TY - JOUR

T1 - The Power of Subsampling in Submodular Maximization

T2 - Mathematics of Operations Research

AU - Harshaw, Christopher

AU - Kazemi, Ehsan

AU - Feldman, Moran

AU - Karbasi, Amin

N1 - doi: 10.1287/moor.2021.1172

PY - 2021

Y1 - 2021

N2 - We propose subsampling as a unified algorithmic technique for submodular maximization in centralized and online settings. The idea is simple: independently sample elements from the ground set and use simple combinatorial techniques (such as greedy or local search) on these sampled elements. We show that this approach leads to optimal/state-of-the-art results despite being much simpler than existing methods. In the usual off-line setting, we present SampleGreedy, which obtains a (p+2+o(1))-approximation for maximizing a submodular function subject to a p-extendible system using O(n+nk/p) evaluation and feasibility queries, where k is the size of the largest feasible set. The approximation ratio improves to p + 1 and p for monotone submodular and linear objectives, respectively. In the streaming setting, we present Sample-Streaming, which obtains a (4p+2?o(1))-approximation for maximizing a submodular function subject to a p-matchoid using O(k) memory and O(km/p) evaluation and feasibility queries per element, and m is the number of matroids defining the p-matchoid. The approximation ratio improves to 4p for monotone submodular objectives. We empirically demonstrate the effectiveness of our algorithms on video summarization, location summarization, and movie recommendation tasks.

AB - We propose subsampling as a unified algorithmic technique for submodular maximization in centralized and online settings. The idea is simple: independently sample elements from the ground set and use simple combinatorial techniques (such as greedy or local search) on these sampled elements. We show that this approach leads to optimal/state-of-the-art results despite being much simpler than existing methods. In the usual off-line setting, we present SampleGreedy, which obtains a (p+2+o(1))-approximation for maximizing a submodular function subject to a p-extendible system using O(n+nk/p) evaluation and feasibility queries, where k is the size of the largest feasible set. The approximation ratio improves to p + 1 and p for monotone submodular and linear objectives, respectively. In the streaming setting, we present Sample-Streaming, which obtains a (4p+2?o(1))-approximation for maximizing a submodular function subject to a p-matchoid using O(k) memory and O(km/p) evaluation and feasibility queries per element, and m is the number of matroids defining the p-matchoid. The approximation ratio improves to 4p for monotone submodular objectives. We empirically demonstrate the effectiveness of our algorithms on video summarization, location summarization, and movie recommendation tasks.

U2 - 10.1287/moor.2021.1172

DO - 10.1287/moor.2021.1172

M3 - Article

SN - 0364-765X

VL - 47

SP - 1365

EP - 1393

JO - Mathematics of Operations Research

JF - Mathematics of Operations Research

IS - 2

ER -