• Corpus ID: 2524522

Combinatorial Approximation Algorithms for MaxCut using Random Walks

@article{Kale2011CombinatorialAA,
  title={Combinatorial Approximation Algorithms for MaxCut using Random Walks},
  author={Satyen Kale and C. Seshadhri},
  journal={ArXiv},
  year={2011},
  volume={abs/1008.3938}
}
We give the first combinatorial approximation algorithm for Maxcut that beats the trivial 0.5 factor by a constant. The main partitioning procedure is very intuitive, natural, and easily described. It essentially performs a number of random walks and aggregates the information to provide the partition. We can control the running time to get an approximation factor-running time tradeoff. We show that for any constant b > 1.5, there is an O(n^{b}) algorithm that outputs a (0.5+delta… 

Figures from this paper

On randomizing two derandomized greedy algorithms

TLDR
It is shown that when executed on a random permutation of the variables, the performance ratio of Johnson’s approximation algorithm for MAX-SAT is improved to 2/3 + c for some c > 0.1 and the hope was that running the greedy algorithm on arandom permutations of the vertices would result in a 1/2+ c approximation algorithm.

Randomized greedy: new variants of some classic approximation algorithms

TLDR
It is shown that when executed on a random permutation of the variables, the performance ratio of Johnson's approximation algorithm for MAX-SAT is improved to 2/3 + c for some c > 0 and the hope was that running the greedy algorithm on arandom permutations of the vertices would result in a 1/2 + c approximation algorithm, but it turns out that in this case the performance of the algorithm remains 1/ 2.

Streaming Lower Bounds for Approximating MAX-CUT

TLDR
It is shown that for any e > 0, any streaming algorithm that obtains a (1 + e)-approximation to the max cut value when edges arrive in adversarial order requires n1−O(e) space, implying that Ω(n) space is necessary to obtain an arbitrarily good approximation to themax cut value.

Improved Combinatorial Approximation Algorithms for MAX CUT in Sparse Graphs

TLDR
A new vertex decomposition of graphs is introduced, which is called tree-bipartite decomposition, which presents a linear-time ( 1 2 + n−1 2m )-approximation algorithm for the Max-Cut problem and a derivative is presented, which solves an open problem in their paper.

Fast Distributed Approximation for Max-Cut

TLDR
These algorithms make non-trivial use of the greedy approach of Buchbinder et al. (SIAM Journal on Computing, 2015) for maximizing an unconstrained (non-monotone) submodular function, which may be of independent interest.

Adapting Sequential Algorithms to the Distributed Setting

TLDR
This paper defines a robust family of local sequential algorithms which can be easily adapted to the distributed setting, and develops algorithms which have the same approximation guarantees as their sequential counterparts, up to a constant additive $\epsilon$ factor.

Adapting Local Sequential Algorithms to the Distributed Setting

TLDR
This paper defines a robust family of local sequential algorithms which can be easily adapted to the distributed setting, and develops algorithms which have the same approximation guarantees as their sequential counterparts, up to a constant additive $\epsilon$ factor.

Cubical coloring - fractional covering by cuts and semidefinite programming

  • Robert Šámal
  • Mathematics, Computer Science
    Discret. Math. Theor. Comput. Sci.
  • 2015
TLDR
A new graph invariant that measures fractional covering of a graph by cuts, useful for study of homomorphisms and tension-continuous mappings is introduced and the value of the defined parameter is found for a family of graphs based on hypercubes.

Oblivious and Non-oblivious Local Search for Combinatorial Optimization

TLDR
This thesis gives a new, randomized approximation algorithm for maximizing a monotone submodular function subject to a matroid constraint, and gives a non-oblivious local search algorithm that delivers improved approximations for a variety of specific problems.

References

SHOWING 1-10 OF 40 REFERENCES

Max cut and the smallest eigenvalue

TLDR
A new approximation algorithm for Max Cut is described, which can be implemented in nearly linear time, and finds a solution that cuts a 1-4√ε + 8ε-o(1) fraction of edges in graphs in which the optimum is 1/2 + ε.

Local Graph Partitioning using PageRank Vectors

TLDR
An improved algorithm for computing approximate PageRank vectors, which allows us to find a cut with conductance at most oslash and approximately optimal balance in time O(m log4 m/oslash) in time proportional to its size.

Improved Bounds for Mixing Rates of Markov Chains and Multicommodity Flow

  • A. Sinclair
  • Computer Science
    Combinatorics, Probability and Computing
  • 1992
TLDR
A new upper bound on the mixing rate is presented, based on the solution to a multicommodity flow problem in the Markov chain viewed as a graph, and improved bounds are obtained for the runtimes of randomised approximation algorithms for various problems, including computing the permanent of a 0–1 matrix, counting matchings in graphs, and computing the partition function of a ferromagnetic Ising system.

Combinatorial 5/6-approximation of Max Cut in graphs of maximum degree 3

Approximation algorithms for unique games

  • L. Trevisan
  • Computer Science
    46th Annual IEEE Symposium on Foundations of Computer Science (FOCS'05)
  • 2005
We present a polynomial time algorithm based on semidefinite programming that, given a unique game of value 1 - O(1/logn), satisfies a constant fraction of constraints, where n is the number of

An algorithm for improving graph partitions

TLDR
It is demonstrated empirically that applying Improve to the output of various graph partitioning algorithms greatly improves the quality of cuts produced without significantly impacting the running time.

A combinatorial algorithm for MAX CSP

Improved approximation algorithms for maximum cut and satisfiability problems using semidefinite programming

TLDR
This algorithm gives the first substantial progress in approximating MAX CUT in nearly twenty years, and represents the first use of semidefinite programming in the design of approximation algorithms.

MAX CUT in cubic graphs

The PCP theorem by gap amplification

  • Irit Dinur
  • Mathematics, Computer Science
    STOC '06
  • 2006
TLDR
A new combinatorial amplification transformation that doubles the unsat-value of a constraint-system, with only a linear blowup in the size of the system, is described.