17,383 research outputs found

    Two Results on Slime Mold Computations

    Full text link
    We present two results on slime mold computations. In wet-lab experiments (Nature'00) by Nakagaki et al. the slime mold Physarum polycephalum demonstrated its ability to solve shortest path problems. Biologists proposed a mathematical model, a system of differential equations, for the slime's adaption process (J. Theoretical Biology'07). It was shown that the process convergences to the shortest path (J. Theoretical Biology'12) for all graphs. We show that the dynamics actually converges for a much wider class of problems, namely undirected linear programs with a non-negative cost vector. Combinatorial optimization researchers took the dynamics describing slime behavior as an inspiration for an optimization method and showed that its discretization can ε\varepsilon-approximately solve linear programs with positive cost vector (ITCS'16). Their analysis requires a feasible starting point, a step size depending linearly on ε\varepsilon, and a number of steps with quartic dependence on opt/(εΦ)\mathrm{opt}/(\varepsilon\Phi), where Φ\Phi is the difference between the smallest cost of a non-optimal basic feasible solution and the optimal cost (opt\mathrm{opt}). We give a refined analysis showing that the dynamics initialized with any strongly dominating point converges to the set of optimal solutions. Moreover, we strengthen the convergence rate bounds and prove that the step size is independent of ε\varepsilon, and the number of steps depends logarithmically on 1/ε1/\varepsilon and quadratically on opt/Φ\mathrm{opt}/\Phi

    Deterministic and Probabilistic Binary Search in Graphs

    Full text link
    We consider the following natural generalization of Binary Search: in a given undirected, positively weighted graph, one vertex is a target. The algorithm's task is to identify the target by adaptively querying vertices. In response to querying a node qq, the algorithm learns either that qq is the target, or is given an edge out of qq that lies on a shortest path from qq to the target. We study this problem in a general noisy model in which each query independently receives a correct answer with probability p>12p > \frac{1}{2} (a known constant), and an (adversarial) incorrect one with probability 1p1-p. Our main positive result is that when p=1p = 1 (i.e., all answers are correct), log2n\log_2 n queries are always sufficient. For general pp, we give an (almost information-theoretically optimal) algorithm that uses, in expectation, no more than (1δ)log2n1H(p)+o(logn)+O(log2(1/δ))(1 - \delta)\frac{\log_2 n}{1 - H(p)} + o(\log n) + O(\log^2 (1/\delta)) queries, and identifies the target correctly with probability at leas 1δ1-\delta. Here, H(p)=(plogp+(1p)log(1p))H(p) = -(p \log p + (1-p) \log(1-p)) denotes the entropy. The first bound is achieved by the algorithm that iteratively queries a 1-median of the nodes not ruled out yet; the second bound by careful repeated invocations of a multiplicative weights algorithm. Even for p=1p = 1, we show several hardness results for the problem of determining whether a target can be found using KK queries. Our upper bound of log2n\log_2 n implies a quasipolynomial-time algorithm for undirected connected graphs; we show that this is best-possible under the Strong Exponential Time Hypothesis (SETH). Furthermore, for directed graphs, or for undirected graphs with non-uniform node querying costs, the problem is PSPACE-complete. For a semi-adaptive version, in which one may query rr nodes each in kk rounds, we show membership in Σ2k1\Sigma_{2k-1} in the polynomial hierarchy, and hardness for Σ2k5\Sigma_{2k-5}

    Finding kk Simple Shortest Paths and Cycles

    Get PDF
    The problem of finding multiple simple shortest paths in a weighted directed graph G=(V,E)G=(V,E) has many applications, and is considerably more difficult than the corresponding problem when cycles are allowed in the paths. Even for a single source-sink pair, it is known that two simple shortest paths cannot be found in time polynomially smaller than n3n^3 (where n=Vn=|V|) unless the All-Pairs Shortest Paths problem can be solved in a similar time bound. The latter is a well-known open problem in algorithm design. We consider the all-pairs version of the problem, and we give a new algorithm to find kk simple shortest paths for all pairs of vertices. For k=2k=2, our algorithm runs in O(mn+n2logn)O(mn + n^2 \log n) time (where m=Em=|E|), which is almost the same bound as for the single pair case, and for k=3k=3 we improve earlier bounds. Our approach is based on forming suitable path extensions to find simple shortest paths; this method is different from the `detour finding' technique used in most of the prior work on simple shortest paths, replacement paths, and distance sensitivity oracles. Enumerating simple cycles is a well-studied classical problem. We present new algorithms for generating simple cycles and simple paths in GG in non-decreasing order of their weights; the algorithm for generating simple paths is much faster, and uses another variant of path extensions. We also give hardness results for sparse graphs, relative to the complexity of computing a minimum weight cycle in a graph, for several variants of problems related to finding kk simple paths and cycles.Comment: The current version includes new results for undirected graphs. In Section 4, the notion of an (m,n) reduction is generalized to an f(m,n) reductio

    Evolution of networks

    Full text link
    We review the recent fast progress in statistical physics of evolving networks. Interest has focused mainly on the structural properties of random complex networks in communications, biology, social sciences and economics. A number of giant artificial networks of such a kind came into existence recently. This opens a wide field for the study of their topology, evolution, and complex processes occurring in them. Such networks possess a rich set of scaling properties. A number of them are scale-free and show striking resilience against random breakdowns. In spite of large sizes of these networks, the distances between most their vertices are short -- a feature known as the ``small-world'' effect. We discuss how growing networks self-organize into scale-free structures and the role of the mechanism of preferential linking. We consider the topological and structural properties of evolving networks, and percolation in these networks. We present a number of models demonstrating the main features of evolving networks and discuss current approaches for their simulation and analytical study. Applications of the general results to particular networks in Nature are discussed. We demonstrate the generic connections of the network growth processes with the general problems of non-equilibrium physics, econophysics, evolutionary biology, etc.Comment: 67 pages, updated, revised, and extended version of review, submitted to Adv. Phy
    corecore