AbstractA simple greedy heuristic algorithm for the multi-criteria set-covering problem is presented. This result is a multi-criteria generalization of the results established previously by Chvatal

Liu, Yi-Hsin

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Appl. Math. L&t. Vol. 6, No. 5, pp. 21-23, 1993 
Printed in Great Britain. All rights reserved 
08939659/93 $6.00 + 0.00 
Copyright@ 1993 Pergamon Press Ltd 
A HEURISTIC ALGORITHM FOR 
THE MULTI-CRITERIA SET-COVERING PROBLEMS 
Y I-HSIN LIU 
Department of Mathematics and Computer Science 
University of Nebraska at Omaha, Omaha, NE 68182-0243, U.S.A. 
(Received May 1993; accepted June 1993) 
Abstract-A simple greedy heuristic algorithm for the multi-criteria set-covering problem is pre 
sented. This result is a multi-criteria generalization of the results established previously by Chvatal. 
1. INTRODUCTION 
In the Multi-Criteria Set-Covering (MCSC) problem, we are given a finite collection P = {Pj : 
1 < j 5 n} of subsets of Z = {1,2 ,..., m}. We associate each Pj E P with a nonnegative 
and nonzero CO& k-vector C: = (Clj,czj,...,Ckj). A subcollection P* of P is called a cover if 
UPjeP* P’ = Z; th e associated cost is a k-vector CPjEP. 3. c. The problem is to find an “efficient” 
cover. That is, to find a cover such that if any improvement of one objective function can be 
achieved it is only at the expense of another. Since it is NP-hard to find an efficient cover, a 
heuristic solution for (MCSC) is important. This note presents a heuristic algorithm to find a 
near optimal solution and a tight bound on the worst case of this relatively simple heuristic. 
An (MCSC) has the following formulation: 
n 
C cljXj 
j=l 
n 
C C2jXj 
mint j=l 
n 
c Ckjxj 
\ j=l 
n 
S.t. C aljXj 2 1 
j=l 
2 amjxj 2 1, Xj E {O,l}, Vj = 1,2 ,..., n. 
j=l 
Where 
and a solution 
1 
Xj = 
if Pj E P* 
0 otherwise. 
We associate a Boolean vector 1= (21,&r,. . . , zn) with a subcollection P of P. 
(MCSC) 
21 
22 Y.-H. LIU 
A feasible solution 3 = (??I, 32,. . . , Zn) of (MCSC) is called an efficient solution. [l] (Pareto 
optimal) if, and only if, there is no other feasible solution z = (~1, x2,. . . cc,) such that 
2 CijXj 5 kcijZj, foralli=1,2 ,..., k, 
j=l j=l 
and 
2 CijXj < ecijZj, for at least one i. 
j=l j=l 
A cover P is called an efficient cover if its associated vector z is efficient. 
2. MAIN RESULTS 
To develop a heuristic algorithm for finding a near optimal solution intuitively in the single 
criterion case, one would select Pj to cover j provided lPj I/Cj, the number of points covered by 
Pj per unit cost, is maximized. However, in a multi-criteria case, cj is a vector, and this intuition 
is not useful. In order to do this, we consider an alternative for a multi-criteria case by choosing 
cj*=max{qj:i=1,2,..., k} > 0. Due to the different scales used in each objective, this choice 
cannot be meaningful unless the coefficients of the objectives are normalized. To do this, without 
loss of generality, we may assume 
72 
c Cij = 1, foralli=1,2 ,..., k. 
j=l 
Then the ratio 1 Pj I/ j c* means the worst case of unit cost for covering j using Pj among all criteria. 
This leads to the following heuristic: 
THE (MCSC) GREEDY HEURISTIC ALGORITHM. 
Step 0 
P* + a 
Step 1 
If Pj = fD’, Vj, Stop. 
P* is a cover, otherwise, find Pk such that ]Pk]/c; = mm{ IPjl/c; : Pj E P - P* and C; = 
msx Cij}. 
l<j<k -- 
Step 2 
P* +- P* u {Pk} 
Pj 6 Pj - Pk, vj # k. 
Go to Step 1. 
Let P* be a cover obtained from the above heuristic, P be any efficient cover, and let x* and 
z be the associated incident vectors of P* and P, repectively. Also, let M = l~Jyn ) Pj I. Then 
in the single objective case (i.e., k = 1) the algorithm is Chvatal’s greedy heuri&, and he has 
shown that “the cost of cover returned by the greedy heuristic is at most cf”=r 9 times the cost 
of an optimal cover.” [2] Th e o f 11 owing general theorem is now presented (k 2 1). 
THEOREM. Let x* be a solution obtained by the (MCSC) greedy heuristic and Z be any efficient 
solution of (MCSC). Then 
A heuristic algorithm 23 
PROOF. For each feasible z = (XI, 22,. . . , G), since cj* = lyyk Cij, we have 
-- 
n n 
c CijXj < c Cj*Xj. 
j=l j=l 
Therefore, by Chvatal’s theorem, for each i = 1,2,. . . , k, 
n n /M.\/n \ 
c CijXJ<C * * 
j=l j=l 
cjxj < (2:) (zGzj) . 
REFERENCES 
1. P.L. Yu, Multiple-Criteria Decision Making, Plenum, New York, NY, (1985). 
2. V. Chvatal, A greedy heuristic for the set-covering problem, Math of O.R. 4 (3), 233-235 (1979). 


A heuristic algorithm for the multi-criteria set-covering problems 

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A heuristic algorithm for the multi-criteria set-covering problems

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