3,499 research outputs found
Typical solution time for a vertex-covering algorithm on finite-connectivity random graphs
In this letter, we analytically describe the typical solution time needed by
a backtracking algorithm to solve the vertex-cover problem on
finite-connectivity random graphs. We find two different transitions: The first
one is algorithm-dependent and marks the dynamical transition from linear to
exponential solution times. The second one gives the maximum computational
complexity, and is found exactly at the threshold where the system undergoes an
algorithm-independent phase transition in its solvability. Analytical results
are corroborated by numerical simulations.Comment: 4 pages, 2 figures, to appear in Phys. Rev. Let
Statistical mechanics of the vertex-cover problem
We review recent progress in the study of the vertex-cover problem (VC). VC
belongs to the class of NP-complete graph theoretical problems, which plays a
central role in theoretical computer science. On ensembles of random graphs, VC
exhibits an coverable-uncoverable phase transition. Very close to this
transition, depending on the solution algorithm, easy-hard transitions in the
typical running time of the algorithms occur.
We explain a statistical mechanics approach, which works by mapping VC to a
hard-core lattice gas, and then applying techniques like the replica trick or
the cavity approach. Using these methods, the phase diagram of VC could be
obtained exactly for connectivities , where VC is replica symmetric.
Recently, this result could be confirmed using traditional mathematical
techniques. For , the solution of VC exhibits full replica symmetry
breaking.
The statistical mechanics approach can also be used to study analytically the
typical running time of simple complete and incomplete algorithms for VC.
Finally, we describe recent results for VC when studied on other ensembles of
finite- and infinite-dimensional graphs.Comment: review article, 26 pages, 9 figures, to appear in J. Phys. A: Math.
Ge
Dynamics of heuristic optimization algorithms on random graphs
In this paper, the dynamics of heuristic algorithms for constructing small
vertex covers (or independent sets) of finite-connectivity random graphs is
analysed. In every algorithmic step, a vertex is chosen with respect to its
vertex degree. This vertex, and some environment of it, is covered and removed
from the graph. This graph reduction process can be described as a Markovian
dynamics in the space of random graphs of arbitrary degree distribution. We
discuss some solvable cases, including algorithms already analysed using
different techniques, and develop approximation schemes for more complicated
cases. The approximations are corroborated by numerical simulations.Comment: 19 pages, 3 figures, version to app. in EPJ
Minimal vertex covers on finite-connectivity random graphs - a hard-sphere lattice-gas picture
The minimal vertex-cover (or maximal independent-set) problem is studied on
random graphs of finite connectivity. Analytical results are obtained by a
mapping to a lattice gas of hard spheres of (chemical) radius one, and they are
found to be in excellent agreement with numerical simulations. We give a
detailed description of the replica-symmetric phase, including the size and the
entropy of the minimal vertex covers, and the structure of the unfrozen
component which is found to percolate at connectivity . The
replica-symmetric solution breaks down at . We give a simple
one-step replica symmetry broken solution, and discuss the problems in
interpretation and generalization of this solution.Comment: 32 pages, 9 eps figures, to app. in PRE (01 May 2001
Message passing for vertex covers
Constructing a minimal vertex cover of a graph can be seen as a prototype for
a combinatorial optimization problem under hard constraints. In this paper, we
develop and analyze message passing techniques, namely warning and survey
propagation, which serve as efficient heuristic algorithms for solving these
computational hard problems. We show also, how previously obtained results on
the typical-case behavior of vertex covers of random graphs can be recovered
starting from the message passing equations, and how they can be extended.Comment: 25 pages, 9 figures - version accepted for publication in PR
Finding long cycles in graphs
We analyze the problem of discovering long cycles inside a graph. We propose
and test two algorithms for this task. The first one is based on recent
advances in statistical mechanics and relies on a message passing procedure.
The second follows a more standard Monte Carlo Markov Chain strategy. Special
attention is devoted to Hamiltonian cycles of (non-regular) random graphs of
minimal connectivity equal to three
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