are in {−1, 0, 1}. This problem, sometimes called Maximum Feasible Subset has been well-studied in the literature [9,3,2]. MAJORITY constraints also play a central role in learning theory [10,17] and in hardness of approximation [7].…”