Abstract
Given an undirected graph \(G=(V,E)\), the k-path partition problem is to find a collection of vertex-disjoint paths containing at most k vertices to cover all the vertices of V. The objective is to minimize the number of paths in the collection. For the k-path partition problem with \(k\ge 3\), we propose a simple local search algorithm, whose approximation ratio improves on the best-known approximation algorithm in Chen (in: Chen, Li, Zhang (eds) Frontiers of algorithmics, Springer, Cham, 2022) for every \(k\ge 4\), especially for \(k=4,5,6,7\). In addition, we give examples to show that our algorithm is tight when k is odd. When k is even, we give almost tight examples.
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Notes
Because any path containing more than three vertices can always be broken into 2-paths and 3-paths by removing properly some edges.
To do this efficiently, we assign a label q to each vertex v if v is an endpoint of some q-path of the current solution. If v is not an endpoint of any path of the current solution, its label is defined as 0. Note that for each execution of Operation 1 we need only update the labels of at most six vertices, i.e. the four endpoints of the two paths being merged and (possilby) another two endpoints of the resulting two new paths.
We assume that w.l.o.g. the edge \(\{u,v\}\) connects a q-path and a \(q'\)-path with \(q,q'\in \{2,3,\ldots ,k-2\}\). If one of \(q,q'\) equals 2, Case 1 of Operation 1 happens since \(q+q'\le 2+(k-2)=k\). Otherwise, one can verify that one of the three cases of Operation 1 occurs.
This inequality holds trivially if \(i\bmod 3 \le k\bmod 3\). Otherwise, it holds that either
\(i\bmod 3=1> k\bmod 3=0\) or \(i\bmod 3=2> k\bmod 3\in \{0,1\}\), which implies \(\left\lfloor \frac{i}{3} \right\rfloor \le \left\lfloor \frac{k}{3} \right\rfloor -1\). Thus,
$$\begin{aligned} 2\cdot \left\lfloor \frac{i}{3} \right\rfloor + i\bmod 3 \le 2\cdot (\left\lfloor \frac{k}{3} \right\rfloor -1) + i \bmod 3 \le 2\cdot \left\lfloor \frac{k}{3} \right\rfloor . \end{aligned}$$
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Acknowledgements
The authors are grateful to the anonymous referees for their valuable and constructive comments. This research is supported by the National Natural Science Foundation of China under Grant Numbers 11671135, 11871213 and the Natural Science Foundation of Shanghai under Grant Number 19ZR1411800.
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Li, S., Yu, W. & Liu, Z. A local search algorithm for the k-path partition problem. Optim Lett 18, 279–290 (2024). https://doi.org/10.1007/s11590-023-01989-8
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DOI: https://doi.org/10.1007/s11590-023-01989-8