Bipartite Perfect Matching in Pseudo-Deterministic NC

We present a pseudo-deterministic NC algorithm for finding perfect matchings in bipartite graphs. Specifically, our algorithm is a randomized parallel algorithm which uses poly(n) processors, poly(log n) depth, poly(log n) random bits, and outputs for each bipartite input graph a unique perfect matching with high probability. That is, on the same graph it returns the same matching for almost all choices of randomness. As an immediate consequence we also find a pseudo-deterministic NC algorithm for constructing a depth first search (DFS) tree. We introduce a method for computing the union of all min-weight perfect matchings of a weighted graph in RNC and a novel set of weight assignments which in combination enable isolating a unique matching in a graph. We then show a way to use pseudo-deterministic algorithms to reduce the number of random bits used by general randomized algorithms. The main idea is that random bits can be reused by successive invocations of pseudo-deterministic randomized algorithms. We use the technique to show an RNC algorithm for constructing a depth first search (DFS) tree using only O(log^2 n) bits whereas the previous best randomized algorithm used O(log^7 n), and a new sequential randomized algorithm for the set-maxima problem which uses fewer random bits than the previous state of the art. Furthermore, we prove that resolving the decision question NC = RNC, would imply an NC algorithm for finding a bipartite perfect matching and finding a DFS tree in NC. This is not implied by previous randomized NC search algorithms for finding bipartite perfect matching, but is implied by the existence of a pseudo-deterministic NC search algorithm

Deterministically Isolating a Perfect Matching in Bipartite Planar Graphs

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 (Mulmuley et al. 1987) 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 (Allender et al. 1999) and $NC^2$ by (Miller and Naor 1995, Mahajan and Varadarajan 2000). 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

Randomized parallel approximations to max flow

The final publication is available at link.springer.comPeer ReviewedPostprint (author's final draft

Analogies between the crossing number and the tangle crossing number

Tanglegrams are special graphs that consist of a pair of rooted binary trees with the same number of leaves, and a perfect matching between the two leaf-sets. These objects are of use in phylogenetics and are represented with straightline drawings where the leaves of the two plane binary trees are on two parallel lines and only the matching edges can cross. The tangle crossing number of a tanglegram is the minimum crossing number over all such drawings and is related to biologically relevant quantities, such as the number of times a parasite switched hosts. Our main results for tanglegrams which parallel known theorems for crossing numbers are as follows. The removal of a single matching edge in a tanglegram with $n$ leaves decreases the tangle crossing number by at most $n-3$, and this is sharp. Additionally, if $\gamma(n)$ is the maximum tangle crossing number of a tanglegram with $n$ leaves, we prove $\frac{1}{2}\binom{n}{2}(1-o(1))\le\gamma(n)<\frac{1}{2}\binom{n}{2}$. Further, we provide an algorithm for computing non-trivial lower bounds on the tangle crossing number in $O(n^4)$ time. This lower bound may be tight, even for tanglegrams with tangle crossing number $\Theta(n^2)$.Comment: 13 pages, 6 figure

Matching Is as Easy as the Decision Problem, in the NC Model

Is matching in NC, i.e., is there a deterministic fast parallel algorithm for it? This has been an outstanding open question in TCS for over three decades, ever since the discovery of randomized NC matching algorithms [KUW85, MVV87]. Over the last five years, the theoretical computer science community has launched a relentless attack on this question, leading to the discovery of several powerful ideas. We give what appears to be the culmination of this line of work: An NC algorithm for finding a minimum-weight perfect matching in a general graph with polynomially bounded edge weights, provided it is given an oracle for the decision problem. Consequently, for settling the main open problem, it suffices to obtain an NC algorithm for the decision problem. We believe this new fact has qualitatively changed the nature of this open problem. All known efficient matching algorithms for general graphs follow one of two approaches: given by Edmonds [Edm65] and Lov\'asz [Lov79]. Our oracle-based algorithm follows a new approach and uses many of the ideas discovered in the last five years. The difficulty of obtaining an NC perfect matching algorithm led researchers to study matching vis-a-vis clever relaxations of the class NC. In this vein, recently Goldwasser and Grossman [GG15] gave a pseudo-deterministic RNC algorithm for finding a perfect matching in a bipartite graph, i.e., an RNC algorithm with the additional requirement that on the same graph, it should return the same (i.e., unique) perfect matching for almost all choices of random bits. A corollary of our reduction is an analogous algorithm for general graphs.Comment: Appeared in ITCS 202

Algorithmic Applications of Baur-Strassen's Theorem: Shortest Cycles, Diameter and Matchings

Consider a directed or an undirected graph with integral edge weights from the set [-W, W], that does not contain negative weight cycles. In this paper, we introduce a general framework for solving problems on such graphs using matrix multiplication. The framework is based on the usage of Baur-Strassen's theorem and of Strojohann's determinant algorithm. It allows us to give new and simple solutions to the following problems: * Finding Shortest Cycles -- We give a simple \tilde{O}(Wn^{\omega}) time algorithm for finding shortest cycles in undirected and directed graphs. For directed graphs (and undirected graphs with non-negative weights) this matches the time bounds obtained in 2011 by Roditty and Vassilevska-Williams. On the other hand, no algorithm working in \tilde{O}(Wn^{\omega}) time was previously known for undirected graphs with negative weights. Furthermore our algorithm for a given directed or undirected graph detects whether it contains a negative weight cycle within the same running time. * Computing Diameter and Radius -- We give a simple \tilde{O}(Wn^{\omega}) time algorithm for computing a diameter and radius of an undirected or directed graphs. To the best of our knowledge no algorithm with this running time was known for undirected graphs with negative weights. * Finding Minimum Weight Perfect Matchings -- We present an \tilde{O}(Wn^{\omega}) time algorithm for finding minimum weight perfect matchings in undirected graphs. This resolves an open problem posted by Sankowski in 2006, who presented such an algorithm but only in the case of bipartite graphs. In order to solve minimum weight perfect matching problem we develop a novel combinatorial interpretation of the dual solution which sheds new light on this problem. Such a combinatorial interpretation was not know previously, and is of independent interest.Comment: To appear in FOCS 201