Approximately Partitioning Vertices into Short Paths

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

💡 Research Summary

The paper studies the k⁻‑path partition (kPP) problem: given a simple undirected graph G = (V,E) and a fixed integer k ≥ 1, partition all vertices into vertex‑disjoint paths each containing at most k vertices, while minimizing the number of paths. Because |V| = |P| + |E(P)| for any k‑pp P, minimizing the number of paths is equivalent to maximizing the total number of edges used by the partition. The authors therefore introduce the complementary maximization problem k‑PPE, which seeks a k‑pp with the largest possible edge count. Lemma 1 shows that an α‑approximation for k‑PPE yields a ((1‑α)·k + α)‑approximation for the original kPP.

The main contributions are two approximation algorithms:

1. For k = 9 or 10 – a (k + 4)/5‑approximation for kPP, which corresponds to a 4/5‑approximation for k‑PPE.
2. For k ≥ 11 – a ((√11 − 2)/7)·k + (9 − √11)/7‑approximation for kPP, i.e., a (9 − √11)/7‑approximation for k‑PPE.

Both algorithms achieve the best known ratios for all k ∈ {9,…,18}.

Core Methodology

1. Maximum triangle‑free path‑cycle cover (F).
   The algorithm first computes a maximum triangle‑free path‑cycle cover F of G in O(n³m²) time (a 2‑matching without 3‑cycles). F spans all vertices, each component being either a path or a cycle of length at least 4. While F contains many edges (|E(F)| ≥ |E(Q)| for any optimal k‑PPE solution Q), it may include cycles or paths longer than k, making it infeasible for kPP.

2. Short cycles as the bottleneck.
   Any cycle in F has length ≥ 4. If F contains no 4‑cycles (for k = 9,10) or no 4‑/5‑cycles (for k ≥ 11), Lemma 3 guarantees that each component can be cut into k‑paths while preserving at least 4/5 (or 5/6) of its edges, directly yielding the desired approximation.

3. Auxiliary graph G′ and weight definition.
   When short cycles exist, the algorithm builds an auxiliary spanning subgraph G′ whose edges join vertices belonging to different connected components of F, with at least one endpoint lying on a short cycle. For a set E′ ⊆ E(G′), define the weight as the number of saturated 4‑cycles (k = 9,10) or the total number of saturated 4‑ and 5‑cycles (k ≥ 11). An edge “saturates” a short cycle if it is incident to any vertex of that cycle.

4. Maximum‑weight path‑cycle cover (W) in G′.
   The problem of finding a maximum‑weight path‑cycle cover W in G′ is reduced to a maximum‑weight **