Parametric monotone function maximization with matroid constraints

• Published in: Journal of global optimization Vol. 75; no. 3; pp. 833 - 849
• Main Authors: Gong, Suning, Nong, Qingqin, Liu, Wenjing, Fang, Qizhi
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.11.2019; Springer; Springer Nature B.V
• Subjects: Algorithms; Analysis; Computer Science; Decision trees; Greedy algorithms; Mathematics; Mathematics and Statistics; Maximization; Operations Research/Decision Theory; Optimization; Real Functions; Rounding; Increasing set function; Approximation algorithm; Generic submodularity ratio; Matroid
• ISSN: 0925-5001; 1573-2916
• DOI: 10.1007/s10898-019-00800-2

We study the problem of maximizing an increasing function f : 2 N → R + subject to matroid constraints. Gruia Calinescu, Chandra Chekuri, Martin Pál and Jan Vondrák have shown that, if f is nondecreasing and submodular, the continuous greedy algorithm and pipage rounding technique can be combined to find a solution with value at least 1 - 1 / e of the optimal value. But pipage rounding technique have strong requirement for submodularity. Chandra Chekuri, Jan Vondrák and Rico Zenklusen proposed a rounding technique called contention resolution schemes. They showed that if f is submodular, the objective value of the integral solution rounding by the contention resolution schemes is at least 1 - 1 / e times of the value of the fractional solution. Let f : 2 N → R + be an increasing function with generic submodularity ratio γ ∈ ( 0 , 1 ] , and let ( N , I ) be a matroid. In this paper, we consider the problem max S ∈ I f ( S ) and provide a γ ( 1 - e - 1 ) ( 1 - e - γ - o ( 1 ) ) -approximation algorithm. Our main tools are the continuous greedy algorithm and contention resolution schemes which are the first time applied to nonsubmodular functions.