18,192 research outputs found
Biased random walks on random graphs
These notes cover one of the topics programmed for the St Petersburg School
in Probability and Statistical Physics of June 2012.
The aim is to review recent mathematical developments in the field of random
walks in random environment. Our main focus will be on directionally transient
and reversible random walks on different types of underlying graph structures,
such as , trees and for .Comment: Survey based one of the topics programmed for the St Petersburg
School in Probability and Statistical Physics of June 2012. 64 pages, 16
figure
Random permutations of a regular lattice
Spatial random permutations were originally studied due to their connections
to Bose-Einstein condensation, but they possess many interesting properties of
their own. For random permutations of a regular lattice with periodic boundary
conditions, we prove existence of the infinite volume limit under fairly weak
assumptions. When the dimension of the lattice is two, we give numerical
evidence of a Kosterlitz-Thouless transition, and of long cycles having an
almost sure fractal dimension in the scaling limit. Finally we comment on
possible connections to Schramm-L\"owner curves.Comment: 23 pages, 8 figure
Fast and Deterministic Approximations for k-Cut
In an undirected graph, a k-cut is a set of edges whose removal breaks the graph into at least k connected components. The minimum weight k-cut can be computed in n^O(k) time, but when k is treated as part of the input, computing the minimum weight k-cut is NP-Hard [Goldschmidt and Hochbaum, 1994]. For poly(m,n,k)-time algorithms, the best possible approximation factor is essentially 2 under the small set expansion hypothesis [Manurangsi, 2017]. Saran and Vazirani [1995] showed that a (2 - 2/k)-approximately minimum weight k-cut can be computed via O(k) minimum cuts, which implies a O~(km) randomized running time via the nearly linear time randomized min-cut algorithm of Karger [2000]. Nagamochi and Kamidoi [2007] showed that a (2 - 2/k)-approximately minimum weight k-cut can be computed deterministically in O(mn + n^2 log n) time. These results prompt two basic questions. The first concerns the role of randomization. Is there a deterministic algorithm for 2-approximate k-cuts matching the randomized running time of O~(km)? The second question qualitatively compares minimum cut to 2-approximate minimum k-cut. Can 2-approximate k-cuts be computed as fast as the minimum cut - in O~(m) randomized time?
We give a deterministic approximation algorithm that computes (2 + eps)-minimum k-cuts in O(m log^3 n / eps^2) time, via a (1 + eps)-approximation for an LP relaxation of k-cut
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