Coupling Variable Fixing Algorithms for the Automatic Recording ...

Coupling Variable Fixing Algorithms for the Automatic Recording Problem ? Meinolf Sellmann and Torsten Fahle University of Paderborn Department of Mathematics and Computer Science Fürstenallee 11, D-33102 Paderborn {sello,tef}@uni-paderborn.de First Version: April 2001 Revised: August 2001

Abstract. Variable fixing is an important technique when solving combinatorial optimization problems. Unique profitable variable values are detected with respect to the objective function and to the constraint structure of the problem. Relying on that specific structure, effective variable fixing algorithms (VFAs) are only suited for the problems they have been designed for. Frequently, new combinatorial optimization problems evolve as a combination of simpler structured problems. For such combinations, we show how VFAs for linear optimization problems can be coupled via Lagrangian relaxation. The method is applied on a multimedia problem incorporating a knapsack and a maximum weighted stable set problem.

1

Introduction

Reduction algorithms are of great importance when combinatorial optimization problems have to be solved exactly. The tightening of problem formulations within a branchand-bound approach improves on the quality of the bounds computed as well as on the approach’s robustness. Given a maximization problem P (x) where x ∈ {0, 1}n , n ∈ IN, the idea of variable fixing is to use upper bound information to detect unique profitable assignments for a variable: If an upper bound on P (x|xi =k ), k ∈ {0, 1}, drops below the best known solution value, then we can set xi ← 1 − k. Frequently, constraints of optimization problems can be grouped such that the overall problem can be viewed as a combination of two or more simpler structured problems. Assuming that efficient variable fixing algorithms (VFAs) for these subproblems exist, their independent application usually does not yield an effective algorithm to perform variable fixing for the combined problem. The reason for this is that tight bounds on the objective cannot be obtained by taking only a subset of the constraints into account. ?

This work was partly supported by the German Science Foundation (DFG) project SFB-376, by the UP-TV project, partially funded by the IST program of the Commission of the European Union as project number 1999-20 751, and by the IST Programme of the EU under contract number IST-1999-14186 (ALCOM-FT).

For a multimedia application incorporating a knapsack problem (KP) and a maximum weighted stable set problem (MWSSP) on an interval graph, we show exemplary how two VFAs for linear optimization problems can be coupled via Lagrangian relaxation to achieve an effective reduction algorithm for the combined problem. The paper is structured as follows: In Section 2, we introduce the Automatic Recording Problem, that can be viewed as a combination of a knapsack problem and a MWSSP on interval graphs. In Section 3, we introduce an efficient VFA for the latter problem. Then, in Section 4, we show how it can be coupled with a previously developed VFA for KP via Lagrangian relaxation. Finally, in Section 5 we give numerical results.

2

The Automatic Recording Problem

The Automatic Recording Problem (ARP) is an example of a problem that is constituted by two simpler substructures. We focus on algorithms that solve the problem exactly and give a tightened formulation of the ARP as an integer program (IP). The technology of digital television offers new possibilities for individualized services that cannot be provided by nowadays analog broadcasts. Additional information like classification of content, or starting and ending times can be submitted within the digital broadcast stream. With those informations at hand, new services can be provided that make use of individual profiles and maximize customer satisfaction. One service – which is available already today – is an "intelligent" digital video recorder that is aware of its users’ preferences and records automatically (see [2]). The recorder tries to match a given user profile with the information submitted by the different TV channels. E.g., a user may be interested in thrillers, the more recent the better. The digital video recorder is supposed to record movies such that the users’ satisfaction is maximized. As the number of channels may be enormous (more than 100 digital channels are possible), a service that automatically provides an individual selection is highly appreciated and subject of current research activities (for example within projects like UP-TV funded by the European Union or the TV-Anytime Forum). In this context, two restrictions have to be met. First, the storage capacity is limited (10h of MPEG-2 video need about 18 GB). And second, only one video can be recorded at a time. More formally, we define the problem as follows: Definition 1. Let n ∈ IN, V = {0, . . . , n−1} the set of movies, start(i) < end(i) ∀ i ∈ V the corresponding starting and ending times. w = (wi )0≤i