Affiliation:
1. Princeton University
2. Microsoft Research New England
3. Cornell University
Abstract
Subexponential time approximation algorithms are presented for the U
nique
G
ames
and S
mall
-S
et
E
xpansion
problems. Specifically, for some absolute constant
c
, the following two algorithms are presented.
(1) An exp(
kn
ϵ
)-time algorithm that, given as input a
k
-alphabet unique game on
n
variables that has an assignment satisfying 1-ϵ
c
fraction of its constraints, outputs an assignment satisfying 1-ϵ fraction of the constraints.
(2) An exp(
n
ϵ
/δ)-time algorithm that, given as input an
n
-vertex regular graph that has a set
S
of δ
n
vertices with edge expansion at most
ϵ
c
, outputs a set
S'
of at most δ
n
vertices with edge expansion at most ϵ.
subexponential algorithm is also presented with improved approximation to M
ax
C
ut
, S
parsest
C
ut
, and V
ertex
C
over
on some interesting subclasses of instances. These instances are graphs with low
threshold rank
, an interesting new graph parameter highlighted by this work.
Khot's Unique Games Conjecture (UGC) states that it is
NP
-hard to achieve approximation guarantees such as ours for U
nique
G
ames
. While the results here stop short of refuting the UGC, they do suggest that U
nique
G
ames
are significantly easier than
NP
-hard problems such as M
ax
3-S
at
, M
ax
3-
Lin
, L
abel
C
over
, and more, which are believed not to have a subexponential algorithm achieving a nontrivial approximation ratio.
Of special interest in these algorithms is a new notion of graph decomposition that may have other applications. Namely, it is shown for every ϵ >0 and every regular
n
-vertex graph
G
, by changing at most δ fraction of
G
's edges, one can break
G
into disjoint parts so that the stochastic adjacency matrix of the induced graph on each part has at most
n
ϵ
eigenvalues larger than 1-η, where η depends polynomially on ϵ. The subexponential algorithm combines this decomposition with previous algorithms for U
nique
G
ames
on graphs with few large eigenvalues [Kolla and Tulsiani 2007; Kolla 2010].
Funder
Sloan Foundation Fellowship
NSF
Packard Fellowship
Publisher
Association for Computing Machinery (ACM)
Subject
Artificial Intelligence,Hardware and Architecture,Information Systems,Control and Systems Engineering,Software
Cited by
40 articles.
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