The sequential sum problem and performance bounds on the greedy algorithm for the on-line Steiner problem

This article is motivated by versions of the dynamic or "on-line" Steiner tree problem (OST) introduced by Imase and Waxman [4]. In this problem one is given an edge-weighted graph G and a sequence sigma = (x(1),...,x(n)) of distinct vertices of G. The requirement is to construct for each i < n a tree T-i spanning the first i vertices of sigma subject to the condition that Ti-1 subset of T-i for all i, where T-i is constructed without knowledge of the remaining vertices x(j), j < i. The goal of the on-line Steiner problem is to minimize the performance ratio; that is, the maximum (over 1 < i < n) of the ratio of the weight of Ti to the weight of the minimum weight tree in G spanning the first i vertices (the latter tree is called the "Steiner tree" for these vertices). In [4] a lower bound of 1 + (1)/(2) [log(2)(n - 1)] was proved for this ratio. The authors further made the interesting conjecture that there is some on-line algorithm for the OST whose performance ratio achieves this lower bound. We show that a strong form of the greedy algorithm achieves a ratio that converges to the conjectured (1)/(2) log(2)(k) + O(1) as the proportion of degree 2 vertices in the instance graph grows. Our results also imply improvements in certain cases on the known upper bound [log(2)(n)] for the performance ratio of the greedy algorithm. Our approach is to study a related graph parameter. For each sequence a as above, define the associated cost L(sigma) = (n)Sigma(i-2)c(i, sigma), where c(i,sigma) = min(1 <= t < i) dist(x(i), x(t)). Then let Opt(n,G) be the maximum of L(sigma) over all such sequences a of length n. The problem of, given n and G, determining Opt(n, G) we call the Sequential Sum Problem (SSP). In this article we analyze the SSP, obtaining exact values and bounds on Opt(n, G) and relating these bounds to the greedy algorithm for the OST. For example, we calculate Opt(n,P) for the path P, and obtain a surprising characterization of all length n sequences a which realize Opt(n, P). By analyzing Opt(n, P) for the "continuous" path, we derive upper bounds on the performance ratio of the greedy algorithm for the OST in arbitrary graphs. On the other hand, generalizing the lower bound argument of [4] we show that there are instances of OST, which can significantly "fool" any on-line algorithm for OST. Specifically, given any tree T normalized to have total edge weight 1, we construct a graph G and a length k <= vertical bar V(T)vertical bar sequence a of vertices of G for which the performance ratio of any on-line algorithm for the OST with input a is lower bounded by Opt(k, T). Finally, we show that the SSP for arbitrary G is NP-complete. (c) 2005 Wiley Periodicals, Inc.