186,193 research outputs found
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]
The maximal energy of classes of integral circulant graphs
The energy of a graph is the sum of the moduli of the eigenvalues of its
adjacency matrix. We study the energy of integral circulant graphs, also called
gcd graphs, which can be characterized by their vertex count and a set
of divisors of in such a way that they have vertex set
and edge set . For a fixed prime power and a fixed divisor set size , we analyze the maximal energy among all matching integral circulant
graphs. Let be the elements of .
It turns out that the differences between the exponents of
an energy maximal divisor set must satisfy certain balance conditions: (i)
either all equal , or at most the two differences
and may occur; %(for a certain depending on and ) (ii)
there are rules governing the sequence of consecutive
differences. For particular choices of and these conditions already
guarantee maximal energy and its value can be computed explicitly.Comment: Discrete Applied Mathematics (2012
The Number of Nowhere-Zero Flows on Graphs and Signed Graphs
A nowhere-zero -flow on a graph is a mapping from the edges of
to the set \{\pm1, \pm2, ..., \pm(k-1)\} \subset \bbZ such that, in
any fixed orientation of , at each node the sum of the labels over the
edges pointing towards the node equals the sum over the edges pointing away
from the node. We show that the existence of an \emph{integral flow polynomial}
that counts nowhere-zero -flows on a graph, due to Kochol, is a consequence
of a general theory of inside-out polytopes. The same holds for flows on signed
graphs. We develop these theories, as well as the related counting theory of
nowhere-zero flows on a signed graph with values in an abelian group of odd
order. Our results are of two kinds: polynomiality or quasipolynomiality of the
flow counting functions, and reciprocity laws that interpret the evaluations of
the flow polynomials at negative integers in terms of the combinatorics of the
graph.Comment: 17 pages, to appear in J. Combinatorial Th. Ser.
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