6,927 research outputs found
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
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
Eight-Fifth Approximation for TSP Paths
We prove the approximation ratio 8/5 for the metric -path-TSP
problem, and more generally for shortest connected -joins.
The algorithm that achieves this ratio is the simple "Best of Many" version
of Christofides' algorithm (1976), suggested by An, Kleinberg and Shmoys
(2012), which consists in determining the best Christofides -tour out
of those constructed from a family \Fscr_{>0} of trees having a convex
combination dominated by an optimal solution of the fractional
relaxation. They give the approximation guarantee for
such an -tour, which is the first improvement after the 5/3 guarantee
of Hoogeveen's Christofides type algorithm (1991). Cheriyan, Friggstad and Gao
(2012) extended this result to a 13/8-approximation of shortest connected
-joins, for .
The ratio 8/5 is proved by simplifying and improving the approach of An,
Kleinberg and Shmoys that consists in completing in order to dominate
the cost of "parity correction" for spanning trees. We partition the edge-set
of each spanning tree in \Fscr_{>0} into an -path (or more
generally, into a -join) and its complement, which induces a decomposition
of . This decomposition can be refined and then efficiently used to
complete without using linear programming or particular properties of
, but by adding to each cut deficient for an individually tailored
explicitly given vector, inherent in .
A simple example shows that the Best of Many Christofides algorithm may not
find a shorter -tour than 3/2 times the incidentally common optima of
the problem and of its fractional relaxation.Comment: 15 pages, corrected typos in citations, minor change
Almost-Tight Distributed Minimum Cut Algorithms
We study the problem of computing the minimum cut in a weighted distributed
message-passing networks (the CONGEST model). Let be the minimum cut,
be the number of nodes in the network, and be the network diameter. Our
algorithm can compute exactly in time. To the best of our knowledge, this is the first paper that
explicitly studies computing the exact minimum cut in the distributed setting.
Previously, non-trivial sublinear time algorithms for this problem are known
only for unweighted graphs when due to Pritchard and
Thurimella's -time and -time algorithms for
computing -edge-connected and -edge-connected components.
By using the edge sampling technique of Karger's, we can convert this
algorithm into a -approximation -time algorithm for any . This improves
over the previous -approximation -time algorithm and
-approximation -time algorithm of Ghaffari and Kuhn. Due to the lower
bound of by Das Sarma et al. which holds for any
approximation algorithm, this running time is tight up to a factor.
To get the stated running time, we developed an approximation algorithm which
combines the ideas of Thorup's algorithm and Matula's contraction algorithm. It
saves an factor as compared to applying Thorup's tree
packing theorem directly. Then, we combine Kutten and Peleg's tree partitioning
algorithm and Karger's dynamic programming to achieve an efficient distributed
algorithm that finds the minimum cut when we are given a spanning tree that
crosses the minimum cut exactly once
Graph Sparsification by Edge-Connectivity and Random Spanning Trees
We present new approaches to constructing graph sparsifiers --- weighted
subgraphs for which every cut has the same value as the original graph, up to a
factor of . Our first approach independently samples each
edge with probability inversely proportional to the edge-connectivity
between and . The fact that this approach produces a sparsifier resolves
a question posed by Bencz\'ur and Karger (2002). Concurrent work of Hariharan
and Panigrahi also resolves this question. Our second approach constructs a
sparsifier by forming the union of several uniformly random spanning trees.
Both of our approaches produce sparsifiers with
edges. Our proofs are based on extensions of Karger's contraction algorithm,
which may be of independent interest
Finite Volume Spaces and Sparsification
We introduce and study finite -volumes - the high dimensional
generalization of finite metric spaces. Having developed a suitable
combinatorial machinery, we define -volumes and show that they contain
Euclidean volumes and hypertree volumes. We show that they can approximate any
-volume with multiplicative distortion. On the other hand, contrary
to Bourgain's theorem for , there exists a -volume that on vertices
that cannot be approximated by any -volume with distortion smaller than
.
We further address the problem of -dimension reduction in the context
of volumes, and show that this phenomenon does occur, although not to
the same striking degree as it does for Euclidean metrics and volumes. In
particular, we show that any metric on points can be -approximated by a sum of cut metrics, improving
over the best previously known bound of due to Schechtman.
In order to deal with dimension reduction, we extend the techniques and ideas
introduced by Karger and Bencz{\'u}r, and Spielman et al.~in the context of
graph Sparsification, and develop general methods with a wide range of
applications.Comment: previous revision was the wrong file: the new revision: changed
(extended considerably) the treatment of finite volumes (see revised
abstract). Inserted new applications for the sparsification technique
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