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Revision #1 to TR09-129 | 29th April 2010 22:45

Subsampling Mathematical Relaxations and Average-case Complexity

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Abstract:

We initiate a study of when the value of mathematical relaxations such as
linear and semi-definite programs for constraint satisfaction problems (CSPs)
is approximately preserved when restricting the instance to a sub-instance
induced by a small random subsample of the variables.

Let $\mathcal{C}$ be a family of CSPs such as 3SAT, Max-Cut, etc., and let
$\Pi$ be a mathematical program that is a \emph{relaxation} for $\mathcal{C}$,
in the sense that for every instance $\mathcal{P}\in\mathcal{C}$, $\Pi(\mathcal{P})$ is a number in
$[0,1]$ upper bounding the maximum fraction of satisfiable constraints of
$\mathcal{P}$. Loosely speaking, we say that \emph{subsampling holds} for $\mathcal{C}$ and
$\Pi$ if for every sufficiently dense instance $\mathcal{P} \in \mathcal{C}$
and every $\epsilon>0$, if we let $\mathcal{P}'$ be the instance obtained by
restricting $\mathcal{P}$ to a sufficiently large constant number of
variables, then $\Pi(\mathcal{P}') \in (1\pm \epsilon)\Pi(\mathcal{P})$. We say that \emph{weak
subsampling} holds if the above guarantee is replaced with $\Pi(\mathcal{P}') =
1-\Theta(\gamma)$ whenever $\Pi(\mathcal{P})= 1-\gamma$, where $\Theta$ hides only
absolute constants. We obtain both positive and negative results, showing
that:

\begin{enumerate}

\item Subsampling holds for the BasicLP and BasicSDP programs. BasicSDP is
a variant of the semi-definite program considered by Raghavendra (2008), who
showed it gives an optimal approximation factor for every
constraint-satisfaction problem under the unique games conjecture. BasicLP is
the linear programming analog of BasicSDP.

\item For tighter versions of BasicSDP obtained by adding additional
constraints from the Lasserre hierarchy, \emph{weak} subsampling holds for
CSPs of unique games type.

\item There are non-unique CSPs for which even weak subsampling fails for the
above tighter semi-definite programs. Also there are unique CSPs for which
(even weak) subsampling fails for the Sherali-Adams \emph{linear programming}
hierarchy.

\end{enumerate}

As a corollary of our weak subsampling for strong semi-definite programs, we
obtain a polynomial-time algorithm to certify that \emph{random geometric
graphs} (of the type considered by Feige and Schechtman, 2002) of max-cut
value $1-\gamma$ have a cut value at most $1-\gamma/10$. More generally, our
results give an approach to obtaining average-case algorithms for CSPs using
semi-definite programming hierarchies.



Changes to previous version:

Several more general results that subsume the previous paper.


Paper:

TR09-129 | 30th November 2009 00:47

Subsampling Semidefinite Programs and Max-Cut on the Sphere





TR09-129
Authors: Boaz Barak, Moritz Hardt, Thomas Holenstein, David Steurer
Publication: 30th November 2009 05:50
Downloads: 2307
Keywords: 


Abstract:

We study the question of whether the value of mathematical programs such as
linear and semidefinite programming hierarchies on a graph $G$, is preserved
when taking a small random subgraph $G'$ of $G$. We show that the value of the
Goemans-Williamson (1995) semidefinite program (SDP) for \maxcut of $G'$ is
approximately equal to the SDP value of $G$. Moreover, this holds even if the
SDP is augmented with any constraint from a large natural family $\mathcal{C}$
of constraints that includes all constraints from the hierarchy of Lasserre
(2001). The subgraph $G'$ is up to a constant factor as small as possible.

In contrast, we show that for linear programs this is \emph{not} the case, and
there are graphs $G$ that have small value for the \maxcut LP with triangle
inequalities, but for which a random subgraph $G'$ has value $1-o(1)$ even for
the LP augmented with $n^{\epsilon}$ rounds of the Sherali-Adams hierarchy.

Our results yield a general approach to use the Lasserre hierarchy to solve
\maxcut on the average on certain types of input distributions. In particular
we show that a natural candidate for hard instances of \maxcut--- the
Feige-Schechtman (2002) graphs that are obtained by sampling a random subset
$S$ of the sphere and placing an edge between pairs of points in $S$ that are
almost antipodal--- is actually easy to solve by SDPs with triangle
inequalities. Such a result can be considered more or less folklore in the
\emph{dense} case, where $S$ is large enough to form a close approximation
of the sphere, but requires quite different techniques to demonstrate in
the sparse case.



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