On the existence of polynomial-time approximation schemes for the reoptimization of discrete optimization problems

Abstract

It is shown that no polynomial-time approximation scheme exists for the reoptimization of the set covering problem in inserting an element into or eliminating it from any set. A similar result is obtained for the minimum graph coloring problem in inserting a vertex with at most two incidence edges and for the minimal bin packing problem in eliminating any element.The concept of reoptimization [1-7] is as follows. Let Ï be some NP-hard (or, possibly, NP-complete) problem, and let I be the initial instance of the problem Ï whose optimal solution is known. A new instance I¢ of the problem Ï is proposed that is obtained as a result of some "insignificant" changes in the instance I. The following question arises: how can knowledge on an optimal solution to I be efficiently used for the computation of an exact or approximate solution to the instance I¢? In using approximate methods, the objective of reoptimization is the application of knowledge on the solution of the initial instance I with a view to achieving a better approximation quality (approximation ratio) of I¢, or creating a more (time) efficient algorithm for the determination of an optimal (or close to the optimal) solution of I¢, or attaining both goals.Investigations on the reoptimization of various discrete optimization problems are performed in [1][2][3][4][5][6][7]. PRELIMINARY INFORMATION ON APPROXIMATE ALGORITHMS AND APPROXIMATION CLASSES OF DISCRETE OPTIMIZATION PROBLEMSWe give some concepts necessary for further presentation and introduced in [7]. Definition 1. An NP-optimization (NPO) problem Ï is defined as a quadruple ( , , , ) E m Sol opt such that · E is the set of instances of Ï recognized in polynomial time; · for a given I E Î , Sol( ) I is the set of feasible solutions to I; for each S I S Î Sol( ), | | (the size of S ) is polynomial with respect to | | I (the size of I); for any given I and any S (polynomial with respect to | | I ), S I Î Sol( ) can be determined in polynomial time;· for given I E Î and S I Î Sol( ), m I S ( , ) is the (numerical) value of S and m is a polynomially computable quantity called an objective function;· opt {min } Î , max is the type of an optimization problem. For an NPO problem Ï { Sol opt} = E m , , , , we denote an optimal solution to an instance I of Ï by S I * ( ) and its value m I S I ( , ( ) ) * by opt ( ) I . Definition 2.For an NPO problem Ï Sol opt = ( , , , ) E m , an approximate (approximation) algorithm A is an algorithm that, for a given instance I of Ï produces a feasible solution S I Î Sol( ) .If A is executed in polynomial time with respect to | | I , then A is called a polynomial-time approximate algorithm for Ï . 368