461 research outputs found
A Landscape Analysis of Constraint Satisfaction Problems
We discuss an analysis of Constraint Satisfaction problems, such as Sphere
Packing, K-SAT and Graph Coloring, in terms of an effective energy landscape.
Several intriguing geometrical properties of the solution space become in this
light familiar in terms of the well-studied ones of rugged (glassy) energy
landscapes. A `benchmark' algorithm naturally suggested by this construction
finds solutions in polynomial time up to a point beyond the `clustering' and in
some cases even the `thermodynamic' transitions. This point has a simple
geometric meaning and can be in principle determined with standard Statistical
Mechanical methods, thus pushing the analytic bound up to which problems are
guaranteed to be easy. We illustrate this for the graph three and four-coloring
problem. For Packing problems the present discussion allows to better
characterize the `J-point', proposed as a systematic definition of Random Close
Packing, and to place it in the context of other theories of glasses.Comment: 17 pages, 69 citations, 12 figure
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
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