In this section, we give an alternative proof to a theorem of Murota [3] on maximum eigensets and the ranks of power products of linking systems.
Let A= (S,S,Λ)be a linking system whose row set and column set are identical. Murota [3] introduced the concept of eigenset of such a linking system and investigated its connection to power products. A subset X⊆S is called an eigenset if
SATORUIWATA 135
(X,X)∈Λ. Let Akdenote the k-th power product A∗ ··· ∗A of A. Then r(Ak)is monotone nonincreasing and convex in k.
Hence there exists`≤ |S|such that r(Ak) =r(Ak+1)holds for k≥`. We denote this rank by r(A∞). The following theorem characterizes r(A∞)in terms of eigensets.
Theorem 18 (Murota [3]) For a linking system A= (S,S,Λ), we have r(A∞) =max{|X| |(X,X)∈Λ}.
We now present an alternative proof of this result using Theorem 17. Consider a matroid pencil (A,I)with diagonal linking system I= (S,S,∆)and denote the rank ofΘk(A,I)byθk(A,I). Then it follows from Lemma 4 that
r(Ak) =θk(A,I)−(k−1)|S|. Therefore, the following lemma completes the proof of Theorem 18.
Lemma 19 For k≥ |S|, we have
θk(A,I) = (k−1)|S|+max{|X| |(X,X)∈Λ}. PROOF: Applying Theorem 17 to(A,I), we obtain
θk(A,I) =max{k|X|+ (k−1)|Z| |(X,Y)∈Λ,X∩Z=/0,Y∩Z=/0}.
Taking(X,Y) = (/0,/0)and Z=S in the right hand side, we observeθk≥(k−1)|S|. If|X|+|Z|<|S|, we have k|X|+ (k− 1)|Z| ≤(k−1)|S| −(k−1) +|X| ≤(k−1)|S|, where the last inequality follows from|X|<|S| ≤k. This implies that the maximum of the right hand side must be attained by some X⊆S and Z=S\X . Thus we obtain
θk(A,I) =max{k|X|+ (k−1)|S\X| |(X,X)∈Λ}, which is obviously equivalent to the desired formula. 2
For a square matrix A, consider a linking system A=L(A). It should be noted that Ak can be different from L(Ak). A theorem of Poljak [5], however, shows that rank Ak=r(Ak)holds if A is a generic matrix, i.e., if the nonzero entries of A are independent parameters. An alternative proof for this theorem is also described in [1].
References
[1] S. IWATA ANDR. SHIMIZU, Combinatorial analysis of generic matrix pencils, Proc. IPCO XI, to appear.
[2] J. P. S. KUNG, Bimatroids and invariants, Adv. Math., 30 (1978), pp. 238–249.
[3] K. MUROTA, Eigensets and power products of bimatroids, Adv. Math., 80 (1990), pp. 78–91.
[4] K. MUROTA, Matrices and Matroids for Systems Analysis, Springer-Verlag, 2000.
[5] S. POLJAK, Maximum rank of powers of a matrix of a given pattern, Proc. Amer. Math. Soc., 106 (1989), pp.1137–1144.
[6] A. SCHRIJVER, Matroids and linking systems, J. Combin. Theory, B26 (1979), pp. 349–369.
On Resource Constrained Optimization Problems
ALPAR´ J ¨UTTNER∗
Department of Operations Research and Egerv´ary Research Group, E¨otv¨os University, P´azm´any P´eter
S´et´any 1/C, Budapest, Hungary, H-1117 alpar@cs.elte.hu
Abstract: This paper shows that a method that has long been used to solve Resource Constrained Optimization Problems and found extremely effective in practice, is effective in the theoretical sense as well, it is proved to be strongly polynomial. In the special case of Resource Constrained Shortest Path Problem a better running time estimation is also presented.
Keywords: Resource constrained Optimization, Lagrangian relaxation, strongly polynomial algorithms
1 Introduction
In order to define the resource constrained optimization problem in general, first let us consider an underlying set E, a cost function c : E−→R+and an abstract optimization problem
min{
∑
e∈P
c(e): P∈ }, (1)
where ⊆2Edenotes the set of the feasible solutions. We refer this problem basic problem in this paper.
The corresponding constrained optimization problem is the following. Let d : E−→R+be another given weighting called delay, and∆∈R+a given constant called delay constraint. With these notations we are looking for the value
min{
∑
e∈P
c(e): P∈ ,
∑
e∈P
d(e)≤∆}. (2)
An important example for this is the Constrained Shortest Path Problem. Assume that a network is given as a directed, connected graph G= (V,E), where V represents the set of nodes, and E represents the set of directed links. Each link e∈E is characterized by two nonnegative values, a cost c(e)and a delay d(e). With a given delay constraint∆∈R+and two given nodes s,t∈V the task is to find a least cost path P between s and t with the side constraint that the delay of the path is less then∆.
One can define the Constrained Minimum Cost Perfect Matching Problem and the Constrained Minimum Cost Spanning Tree Problem in the same way.
Although their unconstrained versions are easy to solve, the three problems mentioned above are -hard (see e.g.
[3]). A usual way to find near optimal solutions to these problems is to get rid of the additional constrain using Lagrangian relaxation. In this way the constrained problem turns into a maximization of a one dimensional concave function (see Section 2).
A simple way to find the optimum of the relaxed problem is to use binary search, which is polynomial for integer costs and delays [28]
Instead of using binary search another a simple and practically even more effective method — described in Section 3 — has been found and applied independently by several authors. After [29] it is sometimes called Handler-Zang method. The same method was used in [11] making some people call it BM method. Other papers aim at further improving either on the running time in practical cases [25] or on the quality of the found solutions [26].
Although this method turned to be particularly efficient in practical applications, the worst case running time of this method was unknown for a long time.
Mehlhorn and Ziegelmann showed that for the Constrained Shortest Path problem the Handler-Zang algorithm is poly-nomial for integer costs and delays. If c(e)∈[0,···,C] and d(e)∈[0,···,R]for each e∈E, then it will terminate after O log(|V|RC)
iterations. They also presented examples showing that this running time is tight for small costs and delays (i.e. if R≤ |V|and C≤ |V|).
∗Research is supported by OTKA T37547
ALPAR´ J ¨UTTNER 137
One may observe that this problem can be transformed to an extension of the LCFO (Least Cost Fractional Optimization) problem discussed in [30]. Moreover, the strongly polynomial solution method proposed in [30] turns out to be equivalent with the Handler-Zang algorithm in this case, showing that the number of the iterations made by the Handler-Zang algorithm does not depend on the range of the cost and the delay functions, so this method is actually strongly polynomial if the basic problem can be solved in strongly polynomial time. This also shows that Mehlhorn’s and Ziegelmann’s bound on the running time is not tight for large costs or delays.
By a more careful application of the technique proposed in [19], in Section 4.1 we prove that the Handler-Zang algorithm takes O |E|2log(|E|)
iterations for arbitrary constrained optimization problem. This improves the bound can be obtained from [30] by a factor O(log(|E|)).
For the Constrained Shortest Path problem an even better bound is shown in Section 4.2, the number of iterations is proved to be O |E|log2(|E|)
in this case.