19 research outputs found
Vector Bin Packing with Multiple-Choice
We consider a variant of bin packing called multiple-choice vector bin
packing. In this problem we are given a set of items, where each item can be
selected in one of several -dimensional incarnations. We are also given
bin types, each with its own cost and -dimensional size. Our goal is to pack
the items in a set of bins of minimum overall cost. The problem is motivated by
scheduling in networks with guaranteed quality of service (QoS), but due to its
general formulation it has many other applications as well. We present an
approximation algorithm that is guaranteed to produce a solution whose cost is
about times the optimum. For the running time to be polynomial we
require and . This extends previous results for vector
bin packing, in which each item has a single incarnation and there is only one
bin type. To obtain our result we also present a PTAS for the multiple-choice
version of multidimensional knapsack, where we are given only one bin and the
goal is to pack a maximum weight set of (incarnations of) items in that bin
Characterizing Locality in Decoder-Based EAs for the Multidimensional Knapsack Problem
The performance of decoder-based evolutionary algorithms (EAs) strongly depends on the locality of the used decoder and operators
Shrinking Maxima, Decreasing Costs: New Online Packing and Covering Problems
International audienceWe consider two new variants of online integer programs that are duals. In the packing problem we are given a set of items and a collection of knapsack constraints over these items that are revealed over time in an online fashion. Upon arrival of a constraint we may need to remove several items (irrevocably) so as to maintain feasibility of the solution. Hence, the set of packed items becomes smaller over time. The goal is to maximize the number, or value, of packed items. The problem originates from a bu↵er-overflow model in communication networks, where items represent information units broken into multiple packets. The other problem considered is online covering: There is a universe to be covered. Sets arrive online, and we must decide for each set whether we add it to the cover or give it up. The cost of a solution is the total cost of sets taken, plus a penalty for each uncovered element. The number of sets in the solution grows over time, but its cost goes down. This problem is motivated by team formation, where the universe consists of skills, and sets represent candidates we may hire. The packing problem was introduced in [8] for the special case where the matrix is binary; in this paper we extend the solution to general matrices with non-negative integer entries. The covering problem is introduced in this paper; we present matching upper and lower bounds on its competitive ratio
Modified choice function heuristic selection for the multidimensional knapsack problem
Hyper-heuristics are a class of high-level search methods used to solve computationally difficult problems, which operate on a search space of low-level heuristics rather than solutions directly. Previous work has shown that selection hyper-heuristics are able to solve many combinatorial optimisation problems, including the multidimensional 0-1 knapsack problem (MKP). The traditional framework for iterative selection hyper-heuristics relies on two key components, a heuristic selection method and a move acceptance criterion. Existing work has shown that a hyper-heuristic using Modified Choice Function heuristic selection can be effective at solving problems in multiple problem domains. Late Acceptance Strategy is a hill climbing metaheuristic strategy often used as a move acceptance criteria in selection hyper-heuristics. This work compares a Modified Choice Function - Late Acceptance Strategy hyper-heuristic to an existing selection hyper-heuristic method from the literature which has previously performed well on standard MKP benchmarks
Approximability of sparse integer programs
The main focus of this paper is a pair of new approximation algorithms for sparse integer programs. First, for covering integer programs {min cx: Ax ≥ b,0 ≤ x ≤ d} where A has at most k nonzeroes per row, we give a k-approximation algorithm. (We assume A, b, c, d are nonnegative.) For any k ≥ 2 and ǫ> 0, if P = NP this ratio cannot be improved to k − 1 − ǫ, and under the unique games conjecture this ratio cannot be improved to k − ǫ. One key idea is to replace individual constraints by others that have better rounding properties but the same nonnegative integral solutions; another critical ingredient is knapsack-cover inequalities. Second, for packing integer programs {max cx: Ax ≤ b,0 ≤ x ≤ d} where A has at most k nonzeroes per column, we give a 2 k k 2-approximation algorithm. This is the first polynomial-time approximation algorithm for this problem with approximation ratio depending only on k, for any k> 1. Our approach starts from iterated LP relaxation, and then uses probabilistic and greedy methods to recover a feasible solution. Note added after publication: This version includes subsequent developments: a O(k 2) approximation for the latter problem using the iterated rounding framework, and several literature reference updates including a O(k)-approximation for the same problem by Bansal et al