An implicit exhaustion method for solving some extremum problems on sets

Abstract

Covering, partitioning, and packing problems are widespread extremum problems on sets. STATEMENTS OF THE PROBLEMSAssume that there are given a finite set V = {vl,..., vm} and a family of its subsets S = {S1,..., Sn}. Let us assign some weight (price) cj to every subset Sj of the family S.Let S' = {Sj~,..., Sj, } be some subfamily of the family S. If every element v~ is contained in no less (no more) than one of the subsets Sj entering into S', then S' is called a covering (a packing) of the set V. A covering that is a packing at the same time is called a partition of the set V. The problem consists of choosing some subfamily of the family S so that it is a covering (a partition or packing) of the set V and the total weight (cost) of this covering or partition is minimal (maximal in the case of a partition or packing). The minimum (maximum) is taken over the set of all possible coverings (partitions or packings).Denote by A = [a~j]~x~ the matrix of incidences of elements of V and subsets Sj: a~j = I if vi E Sj and a~j = 0 if vi r Sj. Every subfamily S' of the family S is given by the vector with the components zj = 1 if the subset Sj is included in S ~ and xj = 0 otherwise.Then mathematical models of these problems can be formulated as follows.