64 research outputs found
How to Play Unique Games against a Semi-Random Adversary
In this paper, we study the average case complexity of the Unique Games
problem. We propose a natural semi-random model, in which a unique game
instance is generated in several steps. First an adversary selects a completely
satisfiable instance of Unique Games, then she chooses an epsilon-fraction of
all edges, and finally replaces ("corrupts") the constraints corresponding to
these edges with new constraints. If all steps are adversarial, the adversary
can obtain any (1-epsilon) satisfiable instance, so then the problem is as hard
as in the worst case. In our semi-random model, one of the steps is random, and
all other steps are adversarial. We show that known algorithms for unique games
(in particular, all algorithms that use the standard SDP relaxation) fail to
solve semi-random instances of Unique Games.
We present an algorithm that with high probability finds a solution
satisfying a (1-delta) fraction of all constraints in semi-random instances (we
require that the average degree of the graph is Omega(log k). To this end, we
consider a new non-standard SDP program for Unique Games, which is not a
relaxation for the problem, and show how to analyze it. We present a new
rounding scheme that simultaneously uses SDP and LP solutions, which we believe
is of independent interest.
Our result holds only for epsilon less than some absolute constant. We prove
that if epsilon > 1/2, then the problem is hard in one of the models, the
result assumes the 2-to-2 conjecture.
Finally, we study semi-random instances of Unique Games that are at most
(1-epsilon) satisfiable. We present an algorithm that with high probability,
distinguishes between the case when the instance is a semi-random instance and
the case when the instance is an (arbitrary) (1-delta) satisfiable instance if
epsilon > c delta
Approximating Unique Games Using Low Diameter Graph Decomposition
We design approximation algorithms for Unique Gmeas when the constraint graph admits good low diameter graph decomposition. For the M2Lin(k) problem in K(r)-minor free graphs, when there is an assignment satisfying 1-eps fraction of constraints, we present an algorithm that produces an assignment satisfying 1-O(r*eps) fraction of constraints, with the approximation ratio independent of the alphabet size. A corollary is an improved approximation algorithm for the Min-UnCut problem for K(r)-minor free graphs. For general Unique Games in K(r)-minor free graphs, we provide another algorithm that produces an assignment satisfying 1-O(r *sqrt(eps)) fraction of constraints.
Our approach is to round a linear programming relaxation to find a minimum subset of edges that intersects all the inconsistent cycles. We show that it is possible to apply the low diameter graph decomposition technique on the constraint graph directly, rather than to work on the label extended graph as in previous algorithms for Unique Games. The same approach applies when the constraint graph is of genus g, and we get similar results with r replaced by log g in the M2Lin(k) problem and by sqrt(log g) in the general problem. The former result generalizes the result of Gupta-Talwar for Unique Games in the M2Lin(k) case, and the latter result generalizes the result of Trevisan for general Unique Games
Dimension-free L_2 maximal inequality for spherical means in the hypercube
We establish the maximal inequality claimed in the title. In combinatorial terms this has the implication that for sufficiently small ε > 0, for all n, any marking of an ε fraction of the vertices of the n-dimensional hypercube necessarily leaves a vertex x such that marked vertices are a minority of every sphere centered at x
Faster generation of random spanning trees
In this paper, we set forth a new algorithm for generating approximately
uniformly random spanning trees in undirected graphs. We show how to sample
from a distribution that is within a multiplicative of uniform in
expected time \TO(m\sqrt{n}\log 1/\delta). This improves the sparse graph
case of the best previously known worst-case bound of , which has stood for twenty years.
To achieve this goal, we exploit the connection between random walks on
graphs and electrical networks, and we use this to introduce a new approach to
the problem that integrates discrete random walk-based techniques with
continuous linear algebraic methods. We believe that our use of electrical
networks and sparse linear system solvers in conjunction with random walks and
combinatorial partitioning techniques is a useful paradigm that will find
further applications in algorithmic graph theory
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