Abstract

Smoothed analysis of multiobjective 0-1 linear optimization has drawn con- siderable attention recently. The goal is to give bounds for the number of Pareto-optimal solutions (i. e., solutions with the property that no other solution is at least as good in all the coordinates and better in at least one) for multiobjective optimization problems. In this article we prove several lower bounds for the expected number of Pareto optima. Our basic result is a lower bound of Wd(n d 1 ) for optimization problems with d objectives and n variables under fairly general conditions on the distributions of the linear objectives. Our proof relates the problem of finding lower bounds for the number of Pareto optima to results in discrete geometry and geometric probability about arrangements of hyperplanes. We use our basic result to derive the following results: (1) To our knowledge, the first lower bound for natural multiobjective optimization problems. We illustrate this on the maximum spanning tree problem with randomly chosen edge weights. Our technique is sufficiently flexible to yield such lower bounds also for other standard objective functions studied in this setting (such as multiobjective shortest path, TSP, matching). (2) A smoothed lower bound of minfWd(n d 1:5 f d ); 2 Qd(n) g for f -smooth instances of the 0-1 knapsack problem with d profits.

Highlights

  • Multiobjective optimization involves scenarios where there is more than one objective function to optimize: when planning a train trip we may want to choose connections that minimize fare, total time, number of train changes, etc

  • The objectives may be in conflict with each other and there may not be a single best solution to the problem. Such multiobjective optimization problems arise in diverse fields ranging from economics to computer science, and have been well-studied

  • The set of Pareto-optimal solutions (Pareto set in short) contains all desirable solutions as any other solution is strictly worse than a solution in the Pareto set

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Summary

Introduction

Multiobjective optimization involves scenarios where there is more than one objective function to optimize: when planning a train trip we may want to choose connections that minimize fare, total time, number of train changes, etc. In the d-objective maximum spanning tree problem on Kn there exists a choice of 2-smooth distributions such that the expected number of Pareto-optimal spanning trees is at least n − 3 d−1 The proof of this theorem utilizes the full power of Theorem 1.1, namely the ability to choose different symmetric distributions. 3. a set S ⊆ {0, 1}n such that if profits v(ji) are chosen independently and uniformly at random in [ai j, bi j], the expected number of Pareto-optimal solutions of the d-dimensional knapsack problem with solution set S is at least min Ωd(nd−1.5φ d), 2Θd(n) This lower bound should be contrasted with the aforementioned upper bounds 2 · (4φ d)d(d+1)/2 · n2d [15] and Od(n2dφ d) [7].

The basic theorem
First proof
Second proof
Lower bound for multiobjective maximum spanning trees
Lower bound for general solution sets
Discussion and conclusion
A Technical proofs
Full Text
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