In this paper, we study network flow algorithms for bipartite networks. A network G = (V; E) is called bipartite if its vertex set V can be partitioned into two subsets V 1 and V 2 such that all edges have one endpoint in V 1 and the other in V 2 . Let n = jV j, n 1 = jV 1 j, n 2 = jV 2 j, m = jEj and assume without loss of generality that n 1 n 2 . We call a bipartite network unbalanced if n 1 ø n 2 and balanced otherwise. (This notion is necessarily imprecise.) We show that several maximum flow algorithms can be substantially sped up when applied to unbalanced networks. The basic idea in these improvements is a two-edge push rule that allows us to "charge" most computation to vertices in V 1 , and hence develop algorithms whose running times depend on n 1 rather than n. For example, we show that the two-edge push version of Goldberg and Tarjan's FIFO preflow push algorithm runs in O(n 1 m + n 3 1 ) time and that the analogous version of Ahuja and Orlin's excess scaling algori...

. We consider the traveling salesman problem when the cities are points in R d for some fixed d and distances are computed according to a polyhedral norm. We show that for any such norm, the problem of finding a tour of maximum length can be solved in polynomial time. If arithmetic operations are assumed to take unit time, our algorithms run in time O(n f \Gamma2 log n), where f is the number of facets of the polyhedron determining the polyhedral norm. Thus for example we have O(n 2 log n) algorithms for the cases of points in the plane under the Rectilinear and Sup norms. This is in contrast to the fact that finding a minimum length tour in each case is NP-hard. 1 Introduction In the Traveling Salesman Problem (TSP), the input consists of a set C of cities together with the distances d(c; c 0 ) between every pair of distinct cities c; c 0 2 C. The goal is to find an ordering or tour of the cities that minimizes (Minimum TSP) or maximizes (Maximum TSP) the total tour leng...

A widely used method for determining the similarity of two labeled trees is to compute a maximum agreement subtree of the two trees. Previous work on this similarity measure is only concerned with the comparison of labeled trees of two special kinds, namely, uniformly labeled trees (i.e., trees with all their nodes labeled by the same symbol) and evolutionary trees (i.e., leaf-labeled trees with distinct symbols for distinct leaves). This paper presents an algorithm for comparing trees that are labeled in an arbitrary manner. In addition to this generality, this algorithm is faster than the previous algorithms. Another contribution of this paper is on maximum weight bipartite matchings. We show how to speed up the best known matching algorithms when the input graphs are node-unbalanced or weight-unbalanced. Based on these enhancements, we obtain an efficient algorithm for a new matching problem called the hierarchical bipartite matching problem, which is at the core of our maximum agreement subtree algorithm. 1

: In this paper, we propose an O(n) time algorithm for the Hitchcock transportation problem with n demand points and fixed number of supply points. When the number of supply points is very small and the number of demand points is much larger than that of supply points, our algorithm is efficient. If we have m supply points, the time complexity of our algorithm is bounded by O((m!) 2 n) . Keywords: transportation problem, median finding problem, linear time algorithm. AMS(MOS) subject classifications: 68Q25 68R10 68U05 1 Introduction This paper deals with the Hitchcock transportation problem. The problem is formulated as a linear programming problem in the following way; minimize X i2U X j2V w ij x ij subject to X j2V x ij = a i 8i 2 U; X i2U x ij = b j 8j 2 V; x ij 0 8(i; j) 2 U 2 V; where U = f1; 2; : : : ; ng and V = f1; 2; : : : ; mg: Each element in U is called demand point and each element in V is called supply point. The classical example of this problem is th...

This paper gives output sensitive parallel algorithms whose performance

A general framework for specifying communication network design problems is given. We analyze the computational complexity of several specific problems within this framework. For fixed multirate traffic requirements, we prove that a particular network analysis problem is np-complete, although several related network design problems are either efficiently solvable or have good approximation algorithms. For the case when we wish the network to operate without blocking any connection requests, we give efficient algorithms for dimensioning the link capacities of the network. This work is supported by the National Science Foundation, Bell Communications Research, Bell Northern Research, Digital Equipment Corporation, Italtel SIT, NEC, NTT, and SynOptics. 1. Introduction Much work has been done on the computational problem of designing low-cost communication networks (see [GN89, GTD + 89, GW90, GK90, KKG91, AKR91] and references therein). The general problem is: given a collection of no...

In this paper we present an experimental study of several maximum flow algorithms in the context of unbalanced bipartite networks. Our experiments are motivated by a real world problem of managing reservation-based inventory in Google content ad systems. We are interested in observing the performance of several push-relabel algorithms on our real world data sets and also on some generated ones. Previous work suggested an important improvement for pushrelabel algorithms on unbalanced bipartite networks: the two-edge push rule. Weshowhowthetwo-edge push rule improves the running time. While no single algorithm dominates the results, we show there is one that has very robust performance in practice. 1

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In this paper we present an experimental study of several maximum flow algorithms in the context of unbalanced bipartite networks. Our experiments are motivated by a real world problem of managing reservation-based inventory in Google content ad systems. We are interested in observing the performance of several push-relabel algorithms on our real world data sets and also on some generated ones. Previous work suggested an important improvement for pushrelabel algorithms on unbalanced bipartite networks: the two-edge push rule. Weshowhowthetwo-edge push rule improves the running time. While no single algorithm dominates the results, we show there is one that has very robust performance in practice. 1