201 research outputs found
Minimum Entropy Orientations
We study graph orientations that minimize the entropy of the in-degree
sequence. The problem of finding such an orientation is an interesting special
case of the minimum entropy set cover problem previously studied by Halperin
and Karp [Theoret. Comput. Sci., 2005] and by the current authors
[Algorithmica, to appear]. We prove that the minimum entropy orientation
problem is NP-hard even if the graph is planar, and that there exists a simple
linear-time algorithm that returns an approximate solution with an additive
error guarantee of 1 bit. This improves on the only previously known algorithm
which has an additive error guarantee of log_2 e bits (approx. 1.4427 bits).Comment: Referees' comments incorporate
Parallel Chip Firing Game associated with n-cube orientations
We study the cycles generated by the chip firing game associated with n-cube
orientations. We show the existence of the cycles generated by parallel
evolutions of even lengths from 2 to on (n >= 1), and of odd
lengths different from 3 and ranging from 1 to on (n >= 4)
Fermions and Loops on Graphs. II. Monomer-Dimer Model as Series of Determinants
We continue the discussion of the fermion models on graphs that started in
the first paper of the series. Here we introduce a Graphical Gauge Model (GGM)
and show that : (a) it can be stated as an average/sum of a determinant defined
on the graph over (binary) gauge field; (b) it is equivalent
to the Monomer-Dimer (MD) model on the graph; (c) the partition function of the
model allows an explicit expression in terms of a series over disjoint directed
cycles, where each term is a product of local contributions along the cycle and
the determinant of a matrix defined on the remainder of the graph (excluding
the cycle). We also establish a relation between the MD model on the graph and
the determinant series, discussed in the first paper, however, considered using
simple non-Belief-Propagation choice of the gauge. We conclude with a
discussion of possible analytic and algorithmic consequences of these results,
as well as related questions and challenges.Comment: 11 pages, 2 figures; misprints correcte
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