On Maximizing Sums of Non-Monotone Submodular and Linear Functions

We study the problem of Regularized Unconstrained Submodular Maximization (RegularizedUSM) as defined by [Bodek and Feldman \u2722]. In this problem, we are given query access to a non-negative submodular function f: 2^N ? ?_{? 0} and a linear function ?: 2^N ? ? over the same ground set N, and the objective is to output a set T ? N approximately maximizing the sum f(T)+?(T). Specifically, an algorithm is said to provide an (?,?)-approximation for RegularizedUSM if it outputs a set T such that E[f(T)+?(T)] ? max_{S ? N}[? ? f(S)+?? ?(S)]. We also study the setting where S and T are constrained to be independent in a given matroid, which we refer to as Regularized Constrained Submodular Maximization (RegularizedCSM). The special case of RegularizedCSM with monotone f has been extensively studied [Sviridenko et al. \u2717, Feldman \u2718, Harshaw et al. \u2719]. On the other hand, we are aware of only one prior work that studies RegularizedCSM with non-monotone f [Lu et al. \u2721], and that work constrains ? to be non-positive. In this work, we provide improved (?,?)-approximation algorithms for both {RegularizedUSM} and {RegularizedCSM} with non-monotone f. In particular, we are the first to provide nontrivial (?,?)-approximations for RegularizedCSM where the sign of ? is unconstrained, and the ? we obtain for RegularizedUSM improves over [Bodek and Feldman \u2722] for all ? ? (0,1). In addition to approximation algorithms, we provide improved inapproximability results for all of the aforementioned cases. In particular, we show that the ? our algorithm obtains for {RegularizedCSM} with unconstrained ? is essentially tight for ? ? e/(e+1). Using similar ideas, we are also able to show 0.478-inapproximability for maximizing a submodular function where S and T are subject to a cardinality constraint, improving a 0.491-inapproximability result due to [Oveis Gharan and Vondrak \u2710]

Submodular Maximization Subject to Matroid Intersection on the Fly

Despite a surge of interest in submodular maximization in the data stream model, there remain significant gaps in our knowledge about what can be achieved in this setting, especially when dealing with multiple constraints. In this work, we nearly close several basic gaps in submodular maximization subject to k matroid constraints in the data stream model. We present a new hardness result showing that super polynomial memory in k is needed to obtain an o(k/(log k))-approximation. This implies near optimality of prior algorithms. For the same setting, we show that one can nevertheless obtain a constant-factor approximation by maintaining a set of elements whose size is independent of the stream size. Finally, for bipartite matching constraints, a well-known special case of matroid intersection, we present a new technique to obtain hardness bounds that are significantly stronger than those obtained with prior approaches. Prior results left it open whether a 2-approximation may exist in this setting, and only a complexity-theoretic hardness of 1.91 was known. We prove an unconditional hardness of 2.69

On the complexity of nonlinear mixed-integer optimization

This is a survey on the computational complexity of nonlinear mixed-integer optimization. It highlights a selection of important topics, ranging from incomputability results that arise from number theory and logic, to recently obtained fully polynomial time approximation schemes in fixed dimension, and to strongly polynomial-time algorithms for special cases.Comment: 26 pages, 5 figures; to appear in: Mixed-Integer Nonlinear Optimization, IMA Volumes, Springer-Verla