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

Learning-Based Optimization of Cache Content in a Small Cell Base Station

Optimal cache content placement in a wireless small cell base station (sBS) with limited backhaul capacity is studied. The sBS has a large cache memory and provides content-level selective offloading by delivering high data rate contents to users in its coverage area. The goal of the sBS content controller (CC) is to store the most popular contents in the sBS cache memory such that the maximum amount of data can be fetched directly form the sBS, not relying on the limited backhaul resources during peak traffic periods. If the popularity profile is known in advance, the problem reduces to a knapsack problem. However, it is assumed in this work that, the popularity profile of the files is not known by the CC, and it can only observe the instantaneous demand for the cached content. Hence, the cache content placement is optimised based on the demand history. By refreshing the cache content at regular time intervals, the CC tries to learn the popularity profile, while exploiting the limited cache capacity in the best way possible. Three algorithms are studied for this cache content placement problem, leading to different exploitation-exploration trade-offs. We provide extensive numerical simulations in order to study the time-evolution of these algorithms, and the impact of the system parameters, such as the number of files, the number of users, the cache size, and the skewness of the popularity profile, on the performance. It is shown that the proposed algorithms quickly learn the popularity profile for a wide range of system parameters.Comment: Accepted to IEEE ICC 2014, Sydney, Australia. Minor typos corrected. Algorithm MCUCB correcte

Multiobjective metaheuristic approaches for mean-risk combinatorial optimisation with applications to capacity expansion

Tese de doutoramento. Engenharia Electrotécnica e de Computadores. Faculdade de Engenharia. Universidade do Porto. 200

The Price of Information in Combinatorial Optimization

Consider a network design application where we wish to lay down a minimum-cost spanning tree in a given graph; however, we only have stochastic information about the edge costs. To learn the precise cost of any edge, we have to conduct a study that incurs a price. Our goal is to find a spanning tree while minimizing the disutility, which is the sum of the tree cost and the total price that we spend on the studies. In a different application, each edge gives a stochastic reward value. Our goal is to find a spanning tree while maximizing the utility, which is the tree reward minus the prices that we pay. Situations such as the above two often arise in practice where we wish to find a good solution to an optimization problem, but we start with only some partial knowledge about the parameters of the problem. The missing information can be found only after paying a probing price, which we call the price of information. What strategy should we adopt to optimize our expected utility/disutility? A classical example of the above setting is Weitzman's "Pandora's box" problem where we are given probability distributions on values of $n$ independent random variables. The goal is to choose a single variable with a large value, but we can find the actual outcomes only after paying a price. Our work is a generalization of this model to other combinatorial optimization problems such as matching, set cover, facility location, and prize-collecting Steiner tree. We give a technique that reduces such problems to their non-price counterparts, and use it to design exact/approximation algorithms to optimize our utility/disutility. Our techniques extend to situations where there are additional constraints on what parameters can be probed or when we can simultaneously probe a subset of the parameters.Comment: SODA 201