Improved approximations for the minimum-cut ratio and the flux

Abstract

LetC be any concurrent flow problem with integer demands and capacities on an undirected graphG. Kleinet al. [5] recently showed that the maximum throughputz ^* is Ω (S/(logC logD)), whereS is the minimum cut ratio,C is the sum of the capacities on the edges, andD is the sum of the demands of the commodities. They also presented a polynomial time algorithm which finds a cut whose ratio isS · O(logC logD). Leighton and Rao [8] have shown that for the concurrent flow problem with a unit demand between every pair of nodes,z ^* is Ω (S/logn), wheren is the number of nodes ofG. This leads to anO(logn) approximation of the flux of a graphG with uniform weights on the nodes. We show that for the problem in [5] the maximum throughputz ^* is Ω(S/(logn logD)), and we find a cut whose ratio isS · O (logn logD). For the special case in which the demands have the form

$$demand between u and v = l(u) \cdot \frac{{l(v)}}{2},$$

wherel(u) is an assignment of a positive integer weight to nodeu, we show that the lower bound onz* can be improved to be Ω(S/logn). This generalizes the result of Leighton and Rao.