Karp's patching algorithm on dense digraphs
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We consider the following question. We are given a dense digraph D with minimum in- and out-degree at least α n, where α>1/2 is a constant. The edges of D are given edge costs C(e),e∈ E(D), where C(e) is an independent copy of the uniform [0,1] random variable U. Let C(i,j),i,j∈[n] be the associated n× n cost matrix where C(i,j)=∞ if (i,j)∉ E(D). We show that w.h.p. the patching algorithm of Karp finds a tour for the asymmetric traveling salesperson problem that is asymptotically equal to that of the associated assignment problem. Karp's algorithm runs in polynomial time.
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