Approximately Partitioning Vertices into Short Paths

TL;DR

This work tackles the $k^-$-path partition problem ($k$PP) by introducing a two-stage augmentation that begins with a maximum triangle-free path-cycle cover and augments it with a maximum-weight path-cycle cover on a carefully constructed graph to produce a feasible $k^-$-path partition. By formulating and leveraging the related $k$PPE problem (maximize the number of edges in a $k$-pp) and showing that a good $k$PPE approximation translates into an $((1-\alpha)k+\alpha)$-approximation for $k$PP, the authors derive two concrete algorithms: a $(k+4)/5$-approximation for $k\in\{9,10\}$ and a $(\frac{\sqrt{11}-2}{7}k + \frac{9-\sqrt{11}}{7})$-approximation for $k\ge 11$. The methods achieve the best-known ratios for $k$ in $\{9,\dots,18\}$, and the analysis combines structural characterizations of components in $\mathcal{F}+\mathcal{W}$, saturation strategies for short cycles, and recursive refinement when necessary. The approach has potential practical impact for routing and monitoring tasks, while also inviting extensions to larger or directed graphs and exploration of inapproximability boundaries for constant $k$. All formulas are presented with their mathematical notation explicitly wrapped in $...$.

Abstract

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

Figures (9)

• Figure 1: The initial steps of our algorithms.
• Figure 2: The left graph shows a possible structure of a connected component $K$ in ${\cal {F}}+{\cal {W}}$ while the right graph shows $K[2]$. Each thick (respectively, dashed) edge is in ${\cal {F}}$ (respectively, ${\cal {W}}$). $K_c$ is a $3$-path and each satellite element of $K$ is a $4$-cycle in ${\cal {F}}$. Moreover, $v_i$ is a $(3-i)$-anchor for $i \in \{ 1, 2, 3 \}$.
• Figure 3: An example connected component $K$ of ${\cal {F}}+{\cal {W}}$ for Case 1 in the proof of Lemma \ref{['lemma08']}. Removing the two edges $\{ v_0, v_1 \} = \{ v_4, v_1 \}, \{ v_1, v_2 \}$ yields a graph whose connected components are $K[1]$ and $K[2, 4]$. $K[1]$ (respectively, $K[2, 4]$) has a $k$-pp with $4$ (respectively, $6$) edges.
• Figure 4: Possible structures of special connected components $K$ of ${\cal {F}}+{\cal {W}}$, where each thick (respectively, dashed) edge is in ${\cal {F}}$ (respectively, ${\cal {W}}$). The top row shows the structure of a type-1, type-2, or type-3 $K$ from left to right. The bottom row shows three possible structures of a type-$4$$K$, where the pair $(i, \ell)$ stated in Condition S4 is $(1, 2)$, $(2, 4)$ and $(4, 4)$, respectively.
• Figure 5: A balanced connected component of ${\cal {F}}+{\cal {W}}$, where each thick and dashed edge is in ${\cal {F}}$ and ${\cal {W}}$, respectively.

Theorems & Definitions (28)

• Lemma 1
• Lemma 2
• Lemma 3
• Definition 1
• Definition 2
• Lemma 4
• Lemma 5
• Definition 3
• Lemma 6
• Definition 4