2 research outputs found
Subexponential LPs Approximate Max-Cut
We show that for every , the degree-
Sherali-Adams linear program (with variables
and constraints) approximates the maximum cut problem within a factor of
, for some . Our
result provides a surprising converse to known lower bounds against all linear
programming relaxations of Max-Cut, and hence resolves the extension complexity
of approximate Max-Cut for approximation factors close to (up to
the function ). Previously, only semidefinite
programs and spectral methods were known to yield approximation factors better
than for Max-Cut in time . We also show that
constant-degree Sherali-Adams linear programs (with variables
and constraints) can solve Max-Cut with approximation factor close to on
graphs of small threshold rank: this is the first connection of which we are
aware between threshold rank and linear programming-based algorithms.
Our results separate the power of Sherali-Adams versus Lov\'asz-Schrijver
hierarchies for approximating Max-Cut, since it is known that
approximation of Max Cut requires
rounds in the Lov\'asz-Schrijver hierarchy.
We also provide a subexponential time approximation for Khot's Unique Games
problem: we show that for every the degree- Sherali-Adams linear program distinguishes instances of Unique Games
of value from instances of value , for
some , where is the alphabet size. Such
guarantees are qualitatively similar to those of previous subexponential-time
algorithms for Unique Games but our algorithm does not rely on semidefinite
programming or subspace enumeration techniques
Linear Programming and Community Detection
The problem of community detection with two equal-sized communities is
closely related to the minimum graph bisection problem over certain random
graph models. In the stochastic block model distribution over networks with
community structure, a well-known semidefinite programming (SDP) relaxation of
the minimum bisection problem recovers the underlying communities whenever
possible. Motivated by their superior scalability, we study the theoretical
performance of linear programming (LP) relaxations of the minimum bisection
problem for the same random models. We show that unlike the SDP relaxation that
undergoes a phase transition in the logarithmic average-degree regime, the LP
relaxation exhibits a transition from recovery to non-recovery in the linear
average-degree regime. We show that in the logarithmic average-degree regime,
the LP relaxation fails in recovering the planted bisection with high
probability.Comment: 35 pages, 3 figure