Abstract
For many backtracking search algorithms, the running time has been found experimentally to have a heavy-tailed distribution, in which it is often much greater than its median. We analyze two natural variants of the Davis-Putnam-Logemann-Loveland (DPLL) algorithm for Graph 3-Coloring on sparse random graphs of average degree c. 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 → c * = 3.847... Then we show that even in the “easy” regime 1 < c < c * where P c (0) > 0 – including just above the degree c=1 where the giant component first appears – the expected number of backtrackings 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.
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Jia, H., Moore, C. (2004). How Much Backtracking Does It Take to Color Random Graphs? Rigorous Results on Heavy Tails. In: Wallace, M. (eds) Principles and Practice of Constraint Programming – CP 2004. CP 2004. Lecture Notes in Computer Science, vol 3258. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-30201-8_58
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DOI: https://doi.org/10.1007/978-3-540-30201-8_58
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