Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: How much backtracking does it take to color random graphs? Rigorous
  results on heavy tails   Many backtracking algorithms exhibit heavy-tailed distributions, in which
their running time is often much longer than their median. We analyze the
behavior of two natural variants of the Davis-Putnam-Logemann-Loveland (DPLL)
algorithm for Graph 3-Coloring on sparse random graphs G(n,p=c/n). Let P_c(b)
be the probability that DPLL backtracks b times. First, we calculate
analytically the probability P_c(0) that these algorithms find a 3-coloring
with no backtracking at all, and show that it goes to zero faster than any
analytic function as c \to c^* = 3.847... Then we show that even in the
``easy'' phase 1 < c < c^* where P_c(0) > 0, including just above the emergence
of the giant component, the expected number of backtracks is exponentially
large with positive probability. To our knowledge this is the first rigorous
proof that the running time of a natural backtracking algorithm has a heavy
tail for graph coloring. Moreover, our results show that these algorithms take
exponential time, not just below the 3-colorability threshold, but just above
the degree c=1 at which the giant component first appears. In addition, we give
experimental evidence and heuristic arguments that this tail takes the form
P_c(b) ~ b^{-1} up to an exponential cutoff.

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The categories of this paper are: ao-sci
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The categories of this paper are: solv-int
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The categories of this paper are: comp-gas
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The categories of this paper are: adap-org
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The categories of this paper are: chao-dyn
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The categories of this paper are: stat
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The categories of this paper are: funct-an
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The categories of this paper are: gr-qc
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The categories of this paper are: nucl-th
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The categories of this paper are: quant-ph
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The categories of this paper are: hep-ex
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The categories of this paper are: q-bio
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The categories of this paper are: math-ph
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The categories of this paper are: cmp-lg
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The categories of this paper are: econ
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The categories of this paper are: eess