8,884 research outputs found
Sudden emergence of q-regular subgraphs in random graphs
We investigate the computationally hard problem whether a random graph of
finite average vertex degree has an extensively large -regular subgraph,
i.e., a subgraph with all vertices having degree equal to . We reformulate
this problem as a constraint-satisfaction problem, and solve it using the
cavity method of statistical physics at zero temperature. For , we find
that the first large -regular subgraphs appear discontinuously at an average
vertex degree c_\reg{3} \simeq 3.3546 and contain immediately about 24% of
all vertices in the graph. This transition is extremely close to (but different
from) the well-known 3-core percolation point c_\cor{3} \simeq 3.3509. For
, the -regular subgraph percolation threshold is found to coincide with
that of the -core.Comment: 7 pages, 5 figure
The Quantum Adiabatic Algorithm applied to random optimization problems: the quantum spin glass perspective
Among various algorithms designed to exploit the specific properties of
quantum computers with respect to classical ones, the quantum adiabatic
algorithm is a versatile proposition to find the minimal value of an arbitrary
cost function (ground state energy). Random optimization problems provide a
natural testbed to compare its efficiency with that of classical algorithms.
These problems correspond to mean field spin glasses that have been extensively
studied in the classical case. This paper reviews recent analytical works that
extended these studies to incorporate the effect of quantum fluctuations, and
presents also some original results in this direction.Comment: 151 pages, 21 figure
Clustering of solutions in hard satisfiability problems
We study the structure of the solution space and behavior of local search
methods on random 3-SAT problems close to the SAT/UNSAT transition. Using the
overlap measure of similarity between different solutions found on the same
problem instance we show that the solution space is shrinking as a function of
alpha. We consider chains of satisfiability problems, where clauses are added
sequentially. In each such chain, the overlap distribution is first smooth, and
then develops a tiered structure, indicating that the solutions are found in
well separated clusters. On chains of not too large instances, all solutions
are eventually observed to be in only one small cluster before vanishing. This
condensation transition point is estimated to be alpha_c = 4.26. The transition
approximately obeys finite-size scaling with an apparent critical exponent of
about 1.7. We compare the solutions found by a local heuristic, ASAT, and the
Survey Propagation algorithm up to alpha_c.Comment: 8 pages, 9 figure
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