Sublinear-time algorithms for monomer-dimer systems on bounded degree graphs

  title={Sublinear-time algorithms for monomer-dimer systems on bounded degree graphs},
  author={Marc Lelarge and Hang Zhou},
  booktitle={Theoretical Computer Science},
  • M. LelargeHang Zhou
  • Published in Theoretical Computer Science 17 August 2012
  • Computer Science, Mathematics

Graph Algorithms: Network Inference and Planar Graph Optimization. (Algorithmes des Graphes: Inférence des Réseaux et Optimisation dans les Graphes Planaires)

For planar graphs, the correlation clustering problem is reduced to two-edge-connected augmentation, and it is shown that both problems, although they are NP-hard, have a polynomial-time approximation scheme.

Bypassing correlation decay for matchings with an application to XORSAT

  • M. Lelarge
  • Mathematics, Computer Science
    2013 IEEE Information Theory Workshop (ITW)
  • 2013
It is shown that monotonicity properties of the maximum matching problem allows us to define solutions for the cavity equations and to identify the `right' solution of these equations and then to compute the asymptotics for the size of a maximum matching.

Topics in random graphs, combinatorial optimization, and statistical inference

The manuscript is made of three chapters presenting three different topics on which I worked with Ph.D. students, and a gentle introduction to the theory of random graphs with an emphasis on contagions on such networks.

Law of large numbers for matchings, extension and applications

The fact that global properties of matchings can be read from local properties of the underlying graph has been rediscovered many times in statistical physics, combinatorics, group theory and



On Approximating the Minimum Vertex Cover in Sublinear Time and the Connection to Distributed Algorithms

A near-optimal sublinear-time algorithm for approximating the minimum vertex cover size

An algorithm is given that outputs a (2, e)-estimate of the size of a minimum vertex cover whose query complexity and running time are O(n) · poly(1/e) and the result is nearly optimal.

Counting stars and other small subgraphs in sublinear time

An algorithm is designed that, given an approximation parameter 0 < ε < 1 and query access to a graph <i>G</i>, outputs an estimate of the number of stars of a certain size such that with high constant probability, (1-ε) vC<i>v</i><sub><i>s</i></sub> ≤ v(1 + ε)</i), where

Approximating the Minimum Spanning Tree Weight in Sublinear Time

A probabilistic algorithm that estimates on the number of components in various subgraphs of G can be used to estimate the weight of its MST and proves a nearly matching lower bound of Ω(dωƐċ2) on the probe and time complexity of any approximation algorithm for MST weight.

The Complexity of Counting in Sparse, Regular, and Planar Graphs

It is proved that the problems of counting matchings, vertex covers, independent sets, and extremal variants of these all remain hard when restricted to planar bipartite graphs of bounded degree or regular graphs of constant degree.

Counting independent sets up to the tree threshold

It is shown that on any graph of maximum degree Δ correlations decay with distance at least as fast as they do on the regular tree of the same degree, which resolves an open conjecture in statistical physics.

Approximation Algorithms for Two-State Anti-Ferromagnetic Spin Systems on Bounded Degree Graphs

The results of this paper indicate a tight relationship between complexity theory and phase transition phenomena in two-state anti-ferromagnetic spin systems on graphs of maximum degree $$d$$d for parameters outside the uniqueness region.

Correlation decay and deterministic FPTAS for counting list-colorings of a graph

The principle insight of the present work is that the correlation decayproperty can be established with respect to certain computation tree, as opposed to the conventional correlation decay property which is typically established withrespect to graph theoretic neighborhoods of a given node.

An improved constant-time approximation algorithm for maximum~matchings

An algorithm to approximate the size of some maximal independent set with additive error ε n whose running time is O(d2) is presented, and it is shown that there are approximation algorithms for many other problems, e.g., the maximum matching problem, the minimum vertex cover problem, and the minimum set cover problems, that run exponentially faster than existing algorithms.

Simple deterministic approximation algorithms for counting matchings

A deterministic fully polynomial time approximationscheme (FPTAS) is constructed for computing the total number of matchings in abounded degree graph and another problem to the small, but growing, class of P-complete problems for which there is now a deterministic FPTAS.