170,189 research outputs found
On the Usefulness of Predicates
Motivated by the pervasiveness of strong inapproximability results for
Max-CSPs, we introduce a relaxed notion of an approximate solution of a
Max-CSP. In this relaxed version, loosely speaking, the algorithm is allowed to
replace the constraints of an instance by some other (possibly real-valued)
constraints, and then only needs to satisfy as many of the new constraints as
possible.
To be more precise, we introduce the following notion of a predicate
being \emph{useful} for a (real-valued) objective : given an almost
satisfiable Max- instance, there is an algorithm that beats a random
assignment on the corresponding Max- instance applied to the same sets of
literals. The standard notion of a nontrivial approximation algorithm for a
Max-CSP with predicate is exactly the same as saying that is useful for
itself.
We say that is useless if it is not useful for any . This turns out to
be equivalent to the following pseudo-randomness property: given an almost
satisfiable instance of Max- it is hard to find an assignment such that the
induced distribution on -bit strings defined by the instance is not
essentially uniform.
Under the Unique Games Conjecture, we give a complete and simple
characterization of useful Max-CSPs defined by a predicate: such a Max-CSP is
useless if and only if there is a pairwise independent distribution supported
on the satisfying assignments of the predicate. It is natural to also consider
the case when no negations are allowed in the CSP instance, and we derive a
similar complete characterization (under the UGC) there as well.
Finally, we also include some results and examples shedding additional light
on the approximability of certain Max-CSPs
Explicit Optimal Hardness via Gaussian stability results
The results of Raghavendra (2008) show that assuming Khot's Unique Games
Conjecture (2002), for every constraint satisfaction problem there exists a
generic semi-definite program that achieves the optimal approximation factor.
This result is existential as it does not provide an explicit optimal rounding
procedure nor does it allow to calculate exactly the Unique Games hardness of
the problem.
Obtaining an explicit optimal approximation scheme and the corresponding
approximation factor is a difficult challenge for each specific approximation
problem. An approach for determining the exact approximation factor and the
corresponding optimal rounding was established in the analysis of MAX-CUT (KKMO
2004) and the use of the Invariance Principle (MOO 2005). However, this
approach crucially relies on results explicitly proving optimal partitions in
Gaussian space. Until recently, Borell's result (Borell 1985) was the only
non-trivial Gaussian partition result known.
In this paper we derive the first explicit optimal approximation algorithm
and the corresponding approximation factor using a new result on Gaussian
partitions due to Isaksson and Mossel (2012). This Gaussian result allows us to
determine exactly the Unique Games Hardness of MAX-3-EQUAL. In particular, our
results show that Zwick algorithm for this problem achieves the optimal
approximation factor and prove that the approximation achieved by the algorithm
is as conjectured by Zwick.
We further use the previously known optimal Gaussian partitions results to
obtain a new Unique Games Hardness factor for MAX-k-CSP : Using the well known
fact that jointly normal pairwise independent random variables are fully
independent, we show that the the UGC hardness of Max-k-CSP is , improving on results of Austrin and Mossel (2009)
A deep cut ellipsoid algorithm for convex programming
This paper proposes a deep cut version of the ellipsoid algorithm for solving a general class of continuous convex programming problems. In each step the algorithm does not require more computational effort to construct these deep cuts than its corresponding central cut version. Rules that prevent some of the numerical instabilities and theoretical drawbacks usually associated with the algorithm are also provided. Moreover, for a large class of convex programs a simple proof of its rate of convergence is given and the relation with previously known results is discussed. Finally some computational results of the deep and central cut version of the algorithm applied to a minĂąâŹâmax stochastic queue location problem are reported.location theory;convex programming;deep cut ellipsoid algorithm;minĂąâŹâmax programming;rate of convergence
Subsampling Mathematical Relaxations and Average-case Complexity
We initiate a study of when the value of mathematical relaxations such as
linear and semidefinite 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 be a family of CSPs such
as 3SAT, Max-Cut, etc., and let be a relaxation for , in the sense
that for every instance , is an upper bound the maximum
fraction of satisfiable constraints of . Loosely speaking, we say that
subsampling holds for and if for every sufficiently dense instance and every , if we let be the instance obtained by
restricting to a sufficiently large constant number of variables, then
. We say that weak subsampling holds if the
above guarantee is replaced with whenever
. We show: 1. Subsampling holds for the BasicLP and BasicSDP
programs. BasicSDP is a variant of the relaxation considered by Raghavendra
(2008), who showed it gives an optimal approximation factor for every CSP under
the unique games conjecture. BasicLP is the linear programming analog of
BasicSDP. 2. For tighter versions of BasicSDP obtained by adding additional
constraints from the Lasserre hierarchy, weak subsampling holds for CSPs of
unique games type. 3. There are non-unique CSPs for which even weak subsampling
fails for the above tighter semidefinite programs. Also there are unique CSPs
for which subsampling fails for the Sherali-Adams linear programming hierarchy.
As a corollary of our weak subsampling for strong semidefinite programs, we
obtain a polynomial-time algorithm to certify that random geometric graphs (of
the type considered by Feige and Schechtman, 2002) of max-cut value
have a cut value at most .Comment: Includes several more general results that subsume the previous
version of the paper
An exact method for a discrete multiobjective linear fractional optimization
Integer linear fractional programming problem with multiple objective MOILFP is an important field of research and has not received as much attention as did multiple objective linear fractional programming. In this work, we develop a branch and cut algorithm based on continuous fractional optimization, for generating the whole integer efficient solutions of the MOILFP problem. The basic idea of the computation phase of the algorithm is to optimize one of the fractional objective functions, then generate an integer feasible solution. Using the reduced gradients of the objective functions, an efficient cut is built and a part of the feasible domain not containing efficient solutions is truncated by adding this cut. A sample problem is solved using this algorithm, and the main practical advantages of the algorithm are indicated
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