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Matching is as Easy as Matrix Inversion
, 1987
"... A new algorithm for finding a maximum matching in a general graph is presented; its special feature being that the only computationally nontrivial 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) algorit ..."
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Cited by 166 (5 self)
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A new algorithm for finding a maximum matching in a general graph is presented; its special feature being that the only computationally nontrivial 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.
A new NCalgorithm for finding a perfect matching in bipartite planar and small genus graphs (Extended Abstract)
, 2000
"... 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 Ptime algorithm for the case of graphs of small genus.) H ..."
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Cited by 8 (2 self)
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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 Ptime 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 NCalgorithm for this problem. Unlike the Miller...
INTERVAL GRAPHS: CANONICAL REPRESENTATIONS IN LOGSPACE ∗
"... Abstract. We present a logspace algorithm for computing a canonical labeling, in fact, a canonical interval representation, for interval graphs. To achieve this, we compute canonical interval representations of interval hypergraphs. This approach also yields a canonical labeling of convex graphs. As ..."
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Cited by 4 (4 self)
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Abstract. We present a logspace algorithm for computing a canonical labeling, in fact, a canonical interval representation, for interval graphs. To achieve this, we compute canonical interval representations of interval hypergraphs. This approach also yields a canonical labeling of convex graphs. As a consequence, the isomorphism and automorphism problems for these graph classes are solvable in logspace. For proper interval graphs we also design logspace algorithms computing their canonical representations by proper and by unit interval systems.
Coloring Permutation Graphs in Parallel
, 2002
"... A coloring of a graph G is an assignment ofcol]B to its vertices so that no two adjacent vertices have the samecol.]B study theproblP ofcolxxPpermutation graphs using certain properties of thele.Pq representation of a permutation andrel#B[.bEPq between permutations, directed acycld graphs and roo ..."
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Cited by 4 (4 self)
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A coloring of a graph G is an assignment ofcol]B to its vertices so that no two adjacent vertices have the samecol.]B study theproblP ofcolxxPpermutation graphs using certain properties of thele.Pq representation of a permutation andrel#B[.bEPq between permutations, directed acycld graphs and rooted trees having speci#c key properties.We propose an e#cient paralnt allnt.P which colh. an nnode permutation graph inO(lO =l.
DETERMINISTICALLY ISOLATING A PERFECT MATCHING IN BIPARTITE PLANAR GRAPHS
"... 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, whe ..."
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Cited by 3 (3 self)
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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 nonuniform 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 nonbipartite planar graphs, which has been open for a long time. Our techniques are elementary and simple. 1.
On the bipartite unique perfect matching problem
 In Proc. 33rd International Colloquium on Automata, Languages and Programming (ICALP
, 2006
"... Abstract. In this note, we give tighter bounds on the complexity of the bipartite unique perfect matching problem, bipartiteUPM. 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 ..."
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Cited by 2 (2 self)
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Abstract. In this note, we give tighter bounds on the complexity of the bipartite unique perfect matching problem, bipartiteUPM. 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 minimumweight perfect matching problem for bipartite graphs is in L C=L and in NL ⊕L. Furthermore, we show that bipartiteUPM is hard for NL. 1
Some perfect matchings and perfect halfintegral matchings in NC ∗
, 2008
"... We show that for any class of bipartite graphs which is closed under edge deletion and where the number of perfect matchings can be counted in NC, there is a deterministic NC algorithm for finding a perfect matching. In particular, a perfect matching can be found in NC for planar bipartite graphs an ..."
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Cited by 2 (2 self)
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We show that for any class of bipartite graphs which is closed under edge deletion and where the number of perfect matchings can be counted in NC, there is a deterministic NC algorithm for finding a perfect matching. In particular, a perfect matching can be found in NC for planar bipartite graphs and K3,3free bipartite graphs via this approach. A crucial ingredient is part of an interiorpoint algorithm due to Goldberg, Plotkin, Shmoys and Tardos. An easy observation allows this approach to handle regular bipartite graphs as well. We show, by a careful analysis of the polynomial time algorithm due to Galluccio and Loebl, that the number of perfect matchings in a graph of small (O(log n)) genus can be counted in NC. So perfect matchings in small genus bipartite graphs can also be found via this approach. We then present a different algorithm for finding a perfect matching in a planar bipartite graph. This algorithm is substantially different from the algorithm described above, and also from the algorithm of Miller and Naor, which predates the approach of Goldberg et al. and tackles the same problem. Our new algorithm extends to small genus bipartite graphs, but not to K3,3free bipartite graphs. We next show that a nontrivial extension of this algorithm allows
Scheduling Interval Orders in Parallel
, 1994
"... Our algorithm is based on a subroutine for computing socalled 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 com ..."
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Our algorithm is based on a subroutine for computing socalled 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 CREWPRAM. 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 Calgorithm for this problem. As another application, it also provides N Calgorithms for some graph problems on interval graphs (which are N Pcomplete in general).
www.stacsconf.org DETERMINISTICALLY ISOLATING A PERFECT MATCHING IN BIPARTITE PLANAR GRAPHS
"... 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, whe ..."
Abstract
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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 nonuniform 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 nonbipartite planar graphs, which has been open for a long time. Our techniques are elementary and simple. 1.