Karp's patching algorithm on dense digraph
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866910971035910144 |
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| author | Frieze, Alan |
| author_facet | Frieze, Alan |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_21645 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Karp's patching algorithm on dense digraph Frieze, Alan Combinatorics Discrete Mathematics 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. |
| title | Karp's patching algorithm on dense digraph |
| topic | Combinatorics Discrete Mathematics |
| url | https://arxiv.org/abs/2505.21645 |