150,805 research outputs found
A Unified View of Graph Regularity via Matrix Decompositions
We prove algorithmic weak and \Szemeredi{} regularity lemmas for several
classes of sparse graphs in the literature, for which only weak regularity
lemmas were previously known. These include core-dense graphs, low threshold
rank graphs, and (a version of) upper regular graphs. More precisely, we
define \emph{cut pseudorandom graphs}, we prove our regularity lemmas for these
graphs, and then we show that cut pseudorandomness captures all of the above
graph classes as special cases.
The core of our approach is an abstracted matrix decomposition, roughly
following Frieze and Kannan [Combinatorica '99] and \Lovasz{} and Szegedy
[Geom.\ Func.\ Anal.\ '07], which can be computed by a simple algorithm by
Charikar [AAC0 '00]. This gives rise to the class of cut pseudorandom graphs,
and using work of Oveis Gharan and Trevisan [TOC '15], it also implies new
PTASes for MAX-CUT, MAX-BISECTION, MIN-BISECTION for a significantly expanded
class of input graphs. (It is NP Hard to get PTASes for these graphs in
general.
A New Regularity Lemma and Faster Approximation Algorithms for Low Threshold Rank Graphs
Kolla and Tulsiani [KT07,Kolla11} and Arora, Barak and Steurer [ABS10]
introduced the technique of subspace enumeration, which gives approximation
algorithms for graph problems such as unique games and small set expansion; the
running time of such algorithms is exponential in the threshold-rank of the
graph.
Guruswami and Sinop [GS11,GS12], and Barak, Raghavendra, and Steurer [BRS11]
developed an alternative approach to the design of approximation algorithms for
graphs of bounded threshold-rank, based on semidefinite programming relaxations
in the Lassere hierarchy and on novel rounding techniques. These algorithms are
faster than the ones based on subspace enumeration and work on a broad class of
problems.
In this paper we develop a third approach to the design of such algorithms.
We show, constructively, that graphs of bounded threshold-rank satisfy a weak
Szemeredi regularity lemma analogous to the one proved by Frieze and Kannan
[FK99] for dense graphs. The existence of efficient approximation algorithms is
then a consequence of the regularity lemma, as shown by Frieze and Kannan.
Applying our method to the Max Cut problem, we devise an algorithm that is
faster than all previous algorithms, and is easier to describe and analyze
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
Approximating Non-Uniform Sparsest Cut via Generalized Spectra
We give an approximation algorithm for non-uniform sparsest cut with the
following guarantee: For any , given cost and demand
graphs with edge weights respectively, we can find a set
with at most
times the optimal non-uniform sparsest cut value,
in time 2^{r/(\delta\epsilon)}\poly(n) provided . Here is the 'th smallest generalized
eigenvalue of the Laplacian matrices of cost and demand graphs; (resp. ) is the weight of edges crossing the
cut in cost (resp. demand) graph and is the
sparsity of the optimal cut. In words, we show that the non-uniform sparsest
cut problem is easy when the generalized spectrum grows moderately fast. To the
best of our knowledge, there were no results based on higher order spectra for
non-uniform sparsest cut prior to this work.
Even for uniform sparsest cut, the quantitative aspects of our result are
somewhat stronger than previous methods. Similar results hold for other
expansion measures like edge expansion, normalized cut, and conductance, with
the 'th smallest eigenvalue of the normalized Laplacian playing the role of
in the latter two cases.
Our proof is based on an l1-embedding of vectors from a semi-definite program
from the Lasserre hierarchy. The embedded vectors are then rounded to a cut
using standard threshold rounding. We hope that the ideas connecting
-embeddings to Lasserre SDPs will find other applications. Another
aspect of the analysis is the adaptation of the column selection paradigm from
our earlier work on rounding Lasserre SDPs [GS11] to pick a set of edges rather
than vertices. This feature is important in order to extend the algorithms to
non-uniform sparsest cut.Comment: 16 page
Structural transition in interdependent networks with regular interconnections
Networks are often made up of several layers that exhibit diverse degrees of
interdependencies. A multilayer interdependent network consists of a set of
graphs that are interconnected through a weighted interconnection matrix , where the weight of each inter-graph link is a non-negative real number . Various dynamical processes, such as synchronization, cascading failures
in power grids, and diffusion processes, are described by the Laplacian matrix
characterizing the whole system. For the case in which the multilayer
graph is a multiplex, where the number of nodes in each layer is the same and
the interconnection matrix , being the identity matrix, it has
been shown that there exists a structural transition at some critical coupling,
. This transition is such that dynamical processes are separated into
two regimes: if , the network acts as a whole; whereas when , the network operates as if the graphs encoding the layers were isolated. In
this paper, we extend and generalize the structural transition threshold to a regular interconnection matrix (constant row and column sum).
Specifically, we provide upper and lower bounds for the transition threshold in interdependent networks with a regular interconnection matrix
and derive the exact transition threshold for special scenarios using the
formalism of quotient graphs. Additionally, we discuss the physical meaning of
the transition threshold in terms of the minimum cut and show, through
a counter-example, that the structural transition does not always exist. Our
results are one step forward on the characterization of more realistic
multilayer networks and might be relevant for systems that deviate from the
topological constrains imposed by multiplex networks.Comment: 13 pages, APS format. Submitted for publicatio
The ising antiferromagnet and max cut on random regular graphs
The Ising antiferromagnet is an important statistical physics model with close connections to the MAX CUT problem. Combining spatial mixing arguments with the method of moments and the interpolation method, we pinpoint the replica symmetry breaking phase transition predicted by physicists. Additionally, we rigorously establish upper bounds on the MAX CUT of random regular graphs predicted by Zdeborová and Boettcher [Journal of Statistical Mechanics 2010]. As an application we prove that the information-theoretic threshold of the disassortative stochastic block model on random regular graphs coincides with the Kesten-Stigum bound
Random Contractions and Sampling for Hypergraph and Hedge Connectivity
We initiate the study of hedge connectivity of undirected graphs, motivated by dependent edge failures in real-world networks. In this model, edges are partitioned into groups called hedges that fail together. The hedge connectivity of a graph is the minimum number of hedges whose removal disconnects the graph. We give a polynomial-time approximation scheme and a quasi-polynomial exact algorithm for hedge connectivity. This provides strong evidence that the hedge connectivity problem is tractable, which contrasts with prior work that established the intractability of the corresponding s−t min-cut problem. Our techniques also yield new combinatorial and algorithmic results in hypergraph connectivity. Next, we study the behavior of hedge graphs under uniform random sampling of hedges. We show that unlike graphs, all cuts in the sample do not converge to their expected value in hedge graphs. Nevertheless, the min-cut of the sample does indeed concentrate around the expected value of the original min-cut. This leads to a sharp threshold on hedge survival probabilities for graph disconnection. To the best of our knowledge, this is the first network reliability analysis under dependent edge failures
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