A simple deterministic algorithm for symmetric submodular maximization subject to a knapsack constraint

Georgios Amanatidis, Georgios Birmpas, Evangelos Markakis

Abstract: Highlights • We study symmetric submodular maximization subject to a knapsack constraint. • Submodular functions become monotone when restricted to their local maxima. • They are “almost” monotone on their approximate local maxima. • There is a greedy approach that is robust under small deviations from monotonicity. • There is a 2 e / ( e − 1 ) -approximation algorithm for symmetric submodular objectives. Abstract We obtain a polynomial-time deterministic ( 2 e e − 1 + ϵ ) -approximation algorithm for maximizing symmetric submodular functions under a budget constraint. Although there exist randomized algorithms with better expected performance, our algorithm achieves the best known factor achieved by a deterministic algorithm, improving on the previously known factor of 6. Furthermore, it is simple, combining two elegant algorithms for related problems; the local search algorithm of Feige, Mirrokni and Vondrák [1] for unconstrained submodular maximization, and the greedy algorithm of Sviridenko [2] for non-decreasing submodular maximization subject to a knapsack constraint.