Max-Balancing Weighted Directed Graphs and Matrix Scaling

Hans Schneider, Michael H. Schneider
1991 Mathematics of Operations Research  
A weighted directed graph G IS a triple (V, A . g) where (V. A) IS a directed graph and g is a n arbitrary real-valued function defined on the arc set A. Let G be a strongly-connected, simple weighted directed graph. We say th a t G is max-balanced if fo r every nontrivial ~ubset of th e vertices W, the maxImum weight over arcs leavin g W equals th e maximum weIght over arcs e ntering W. We show that there ex ists a (up to an addItIve con~tant) un iq ue potential p, for ( E V such that (V, A,
more » ... such that (V, A, g") IS max-b alanced where g/: = P" + g o -PI for a = (U , I ) EA. We describe an O(1V 1 2 IAI) algorithm for computlJ1g P using an a lgorithm for computing the tnaxmwm cycle-mean of C. Fmally. we apply our principal res ult to th e similarity scaling of nonnegatIve matrices.
doi:10.1287/moor.16.1.208 fatcat:xg4pmchy4vez5nn2vlwspzbjxu