722,501 research outputs found
Covering Problems for Partial Words and for Indeterminate Strings
We consider the problem of computing a shortest solid cover of an
indeterminate string. An indeterminate string may contain non-solid symbols,
each of which specifies a subset of the alphabet that could be present at the
corresponding position. We also consider covering partial words, which are a
special case of indeterminate strings where each non-solid symbol is a don't
care symbol. We prove that indeterminate string covering problem and partial
word covering problem are NP-complete for binary alphabet and show that both
problems are fixed-parameter tractable with respect to , the number of
non-solid symbols. For the indeterminate string covering problem we obtain a
-time algorithm. For the partial word covering
problem we obtain a -time algorithm. We
prove that, unless the Exponential Time Hypothesis is false, no
-time solution exists for either problem, which shows
that our algorithm for this case is close to optimal. We also present an
algorithm for both problems which is feasible in practice.Comment: full version (simplified and corrected); preliminary version appeared
at ISAAC 2014; 14 pages, 4 figure
Analysis of the Min-Sum Algorithm for Packing and Covering Problems via Linear Programming
Message-passing algorithms based on belief-propagation (BP) are successfully
used in many applications including decoding error correcting codes and solving
constraint satisfaction and inference problems. BP-based algorithms operate
over graph representations, called factor graphs, that are used to model the
input. Although in many cases BP-based algorithms exhibit impressive empirical
results, not much has been proved when the factor graphs have cycles.
This work deals with packing and covering integer programs in which the
constraint matrix is zero-one, the constraint vector is integral, and the
variables are subject to box constraints. We study the performance of the
min-sum algorithm when applied to the corresponding factor graph models of
packing and covering LPs.
We compare the solutions computed by the min-sum algorithm for packing and
covering problems to the optimal solutions of the corresponding linear
programming (LP) relaxations. In particular, we prove that if the LP has an
optimal fractional solution, then for each fractional component, the min-sum
algorithm either computes multiple solutions or the solution oscillates below
and above the fraction. This implies that the min-sum algorithm computes the
optimal integral solution only if the LP has a unique optimal solution that is
integral.
The converse is not true in general. For a special case of packing and
covering problems, we prove that if the LP has a unique optimal solution that
is integral and on the boundary of the box constraints, then the min-sum
algorithm computes the optimal solution in pseudo-polynomial time.
Our results unify and extend recent results for the maximum weight matching
problem by [Sanghavi et al.,'2011] and [Bayati et al., 2011] and for the
maximum weight independent set problem [Sanghavi et al.'2009]
On the Optimality of a Class of LP-based Algorithms
In this paper we will be concerned with a class of packing and covering
problems which includes Vertex Cover and Independent Set. Typically, one can
write an LP relaxation and then round the solution. In this paper, we explain
why the simple LP-based rounding algorithm for the \\VC problem is optimal
assuming the UGC. Complementing Raghavendra's result, our result generalizes to
a class of strict, covering/packing type CSPs
- …