Karp's patching algorithm on dense digraph
Alan Frieze
TL;DR
The paper extends Karp's patching approach to dense digraphs with high minimum degree, showing that, for acceptable i.i.d. edge costs, the ATSP tour produced by a patched AP solution satisfies $v(AP)\le v(ATSP)\le \widehat{v}(ATSP)\le (1+o(1))v(AP)$ with high probability and runs in polynomial time. The authors formulate AP as an LP, analyze its dual, and bound dual variables to ensure the patching steps introduce only negligible excess cost. Central tools include alternating-path arguments, a careful partition of low/high-cost edges, and a Broder-chain analysis to bound the cycle structure of matchings, along with generalizations to broader distributions. The results substantially extend prior complete-graph findings to dense, non-complete digraphs and illuminate when asymptotic optimality via patching is feasible in random-cost ATSP settings. These insights have practical implications for designing efficient heuristics in dense network routing and related combinatorial optimization tasks under random cost structures.
Abstract
We consider the following question. We are given a dense digraph $D$ with $n$ vertices and minimum in- and out-degree at least $αn$, where $α>1/2$ is a constant. The edges $E(D)$ of $D$ are given independent edge costs $C(e),e\in E(D)$, such that (i) $C$ has a density $f$ that satisfies $f(x)=a+bx+O(x^2)$, for constants $a>0,b$ as $x\to 0$ and such that in general either (ii) $\Pr(C\geq x)\leq \a e^{-\b x}$ for constants $\a,\b>0$, or $f(x)=0$ for $x>\n$ for some constant $\n>0$. Let $C(i,j),i,j\in[n]$ be the associated $n\times n$ cost matrix where $C(i,j)=\infty$ if $(i,j)\notin E$. We show that w.h.p. (a small modification to) 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. The algorithm runs in polynomial time.