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Combinatorial approaches to multiflow problems. (English) Zbl 0598.90036
Let V be a finite set and denote by (V) and [V] the sets of all ordered and unordered pairs (x,y) and [x,y] (x,y$$\in V)$$. A multiflow is a family $$F=\{f_ u| u\in U\}$$ where $$U\subseteq [V]$$ and for $$u=[s,t]$$ $$f_ u$$ is a flow with terminals s and t (i.e. either a flow from s to t or from t to s) in the complete directed graph $$G=(V,(V)).$$
Define $$\zeta_ F$$, $$\delta_ F\in {\mathbb{R}}_+^{[V]}$$ by $\zeta_ F[x,y]=\sum \{f(x,y)+f(y,x)| f\in F\}\quad and$ $\delta_ F[x,y]=\| f_{[x,y]}\|,\quad if\quad [x,y]\in U,\quad and\quad \delta_ F[x,y]=0,\quad otherwise$ where $$\| f\|$$ denotes the value of flow f. The author discusses theoretical properties of the following two problems. (a) The feasibility problem: Given $$c,d\in {\mathbb{R}}_+^{[V]}$$, find a multiflow F satisfying (1) $$\zeta_ F\leq c$$ and $$\delta_ F\geq d$$. (b) A general extremal multiflow problem: Given $$a,b,c,d\in {\mathbb{R}}_+^{[V]}$$, find a multiflow F maximizing $$a\delta_ F-b\zeta_ F$$ subject to (1).
Deriving these results a ’combinatorial’ approach is used rather than a ’linear programming’ approach.
Reviewer: P.Bruckner

##### MSC:
 90B10 Deterministic network models in operations research
##### Keywords:
multicommodity network flow; A multiflow
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##### References:
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