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...

This paper deals with probabilistic and nondeterministic processes represented by a variant of labelled transition systems where any outgoing transition of a state s is augmented with probabilities for the possible successor states. Our main contribution are algorithms for computing the bisimulation equivalence classes as introduced by Larsen & Skou [44] and the simulation preorder `a la Segala & Lynch [57]. The algorithm for deciding bisimilarity is based on a variant of the traditional partitioning technique [43, 51] and runs in time O(mn(log m+ log n)) where m is the number of transitions and n the number of states. The main idea for computing the simulation preorder is the reduction to maximum flow problems in suitable networks. Using the method of Cheriyan, Hagerup & Mehlhorn [15] for computing the maximum flow, the algorithm runs in time O((mn 6 +m 2 n 3 )= log n). Moreover, we show that the network-based technique is also applicable to compute the simulation-like relation...

. Various models and equivalence relations or preorders for probabilistic processes are proposed in the literature. This paper deals with a model based on labelled transition systems extended to the probabalistic setting and gives an O(n 2 \Delta m) algorithm for testing probabilistic bisimulation and an O(n 5 \Delta m 2 ) algorithm for testing probabilistic simulation where n is the number of states and m the number of transitions in the underlying probabilistic transition systems. 1 Introduction Transition systems have proved to be very useful for modelling concurrent processes. A variety of widely accepted equivalence relations and preorders for such systems support the use of transition systems for the design and verification of concurrent systems. In this context, testing equivalences and preorders become important and have been studied e.g. in [3, 4, 8, 11, 17]. For instance, (strong) bisimulation can be decided in time O(m \Delta log n) [22], weak bisimulation in t...

The paper has two parts. In the algorithmic part integer inequality systems of packing types and their application to integral multicommodity flow problems are considered. We give 1 \Gamma ffl approximation algorithms using the randomized rounding/derandomiztion scheme provided that the components of the right hand side vector resp. the capacities are in \Omega\Gamma ffl \Gamma2 log m) where m is the number of constraints resp. the number of edges. In the complexity-theoretic part it is shown that the approximable instances above build hard problems. Extending a result of Garg, Vazirani and Yannakakis (1993), the Mazsnp hardness of the maximum integral multicommodity flow problem for trees with large capacities (in particular c 2\Omega\Gamma390 m)) is proved. Furthermore, for every fixed non-negative integer K the problem with specified demand function r \Gamma K is NP -hard even if c is any function polynomially bounded in n and if the problem with demand function r is fractionally...