Bounds on Double-Sided Myopic Algorithms for Unconstrained Non-monotone Submodular Maximization

TL;DR

This paper investigates the limitations of double-sided myopic algorithms for unconstrained non-monotone submodular maximization, establishing bounds on their approximation ratios and contrasting deterministic and randomized approaches.

Contribution

The paper introduces a formal model called double-sided myopic algorithms and proves bounds on their performance, highlighting differences between deterministic and randomized algorithms.

Findings

Randomized algorithms can achieve a 1/2 approximation ratio.

Deterministic double-sided myopic algorithms cannot match this ratio.

Connections are made to existing algorithms for Max-Di-Cut and related problems.

Abstract

Unconstrained submodular maximization captures many NP-hard combinatorial optimization problems, including Max-Cut, Max-Di-Cut, and variants of facility location problems. Recently, Buchbinder et al. presented a surprisingly simple linear time randomized greedy-like online algorithm that achieves a constant approximation ratio of 1/2, matching optimally the hardness result of Feige et al.. Motivated by the algorithm of Buchbinder et al., we introduce a precise algorithmic model called double-sided myopic algorithms. We show that while the algorithm of Buchbinder et al. can be realized as a randomized online double-sided myopic algorithm, no such deterministic algorithm, even with adaptive ordering, can achieve the same approximation ratio. With respect to the Max-Di-Cut problem, we relate the Buchbinder et al. algorithm and our myopic framework to the online algorithm and inapproximation of Bar-Noy and Lampis.