ACM Transactions on Algorithms最新文献

Coalescing random walks is a fundamental distributed process, where a set of particles perform independent discrete-time random walks on an undirected graph. Whenever two or more particles meet at a given node, they merge and continue as a single random walk. The coalescence time is defined as the expected time until only one particle remains, starting from one particle at every node. Despite recent progress such as by Cooper et al. [14] and Cooper et al. [19], the coalescence time for graphs such as binary trees, d-dimensional tori, hypercubes and more generally, vertex-transitive graphs, remains unresolved.

We provide a powerful toolkit that results in tight bounds for various topologies including the aforementioned ones. The meeting time is defined as the worst-case expected time required for two random walks to arrive at the same node at the same time. As a general result, we establish that for graphs whose meeting time is only marginally larger than the mixing time (a factor of log ²n), the coalescence time of n random walks equals the meeting time up to constant factors. This upper bound is complemented by the construction of a graph family demonstrating that this result is the best possible up to constant factors. Finally, we prove a tight worst case bound for the coalescence time of O(n³). By duality, our results yield identical bounds on the voter model.

Our techniques also yield a new bound on the hitting time and cover time of regular graphs, improving and tightening previous results by Broder and Karlin [12], as well as those by Aldous and Fill [2].

We prove the first polynomial bound on the number of monotonic homotopy moves required to tighten a collection of closed curves on any compact orientable surface, where the number of crossings in the curve is not allowed to increase at any time during the process. The best known upper bound before was exponential, which can be obtained by combining the algorithm of De Graaf and Schrijver [J. Comb. Theory Ser. B, 1997] together with an exponential upper bound on the number of possible surface maps. To obtain the new upper bound, we apply tools from hyperbolic geometry, as well as operations in graph drawing algorithms—the cluster and pipe expansions—to the study of curves on surfaces.

As corollaries, we present two efficient algorithms for curves and graphs on surfaces. First, we provide a polynomial-time algorithm to convert any given multicurve on a surface into minimal position. Such an algorithm only existed for single closed curves, and it is known that previous techniques do not generalize to the multicurve case. Second, we provide a polynomial-time algorithm to reduce any k-terminal plane graph (and more generally, surface graph) using degree-1 reductions, series-parallel reductions, and Δ Y-transformations for arbitrary integer k. Previous algorithms only existed in the planar setting when k ≤ 4, and all of them rely on extensive case-by-case analysis based on different values of k. Our algorithm makes use of the connection between electrical transformations and homotopy moves and thus solves the problem in a unified fashion.

This article considers the question of how to succinctly approximate a multidimensional convex body by a polytope. Given a convex body K of unit diameter in Euclidean d-dimensional space (where d is a constant) and an error parameter ε > 0, the objective is to determine a convex polytope of low combinatorial complexity whose Hausdorff distance from K is at most ε. By combinatorial complexity, we mean the total number of faces of all dimensions. Classical constructions by Dudley and Bronshteyn/Ivanov show that O(1/ε^(d-1)/2) facets or vertices are possible, respectively, but neither achieves both bounds simultaneously. In this article, we show that it is possible to construct a polytope with O(1/ε^(d-1)/2) combinatorial complexity, which is optimal in the worst case.

Our result is based on a new relationship between ε-width caps of a convex body and its polar body. Using this relationship, we are able to obtain a volume-sensitive bound on the number of approximating caps that are “essentially different.” We achieve our main result by combining this with a variant of the witness-collector method and a novel variable-thickness layered construction of the economical cap covering.

No abstract available.

On hypergraphs with m hyperedges and n vertices, where p denotes the total size of the hyperedges, we provide the following results:

We consider the problem of finding a cycle in a sparse directed graph G that is promised to be far from acyclic, meaning that the smallest feedback arc set, i.e., a subset of edges whose deletion results in an acyclic graph, in G is large. We prove an information-theoretic lower bound, showing that for N-vertex graphs with constant outdegree, any algorithm for this problem must make Ω̄(N^5/9) queries to an adjacency list representation of G. In the language of property testing, our result is an Ω̄(N^5/9) lower bound on the query complexity of one-sided algorithms for testing whether sparse digraphs with constant outdegree are far from acyclic. This is the first improvement on the Ω (√ N) lower bound, implicit in the work of Bender and Ron, which follows from a simple birthday paradox argument.

In the Feedback Vertex Set (FVS) problem, one is given an undirected graph G and an integer k, and one needs to determine whether there exists a set of k vertices that intersects all cycles of G (a so-called feedback vertex set). Feedback Vertex Set is one of the most central problems in parameterized complexity: It served as an excellent testbed for many important algorithmic techniques in the field such as Iterative Compression [Guo et al. (JCSS’06)], Randomized Branching [Becker et al. (J. Artif. Intell. Res’00)] and Cut&Count [Cygan et al. (FOCS’11)]. In particular, there has been a long race for the smallest dependence f(k) in run times of the type O^⋆ (f(k)), where the O^⋆ notation omits factors polynomial in n. This race seemed to have reached a conclusion in 2011, when a randomized O^⋆ (3^k) time algorithm based on Cut&Count was introduced.

In this work, we show the contrary and give a O^⋆ (2.7k) time randomized algorithm. Our algorithm combines all mentioned techniques with substantial new ideas: First, we show that, given a feedback vertex set of size k of bounded average degree, a tree decomposition of width (1-Ω (1))k can be found in polynomial time. Second, we give a randomized branching strategy inspired by the one from [Becker et al. (J. Artif. Intell. Res’00)] to reduce to the aforementioned bounded average degree setting. Third, we obtain significant run time improvements by employing fast matrix multiplication.