1,729 research outputs found
Minimum Cuts in Near-Linear Time
We significantly improve known time bounds for solving the minimum cut
problem on undirected graphs. We use a ``semi-duality'' between minimum cuts
and maximum spanning tree packings combined with our previously developed
random sampling techniques. We give a randomized algorithm that finds a minimum
cut in an m-edge, n-vertex graph with high probability in O(m log^3 n) time. We
also give a simpler randomized algorithm that finds all minimum cuts with high
probability in O(n^2 log n) time. This variant has an optimal RNC
parallelization. Both variants improve on the previous best time bound of O(n^2
log^3 n). Other applications of the tree-packing approach are new, nearly tight
bounds on the number of near minimum cuts a graph may have and a new data
structure for representing them in a space-efficient manner
Exact arborescences, matchings and cycles
AbstractSuppose we are given a graph in which edge has an integral weight. An ‘exact’ problem is to determine whether a desired structure exists for which the sum of the edge weights is exactly k for some prescribed k.We consider the special case of the problem in which all costs are zero or one for arborescences and show that a ‘continuity’ property is prossessed similar to that possessed by matroids. This enables us to determine in polynomial time the complete set of values of k for which a solution exists. We also give a minmax theorem for the maximum possible value of k, in terms of a packing of certain directed cuts in the graph.We also show how enumerative techniques can be used to solve the general exact problem for arborescences (implying spanning trees), perfect matchings in planar graphs and sets of disjoint cycles in a class of planar directed graphs which includes those of degree three. For these problems, we thereby obtain polynomial algorithms provided that the weights are bounded by a constant or encoded in unary
Approximating the Held-Karp Bound for Metric TSP in Nearly Linear Time
We give a nearly linear time randomized approximation scheme for the
Held-Karp bound [Held and Karp, 1970] for metric TSP. Formally, given an
undirected edge-weighted graph on edges and , the
algorithm outputs in time, with high probability, a
-approximation to the Held-Karp bound on the metric TSP instance
induced by the shortest path metric on . The algorithm can also be used to
output a corresponding solution to the Subtour Elimination LP. We substantially
improve upon the running time achieved previously
by Garg and Khandekar. The LP solution can be used to obtain a fast randomized
-approximation for metric TSP which improves
upon the running time of previous implementations of Christofides' algorithm
New Exact and Approximation Algorithms for the Star Packing Problem in Undirected Graphs
By a T-star we mean a complete bipartite graph K_{1,t} for some t <= T. For an undirected graph G, a T-star packing is a collection of node-disjoint T-stars in G.
For example, we get ordinary matchings for and packings of paths of length 1 and 2 for . Hereinafter we assume that T >= 2.
Hell and Kirkpatrick devised an ad-hoc augmenting algorithm that finds a T-star packing covering the maximum number of nodes. The latter algorithm also yields a min-max formula.
We show that T-star packings are reducible to network flows, hence the above problem is solvable in time (hereinafter n denotes the number of nodes in G, and m --- the number of edges).
For the edge-weighted case (in which weights may be assumed positive) finding a maximum -packing is NP-hard. A novel 9/4 T/(T + 1)-factor approximation algorithm is presented.
For non-negative node weights the problem reduces to a special case of a max-cost flow. We develop a divide-and-conquer approach that solves it in O(m sqrt(n) log(n)) time. The node-weighted problem with arbitrary weights is more difficult. We prove that it is NP-hard for T >= 3 and is solvable in strongly-polynomial time for T = 2
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