54,839 research outputs found
Practical Minimum Cut Algorithms
The minimum cut problem for an undirected edge-weighted graph asks us to
divide its set of nodes into two blocks while minimizing the weight sum of the
cut edges. Here, we introduce a linear-time algorithm to compute near-minimum
cuts. Our algorithm is based on cluster contraction using label propagation and
Padberg and Rinaldi's contraction heuristics [SIAM Review, 1991]. We give both
sequential and shared-memory parallel implementations of our algorithm.
Extensive experiments on both real-world and generated instances show that our
algorithm finds the optimal cut on nearly all instances significantly faster
than other state-of-the-art algorithms while our error rate is lower than that
of other heuristic algorithms. In addition, our parallel algorithm shows good
scalability
The Peculiar Phase Structure of Random Graph Bisection
The mincut graph bisection problem involves partitioning the n vertices of a
graph into disjoint subsets, each containing exactly n/2 vertices, while
minimizing the number of "cut" edges with an endpoint in each subset. When
considered over sparse random graphs, the phase structure of the graph
bisection problem displays certain familiar properties, but also some
surprises. It is known that when the mean degree is below the critical value of
2 log 2, the cutsize is zero with high probability. We study how the minimum
cutsize increases with mean degree above this critical threshold, finding a new
analytical upper bound that improves considerably upon previous bounds.
Combined with recent results on expander graphs, our bound suggests the unusual
scenario that random graph bisection is replica symmetric up to and beyond the
critical threshold, with a replica symmetry breaking transition possibly taking
place above the threshold. An intriguing algorithmic consequence is that
although the problem is NP-hard, we can find near-optimal cutsizes (whose ratio
to the optimal value approaches 1 asymptotically) in polynomial time for
typical instances near the phase transition.Comment: substantially revised section 2, changed figures 3, 4 and 6, made
minor stylistic changes and added reference
Acyclic Subgraphs of Planar Digraphs
An acyclic set in a digraph is a set of vertices that induces an acyclic
subgraph. In 2011, Harutyunyan conjectured that every planar digraph on
vertices without directed 2-cycles possesses an acyclic set of size at least
. We prove this conjecture for digraphs where every directed cycle has
length at least 8. More generally, if is the length of the shortest
directed cycle, we show that there exists an acyclic set of size at least .Comment: 9 page
A Polynomial-time Bicriteria Approximation Scheme for Planar Bisection
Given an undirected graph with edge costs and node weights, the minimum
bisection problem asks for a partition of the nodes into two parts of equal
weight such that the sum of edge costs between the parts is minimized. We give
a polynomial time bicriteria approximation scheme for bisection on planar
graphs.
Specifically, let be the total weight of all nodes in a planar graph .
For any constant , our algorithm outputs a bipartition of the
nodes such that each part weighs at most and the total cost
of edges crossing the partition is at most times the total
cost of the optimal bisection. The previously best known approximation for
planar minimum bisection, even with unit node weights, was . Our
algorithm actually solves a more general problem where the input may include a
target weight for the smaller side of the bipartition.Comment: To appear in STOC 201
Measuring and Understanding Throughput of Network Topologies
High throughput is of particular interest in data center and HPC networks.
Although myriad network topologies have been proposed, a broad head-to-head
comparison across topologies and across traffic patterns is absent, and the
right way to compare worst-case throughput performance is a subtle problem.
In this paper, we develop a framework to benchmark the throughput of network
topologies, using a two-pronged approach. First, we study performance on a
variety of synthetic and experimentally-measured traffic matrices (TMs).
Second, we show how to measure worst-case throughput by generating a
near-worst-case TM for any given topology. We apply the framework to study the
performance of these TMs in a wide range of network topologies, revealing
insights into the performance of topologies with scaling, robustness of
performance across TMs, and the effect of scattered workload placement. Our
evaluation code is freely available
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