Small Extended Formulation for Knapsack Cover Inequalities from Monotone Circuits

Initially developed for the min-knapsack problem, the knapsack cover inequalities are used in the current best relaxations for numerous combinatorial optimization problems of covering type. In spite of their widespread use, these inequalities yield linear programming (LP) relaxations of exponential size, over which it is not known how to optimize exactly in polynomial time. In this paper we address this issue and obtain LP relaxations of quasi-polynomial size that are at least as strong as that given by the knapsack cover inequalities. For the min-knapsack cover problem, our main result can be stated formally as follows: for any $\varepsilon >0$, there is a $(1/\varepsilon)^{O(1)}n^{O(\log n)}$-size LP relaxation with an integrality gap of at most $2+\varepsilon$, where $n$ is the number of items. Prior to this work, there was no known relaxation of subexponential size with a constant upper bound on the integrality gap. Our construction is inspired by a connection between extended formulations and monotone circuit complexity via Karchmer-Wigderson games. In particular, our LP is based on $O(\log^2 n)$-depth monotone circuits with fan-in~$2$ for evaluating weighted threshold functions with $n$ inputs, as constructed by Beimel and Weinreb. We believe that a further understanding of this connection may lead to more positive results complementing the numerous lower bounds recently proved for extended formulations.Comment: 21 page

The Knapsack Problem with Neighbour Constraints

We study a constrained version of the knapsack problem in which dependencies between items are given by the adjacencies of a graph. In the 1-neighbour knapsack problem, an item can be selected only if at least one of its neighbours is also selected. In the all-neighbours knapsack problem, an item can be selected only if all its neighbours are also selected. We give approximation algorithms and hardness results when the nodes have both uniform and arbitrary weight and profit functions, and when the dependency graph is directed and undirected.Comment: Full version of IWOCA 2011 pape

A decomposition approach for multidimensional knapsacks with family-split penalties

The optimization of Multidimensional Knapsacks with Family-Split Penalties has been introduced in the literature as a variant of the more classical Multidimensional Knapsack and Multi-Knapsack problems. This problem deals with a set of items partitioned in families, and when a single item is picked to maximize the utility, then all items in its family must be picked. Items from the same family can be assigned to different knapsacks, and in this situation split penalties are paid. This problem arises in real applications in various fields. This paper proposes a new exact and fast algorithm based on a specific Combinatorial Benders Cuts scheme. An extensive experimental campaign computationally shows the validity of the proposed method and its superior performance compared to both commercial solvers and state-of-the-art approaches. The paper also addresses algorithmic flexibility and scalability issues, investigates challenging cases, and analyzes the impact of problem parameters on the algorithm behavior. Moreover, it shows the applicability of the proposed approach to a wider class of realistic problems, including fixed costs related to each knapsack utilization. Finally, further possible research directions are considered

Online Learning and Profit Maximization from Revealed Preferences

We consider the problem of learning from revealed preferences in an online setting. In our framework, each period a consumer buys an optimal bundle of goods from a merchant according to her (linear) utility function and current prices, subject to a budget constraint. The merchant observes only the purchased goods, and seeks to adapt prices to optimize his profits. We give an efficient algorithm for the merchant's problem that consists of a learning phase in which the consumer's utility function is (perhaps partially) inferred, followed by a price optimization step. We also consider an alternative online learning algorithm for the setting where prices are set exogenously, but the merchant would still like to predict the bundle that will be bought by the consumer for purposes of inventory or supply chain management. In contrast with most prior work on the revealed preferences problem, we demonstrate that by making stronger assumptions on the form of utility functions, efficient algorithms for both learning and profit maximization are possible, even in adaptive, online settings