Approximation for maximizing monotone non-decreasing set functions with a greedy method

• Published in: Journal of combinatorial optimization Vol. 31; no. 1; pp. 29 - 43
• Main Authors: Wang, Zengfu, Moran, Bill, Wang, Xuezhi, Pan, Quan
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.01.2016
• Subjects: Combinatorics; Convex and Discrete Geometry; Mathematical Modeling and Industrial Mathematics; Mathematics; Mathematics and Statistics; Operations Research/Decision Theory; Optimization; Theory of Computation; Approximation algorithm; Monotone submodular set function; Matroid
• ISSN: 1382-6905; 1573-2886
• DOI: 10.1007/s10878-014-9707-3

We study the problem of maximizing a monotone non-decreasing function f subject to a matroid constraint. Fisher, Nemhauser and Wolsey have shown that, if f is submodular, the greedy algorithm will find a solution with value at least 1 2 of the optimal value under a general matroid constraint and at least 1 - 1 e of the optimal value under a uniform matroid ( M = ( X , I ) , I = { S ⊆ X : | S | ≤ k } ) constraint. In this paper, we show that the greedy algorithm can find a solution with value at least 1 1 + μ of the optimum value for a general monotone non-decreasing function with a general matroid constraint, where μ = α , if 0 ≤ α ≤ 1 ; μ = α K ( 1 - α K ) K ( 1 - α ) if α > 1 ; here α is a constant representing the “elemental curvature” of f , and K is the cardinality of the largest maximal independent sets. We also show that the greedy algorithm can achieve a 1 - ( α + ⋯ + α k - 1 1 + α + ⋯ + α k - 1 ) k approximation under a uniform matroid constraint. Under this unified α -classification, submodular functions arise as the special case 0 ≤ α ≤ 1 .