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

Popular matchings in the marriage and roommates problems

Popular matchings have recently been a subject of study in the context of the so-called House Allocation Problem, where the objective is to match applicants to houses over which the applicants have preferences. A matching M is called popular if there is no other matching M′ with the property that more applicants prefer their allocation in M′ to their allocation in M. In this paper we study popular matchings in the context of the Roommates Problem, including its special (bipartite) case, the Marriage Problem. We investigate the relationship between popularity and stability, and describe efficient algorithms to test a matching for popularity in these settings. We also show that, when ties are permitted in the preferences, it is NP-hard to determine whether a popular matching exists in both the Roommates and Marriage cases

Polycystic kidney disease: Clues to pathogenesis

Autosomal-dominant polycystic kidney disease (ADPKD), largely neglected for several decades, has emerged in recent years as the renal disease most likely to be understood from the gene to the patient. Major breakthroughs have occurred in the genetics of the disorder and new experimental data is providing insights in the pathobiology of cyst formation (Table 1)

Covering Pairs in Directed Acyclic Graphs

The Minimum Path Cover problem on directed acyclic graphs (DAGs) is a classical problem that provides a clear and simple mathematical formulation for several applications in different areas and that has an efficient algorithmic solution. In this paper, we study the computational complexity of two constrained variants of Minimum Path Cover motivated by the recent introduction of next-generation sequencing technologies in bioinformatics. The first problem (MinPCRP), given a DAG and a set of pairs of vertices, asks for a minimum cardinality set of paths "covering" all the vertices such that both vertices of each pair belong to the same path. For this problem, we show that, while it is NP-hard to compute if there exists a solution consisting of at most three paths, it is possible to decide in polynomial time whether a solution consisting of at most two paths exists. The second problem (MaxRPSP), given a DAG and a set of pairs of vertices, asks for a path containing the maximum number of the given pairs of vertices. We show its NP-hardness and also its W[1]-hardness when parametrized by the number of covered pairs. On the positive side, we give a fixed-parameter algorithm when the parameter is the maximum overlapping degree, a natural parameter in the bioinformatics applications of the problem

An O(EV log V) Algorithm for Finding a Maximal Weighted Matching in General Graphs

We define two generalized types of a priority queue by allowing some forms of changing the priorities of the elements in the queue. We show that they can be implemented efficiently. Consequently, each operation takes O (log n) time. We use these generalized priority queues to construct an O (EV log V) algorithm for finding a maximal weighted matching in general graphs

Palgol: A High-Level DSL for Vertex-Centric Graph Processing with Remote Data Access

Pregel is a popular distributed computing model for dealing with large-scale graphs. However, it can be tricky to implement graph algorithms correctly and efficiently in Pregel's vertex-centric model, especially when the algorithm has multiple computation stages, complicated data dependencies, or even communication over dynamic internal data structures. Some domain-specific languages (DSLs) have been proposed to provide more intuitive ways to implement graph algorithms, but due to the lack of support for remote access --- reading or writing attributes of other vertices through references --- they cannot handle the above mentioned dynamic communication, causing a class of Pregel algorithms with fast convergence impossible to implement. To address this problem, we design and implement Palgol, a more declarative and powerful DSL which supports remote access. In particular, programmers can use a more declarative syntax called chain access to naturally specify dynamic communication as if directly reading data on arbitrary remote vertices. By analyzing the logic patterns of chain access, we provide a novel algorithm for compiling Palgol programs to efficient Pregel code. We demonstrate the power of Palgol by using it to implement several practical Pregel algorithms, and the evaluation result shows that the efficiency of Palgol is comparable with that of hand-written code.Comment: 12 pages, 10 figures, extended version of APLAS 2017 pape

Finding 2-Edge and 2-Vertex Strongly Connected Components in Quadratic Time

We present faster algorithms for computing the 2-edge and 2-vertex strongly connected components of a directed graph, which are straightforward generalizations of strongly connected components. While in undirected graphs the 2-edge and 2-vertex connected components can be found in linear time, in directed graphs only rather simple $O(m n)$-time algorithms were known. We use a hierarchical sparsification technique to obtain algorithms that run in time $O(n^2)$. For 2-edge strongly connected components our algorithm gives the first running time improvement in 20 years. Additionally we present an $O(m^2 / \log{n})$-time algorithm for 2-edge strongly connected components, and thus improve over the $O(m n)$ running time also when $m = O(n)$. Our approach extends to k-edge and k-vertex strongly connected components for any constant k with a running time of $O(n^2 \log^2 n)$ for edges and $O(n^3)$ for vertices