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]

Locally Adaptive Optimization: Adaptive Seeding for Monotone Submodular Functions

The Adaptive Seeding problem is an algorithmic challenge motivated by influence maximization in social networks: One seeks to select among certain accessible nodes in a network, and then select, adaptively, among neighbors of those nodes as they become accessible in order to maximize a global objective function. More generally, adaptive seeding is a stochastic optimization framework where the choices in the first stage affect the realizations in the second stage, over which we aim to optimize. Our main result is a $(1-1/e)^2$-approximation for the adaptive seeding problem for any monotone submodular function. While adaptive policies are often approximated via non-adaptive policies, our algorithm is based on a novel method we call \emph{locally-adaptive} policies. These policies combine a non-adaptive global structure, with local adaptive optimizations. This method enables the $(1-1/e)^2$-approximation for general monotone submodular functions and circumvents some of the impossibilities associated with non-adaptive policies. We also introduce a fundamental problem in submodular optimization that may be of independent interest: given a ground set of elements where every element appears with some small probability, find a set of expected size at most $k$ that has the highest expected value over the realization of the elements. We show a surprising result: there are classes of monotone submodular functions (including coverage) that can be approximated almost optimally as the probability vanishes. For general monotone submodular functions we show via a reduction from \textsc{Planted-Clique} that approximations for this problem are not likely to be obtainable. This optimization problem is an important tool for adaptive seeding via non-adaptive policies, and its hardness motivates the introduction of \emph{locally-adaptive} policies we use in the main result

Budget Feasible Mechanism Design: From Prior-Free to Bayesian

Budget feasible mechanism design studies procurement combinatorial auctions where the sellers have private costs to produce items, and the buyer(auctioneer) aims to maximize a social valuation function on subsets of items, under the budget constraint on the total payment. One of the most important questions in the field is "which valuation domains admit truthful budget feasible mechanisms with `small' approximations (compared to the social optimum)?" Singer showed that additive and submodular functions have such constant approximations. Recently, Dobzinski, Papadimitriou, and Singer gave an O(log^2 n)-approximation mechanism for subadditive functions; they also remarked that: "A fundamental question is whether, regardless of computational constraints, a constant-factor budget feasible mechanism exists for subadditive functions." We address this question from two viewpoints: prior-free worst case analysis and Bayesian analysis. For the prior-free framework, we use an LP that describes the fractional cover of the valuation function; it is also connected to the concept of approximate core in cooperative game theory. We provide an O(I)-approximation mechanism for subadditive functions, via the worst case integrality gap I of LP. This implies an O(log n)-approximation for subadditive valuations, O(1)-approximation for XOS valuations, and for valuations with a constant I. XOS valuations are an important class of functions that lie between submodular and subadditive classes. We give another polynomial time O(log n/loglog n) sub-logarithmic approximation mechanism for subadditive valuations. For the Bayesian framework, we provide a constant approximation mechanism for all subadditive functions, using the above prior-free mechanism for XOS valuations as a subroutine. Our mechanism allows correlations in the distribution of private information and is universally truthful.Comment: to appear in STOC 201