Approximately Partitioning Vertices into Short Paths
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866911420841459712 |
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| author | Gong, Mingyang Chen, Zhi-Zhong Mumey, Brendan |
| author_facet | Gong, Mingyang Chen, Zhi-Zhong Mumey, Brendan |
| contents | Given a fixed positive integer $k$ and a simple undirected graph $G = (V, E)$, the {\em $k^-$-path partition} problem, denoted by $k$PP for short, aims to find a minimum collection $\cal{P}$ of vertex-disjoint paths in $G$ such that each path in $\cal{P}$ has at most $k$ vertices and each vertex of $G$ appears in one path in $\cal{P}$. In this paper, we present a $\frac {k+4}5$-approximation algorithm for $k$PP when $k\in\{9,10\}$ and an improved $(\frac{\sqrt{11}-2}7 k + \frac {9-\sqrt{11}}7)$-approximation algorithm when $k \ge 11$. Our algorithms achieve the current best approximation ratios for $k \in \{ 9, 10, \ldots, 18 \}$.
Our algorithms start with a maximum triangle-free path-cycle cover $\cal{F}$, which may not be feasible because of the existence of cycles or paths with more than $k$ vertices. We connect as many cycles in $\cal{F}$ with $4$ or $5$ vertices as possible by computing another maximum-weight path-cycle cover in a suitably constructed graph so that $\cal{F}$ can be transformed into a $k^-$-path partition of $G$ without losing too many edges.
Keywords: $k^-$-path partition; Triangle-free path-cycle cover; $[f, g]$-factor; Approximation algorithm |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2602_03991 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Approximately Partitioning Vertices into Short Paths Gong, Mingyang Chen, Zhi-Zhong Mumey, Brendan Data Structures and Algorithms F.2.2 Given a fixed positive integer $k$ and a simple undirected graph $G = (V, E)$, the {\em $k^-$-path partition} problem, denoted by $k$PP for short, aims to find a minimum collection $\cal{P}$ of vertex-disjoint paths in $G$ such that each path in $\cal{P}$ has at most $k$ vertices and each vertex of $G$ appears in one path in $\cal{P}$. In this paper, we present a $\frac {k+4}5$-approximation algorithm for $k$PP when $k\in\{9,10\}$ and an improved $(\frac{\sqrt{11}-2}7 k + \frac {9-\sqrt{11}}7)$-approximation algorithm when $k \ge 11$. Our algorithms achieve the current best approximation ratios for $k \in \{ 9, 10, \ldots, 18 \}$. Our algorithms start with a maximum triangle-free path-cycle cover $\cal{F}$, which may not be feasible because of the existence of cycles or paths with more than $k$ vertices. We connect as many cycles in $\cal{F}$ with $4$ or $5$ vertices as possible by computing another maximum-weight path-cycle cover in a suitably constructed graph so that $\cal{F}$ can be transformed into a $k^-$-path partition of $G$ without losing too many edges. Keywords: $k^-$-path partition; Triangle-free path-cycle cover; $[f, g]$-factor; Approximation algorithm |
| title | Approximately Partitioning Vertices into Short Paths |
| topic | Data Structures and Algorithms F.2.2 |
| url | https://arxiv.org/abs/2602.03991 |