An efficient algorithm for the parametric resource allocation problem

Authors:
Naoki Katoh;Toshihide Ibaraki
Affiliations:
Department of Management Science, Kobe University of Commerce, Tarumi, Kobe, Japan 655;Department of Information and Computer Sciences, Toyohashi University of Technology, Tempaku-cho, Toyohashi, Japan 440
Venue:
Discrete Applied Mathematics
Year:
1985

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Abstract

The parametric resource allocation problem asks to minimize the sum of separable single-variable convex functions containing a parameter @l, @S"i" "=" "1^n (f"i(x"i + @lg"i(x"i)), under simple constraints @S"i" "=" "1^nx"i = M, l"i@?x"i@?u"i and x"i: nonnegative integers for i = 1, 2, ..., n, where M is a given positive integer, and l"i and u"i are given lower and upper bounds on x"i. This paper presents an efficient algorithm for computing the sequence of all optimal solutions when @l is continuously changed from 0 to ~. The required time is O(G@/M log^2n + nlogn + nlog(M/n)), where G = @S"i" "=" "1^nu"i - @S"i" "=" "1"nl"i and an evaluation of f"i(.)or g"i(.) is assumed to be done in constant time.