Modular and Submodular Optimization with Multiple Knapsack Constraints via Fractional Grouping

A multiple knapsack constraint over a set of items is defined by a set of bins of arbitrary capacities, and a weight for each of the items. An assignment for the constraint is an allocation of subsets of items to the bins which adheres to bin capacities. In this paper we present a unified algorithm that yields efficient approximations for a wide class of submodular and modular optimization problems involving multiple knapsack constraints. One notable example is a polynomial time approximation scheme for Multiple-Choice Multiple Knapsack, improving upon the best known ratio of 2. Another example is Non-monotone Submodular Multiple Knapsack, for which we obtain a (0.385-?)-approximation, matching the best known ratio for a single knapsack constraint. The robustness of our algorithm is achieved by applying a novel fractional variant of the classical linear grouping technique, which is of independent interest

Submodular utility optimization in sensor networks for capacity constraints

With the fast development of wireless sensor network (WSN) technologies, WSNs have widely shifted from a specialized platform for a single application to an integrated infrastructure supporting multiple applications. It is hence a critical problem to allocate multiple applications to multiple sensors in order to maximize user utility subject to various resource constraints. The resulting constrained optimization problem is difficult since it is discrete, nonlinear, and not in closed-form. In this report, we develop an efficient optimization algorithm with rigorous approximation bounds for submodular monotonic optimization with multiple knapsack constraints. Based on a variance reduction formulation, we prove several important theoretical properties, including the monotonicity and submodularity of functions and the multiple knapsack structure of constraints. Then, by exploiting these properties, we propose a local search algorithm with fractional relaxation of constraints and prove the approximation bound that is better than any known results. Experimentally, we verify the theoretical properties on a large dataset from the Intel Berkeley Lab. Comparison against other constrained search algorithms show that our algorithm is superior in both solution time and quality, making it a practical choice for WSN design

Algorithms to Approximate Column-Sparse Packing Problems

Column-sparse packing problems arise in several contexts in both deterministic and stochastic discrete optimization. We present two unifying ideas, (non-uniform) attenuation and multiple-chance algorithms, to obtain improved approximation algorithms for some well-known families of such problems. As three main examples, we attain the integrality gap, up to lower-order terms, for known LP relaxations for k-column sparse packing integer programs (Bansal et al., Theory of Computing, 2012) and stochastic k-set packing (Bansal et al., Algorithmica, 2012), and go "half the remaining distance" to optimal for a major integrality-gap conjecture of Furedi, Kahn and Seymour on hypergraph matching (Combinatorica, 1993).Comment: Extended abstract appeared in SODA 2018. Full version in ACM Transactions of Algorithm

Ranking with submodular functions on a budget

Submodular maximization has been the backbone of many important machine-learning problems, and has applications to viral marketing, diversification, sensor placement, and more. However, the study of maximizing submodular functions has mainly been restricted in the context of selecting a set of items. On the other hand, many real-world applications require a solution that is a ranking over a set of items. The problem of ranking in the context of submodular function maximization has been considered before, but to a much lesser extent than item-selection formulations. In this paper, we explore a novel formulation for ranking items with submodular valuations and budget constraints. We refer to this problem as max-submodular ranking (MSR). In more detail, given a set of items and a set of non-decreasing submodular functions, where each function is associated with a budget, we aim to find a ranking of the set of items that maximizes the sum of values achieved by all functions under the budget constraints. For the MSR problem with cardinality- and knapsack-type budget constraints we propose practical algorithms with approximation guarantees. In addition, we perform an empirical evaluation, which demonstrates the superior performance of the proposed algorithms against strong baselines.Peer reviewe

Test Score Algorithms for Budgeted Stochastic Utility Maximization

Motivated by recent developments in designing algorithms based on individual item scores for solving utility maximization problems, we study the framework of using test scores, defined as a statistic of observed individual item performance data, for solving the budgeted stochastic utility maximization problem. We extend an existing scoring mechanism, namely the replication test scores, to incorporate heterogeneous item costs as well as item values. We show that a natural greedy algorithm that selects items solely based on their replication test scores outputs solutions within a constant factor of the optimum for a broad class of utility functions. Our algorithms and approximation guarantees assume that test scores are noisy estimates of certain expected values with respect to marginal distributions of individual item values, thus making our algorithms practical and extending previous work that assumes noiseless estimates. Moreover, we show how our algorithm can be adapted to the setting where items arrive in a streaming fashion while maintaining the same approximation guarantee. We present numerical results, using synthetic data and data sets from the Academia.StackExchange Q&A forum, which show that our test score algorithm can achieve competitiveness, and in some cases better performance than a benchmark algorithm that requires access to a value oracle to evaluate function values