A new algorithm for finding a maximum matching in a general graph is presented; its special feature being that the only computationally non-trivial step required in its execution is the inversion of a single integer matrix. Since this step can be parallelized, we get a simple parallel (RNC2) algorithm. At the heart of our algorithm lies a probabilistic lemma, the isolating lemma. We show applications of this lemma to parallel computation and randomized reductions.

It has been known for a long time now that the problem of counting the number of perfect matchings in a planar graph is in NC. This result is based on the notion of a pfaffian orientation of a graph. (Recently, Galluccio and Loebl [7] gave a P-time algorithm for the case of graphs of small genus.) However, it is not known if the corresponding search problem, that of finding one perfect matching in a planar graph, is in NC. This situation is intriguing as it seems to contradict our intuition that search should be easier than counting. For the case of planar bipartite graphs, Miller and Naor [22] showed that a perfect matching can indeed be found using an NC algorithm. We present a very different NC-algorithm for this problem. Unlike the Miller...

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Abstract. In this note, we give tighter bounds on the complexity of the bipartite unique perfect matching problem, bipartite-UPM. We show that the problem is in C=L and in NL ⊕L, both subclasses of NC 2. We also consider the (unary) weighted version of the problem. We show that testing uniqueness of the minimum-weight perfect matching problem for bipartite graphs is in L C=L and in NL ⊕L. Furthermore, we show that bipartite-UPM is hard for NL. 1

Abstract. We present a deterministic way of assigning small (log bit) weights to the edges of a bipartite planar graph so that the minimum weight perfect matching becomes unique. The isolation lemma as described in [MVV87] achieves the same for general graphs using a randomized weighting scheme, whereas we can do it deterministically when restricted to bipartite planar graphs. As a consequence, we reduce both decision and construction versions of the matching problem to testing whether a matrix is singular, under the promise that its determinant is 0 or 1, thus obtaining a highly parallel SPL algorithm for bipartite planar graphs. This improves the earlier known bounds of non-uniform SPL by [ARZ99] and NC 2 by [MN95, MV00]. It also rekindles the hope of obtaining a deterministic parallel algorithm for constructing a perfect matching in non-bipartite planar graphs, which has been open for a long time. Our techniques are elementary and simple. 1.

Our algorithm is based on a subroutine for computing so-called scheduling distances, i.e., the minimal number of time steps needed to schedule all those tasks succeeding some given task t and preceding some other task t0. For a given interval order with n tasks, these scheduling distances can be computed using n3 processors and O(log 2 n) time on a CREW-PRAM. We then give an incremental version of the scheduling distance algorithm, which can be used to compute the empty slots in an optimal schedule. From these, we derive the optimal schedule, using no more resources than for the initial scheduling distance computation and considerably improving on previous work by Sunder and He. The algorithm can also be extended to handle task systems which, in addition to interval order precedence constraints, have individual deadlines and/or release times for the tasks. Our algorithm is the first N C-algorithm for this problem. As another application, it also provides N C-algorithms for some graph problems on interval graphs (which are N P-complete in general).

Abstract. We present a deterministic way of assigning small (log bit) weights to the edges of a bipartite planar graph so that the minimum weight perfect matching becomes unique. The isolation lemma as described in [MVV87] achieves the same for general graphs using a randomized weighting scheme, whereas we can do it deterministically when restricted to bipartite planar graphs. As a consequence, we reduce both decision and construction versions of the matching problem to testing whether a matrix is singular, under the promise that its determinant is 0 or 1, thus obtaining a highly parallel SPL algorithm for bipartite planar graphs. This improves the earlier known bounds of non-uniform SPL by [ARZ99] and NC 2 by [MN95, MV00]. It also rekindles the hope of obtaining a deterministic parallel algorithm for constructing a perfect matching in non-bipartite planar graphs, which has been open for a long time. Our techniques are elementary and simple. 1.