Short-Length Menger Theorems

We give short and simple proofs of the following two theorems by Galil and Yu [3]. Let s and t be two vertices in an n-node graph G. (1) There exist k edge-disjoint s-t paths of total length O(n √ k). (2) If we additionally assume that the minimum degree of G is at least k, then there exist k edge-disjoint s-t paths, each of length O(n/k). Let G = (V , E) be an undirected n-node graph, with no parallel edges, and let s and t be two vertices of G such that there exist k edge-disjoint s-t paths. Our goal is to give short proofs of the following two theorems of Galil and Yu [3]. Theorem 1 There exist k edge-disjoint s-t paths of total length O(n √ k). Theorem 2 If we additionally assume that the minimum degree of G is at least k, then there exist k edge-disjoint s-t paths, each of length O(n/k). We view G as a directed graph by replacing each undirected edge by two oppositely oriented directed edges. Our proof of the first theorem is based on a maximum flow algorithm of Even and Tarjan [2]; for our purposes, we need only consider its global structure. The algorithm of [2] runs in phases numbered 1, 2, . . .. In phase d, a residual graph is maintained as a layered directed graph: the endpoints of each edge lie either in the same layer or in adjacent layers, and the distance from s to t is equal to d. The algorithm finds augmenting s-t paths of length d in this layered graph until there exist no more such paths of length at most d; the phase then ends. Proof of Theorem 1. We analyze the behavior of the Even-Tarjan algorithm for producing a flow of value k in G. We prove an upper bound on the total length of all augmenting paths found; this also upper bounds the total length of the flow paths. We set l = 2nk−1/2. We say that an augmenting path is of type 0 if its length is at most l, and of type i (i ≥ 1) if its length is between 2i−1 · l and 2i · l. The total length of all type 0 paths is at most kl = 2n √ k. To bound the total length of all type i paths, for i ≥ 1, note that at the start of phase 2i−1 · l of the Even-Tarjan algorithm, there is some pair of adjacent layers in the residual graph whose union contains at most n/(2i−2 · l) vertices. Between this pair of layers there can be at most n2/(22i−2 · l2) edges, and hence at most this many augmenting paths can be produced from phase 2i−1 · l onward. Thus the total length of all type i paths is at most (n22−2i+2l−2) · (2il) = 4n2l−12−i = 2n √ k2−i , and so the total length of all augmenting paths is at most 2n √ k + 2n √ k ∑ i≥1 2 −i = 4n √ k. For the proof of the second theorem, we consider the problem of finding a set of k edge-disjoint s-t paths in G whose total length is minimum. Let P1, . . . , Pk