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Layering and reverse extension for weighted $k$-path vertex cover
Jiajing Wu, An Zhang, Shiming Li, Guohui Lin
TL;DR
Weighted k-path vertex cover asks for a minimum-weight set of vertices meeting every simple path on k vertices. This paper designs local cost functions within Layering and Reverse Extension, obtaining parity-dependent approximation ratios that match the known special-case ratios for k = 3 and k = 4.
Problem
Weighted k-path vertex cover seeks a minimum-weight vertex set intersecting every simple path on k vertices.
Source: §1 · p. 2Method
The algorithm applies carefully designed local cost functions in the Layering and Reverse Extension local-ratio framework.
Source: §1.2 · p. 3, §1.2.1 · p. 3Results
The algorithm has ratio 3k−1 over 4 for odd k ≥3 and 3k over 4 for even k ≥4, matching the state of the art at k = 3 and k = 4.
Source: §1.2.2 · p. 3, §1.2 · p. 3Takeaways & Limitations
For fixed k ≥3, the paper gives a polynomial-time approximation algorithm for weighted k-path vertex cover.
Source: §1.2.2 · p. 3Both the dynamic programming and Layering and Reverse Extension algorithms have running times exponential in k, so k must be fixed.
Source: §5 · p. 12
Abstract
from arXiv · showhide
We study the weighted $k$-path vertex cover problem for any fixed integer $k \ge 2$, in which a vertex non-negatively weighted simple graph $G = (V, E)$ is given and one wants to find a minimum-weight vertex subset $C \subseteq V$ such that every simple path on $k$ vertices in $G$ contains at least one vertex of $C$. The problem generalizes the classic weighted vertex cover problem, that is, $k = 2$, and it admits a $k$-approximation algorithm of Hochbaum~[SIAM J. Comput., 1982] and a $(k - \frac 12)$-approximation algorithm of Br\"ustle et al.~[IWOCA 2021]. In this paper, we carefully design local weight functions in the Layering and Reverse Extension local ratio framework of Bar-Yehuda et al.~[ACM Comput. Surv., 2004], leading to an approximation algorithm of ratio $\frac {3k-1}4$ for odd $k \ge 3$ and of ratio $\frac {3k}4$ for even $k \ge 4$. When $k = 3$ and $k = 4$, these ratios are $2$ and $3$, respectively, matching the state-of-the-art approximation ratios for the corresponding special cases.
1 Introduction
The paper studies minimum-weight vertex sets intersecting every k-vertex path and develops a Layering and Reverse Extension approximation algorithm for fixed k. Its parity-dependent bounds improve the general prior ratio and match the known ratios for k = 3 and k = 4.
1 Introduction
- For fixed integer k ≥3, weighted kPVC minimizes the total weight of its selected vertices, and the weighted problem is NP-hard for every k ≥3.Source: §1 · p. 2, §1.1 · p. 2
1.1 Previous work
- Earlier general algorithms achieved ratios k and k−1/2, the latter being the only improvement since the weighted problem's formal study.The k-approximation follows from low-frequency weighted set cover.Source: §1.1 · p. 2
1.2 Our contribution
- The contribution is a local-ratio algorithm using carefully designed local cost functions, with ratios 2 for k = 3 and 3 for k = 4, matching the known special-case state of the art.The framework adopted is Layering and Reverse Extension.Source: §1.2 · p. 3
1.2.1 The algorithm framework
- Layering recursively subtracts locally weighted maximal-arm-spider layers, then reverse extension restores removed vertices after the remaining spider-free graph is solved exactly.The local-cost analysis bounds each layer's solution weight relative to its local optimum.Source: §1.2.1 · p. 3
1.2.2 The main result and paper organization
- The algorithm achieves a 3k−1 over 4 approximation ratio for odd k and 3k over 4 for even k, for every fixed k ≥3.Theorem 1 states the algorithm runs in polynomial time.Source: §1.2.2 · p. 3
2 Preliminaries
The preliminaries define the spider structures used to build layers and establish structural facts that support local-cost design. They also define minimal covers through witness paths.
Notation
- OPT(G, w) denotes the minimum total weight among all k-path vertex covers of G.Source: §2 · p. 3
Three-arm spider
- The base three-arm spider Tk is S(m, m, m) for odd k and S(m−1, m, m) for even k; graphs without Tk are called spider-free.The spider is a three-path tree whose paths share one endpoint, the center.Source: §2 · p. 4
Maximal-arm spider
- A maximal-arm spider starts from a three-arm spider and greedily appends eligible paths ending at its center until no further arm can be added.Its construction need not be unique, and the number of arms can depend on which arms are found.Source: §2 · p. 4
Structural properties
- If a cover omits the center of a p-arm spider, it must include a vertex from at least p−1 arms.For a positively weighted maximal-arm spider, a cover containing all its vertices can be reduced by removing at least one spider vertex.Source: §2 · pp. 4–5
Minimal kPVC
- A k-path vertex cover is minimal exactly when every selected vertex has a witness k-path intersecting the cover only at that vertex.Removing that vertex then leaves its witness path uncovered.Source: §2 · p. 5
3 A polynomial-time exact algorithm for spider-free graphs
The paper gives an exact polynomial-time algorithm for weighted k-path vertex cover on spider-free graphs. It bounds their pathwidth as a function of k and uses dynamic programming on a bounded-width path decomposition.
3.2 A dynamic programming exact algorithm
- A bounded-width path decomposition supports a dynamic program that tracks deleted vertices in the current bag and valid partial k-paths in the residual graph.The algorithm updates these states as vertices are introduced or forgotten, retaining minimum-weight representatives for backtracking.Source: §3 · p. 5, §3.2 · pp. 7–8
3.1 Bounded pathwidth and a path decomposition
- Spider-free graphs exclude the three-arm spider Tk as a minor, yielding an explicit pathwidth bound depending only on k.The bound is (3k−5)/2 for odd k and (3k−4)/2 for even k.Source: §3.1 · pp. 5–6
3.2 A dynamic programming exact algorithm
- The algorithm solves weighted k-path vertex cover exactly on spider-free graphs in n·f(k) time for fixed k.The dynamic program returns a minimum-weight cover by backtracking from a minimum-weight state at the final empty bag.Source: §3.2 · p. 8
4 Algorithmic framework of layering and reverse extension
The LRE algorithm repeatedly removes weighted maximal-arm spider layers, solves the spider-free remainder exactly, then restores vertices to obtain a minimal kPVC. Its local-ratio analysis gives the stated approximation guarantee for fixed k.
Spider-free case
- When the residual graph is spider-free, the algorithm uses dynamic programming to solve that instance exactly.The exact algorithm is based on a path decomposition of bounded width.Source: §4 · pp. 9, 11
Layering and reverse extension
- LRE removes a maximal-arm spider layer, recursively solves the residual graph, then restores zero-weight vertices and adds a restored vertex only when needed for feasibility.The reverse-extension process returns a minimal kPVC for the original graph.Source: §4 · p. 9
Reverse extension
- Reverse extension preserves minimality: each restored vertex is added only if the current set would otherwise fail to be a kPVC.The resulting set remains a minimal kPVC after each restoration.Source: §4 · p. 9
Local spider bounds
- A p-arm spider has local optimum at least p−1, while every minimal global kPVC uses local cost at most c_H(V(H))−1 on the layer, supporting the good-layer ratio.The local-ratio theorem combines this layer condition with exact solution of spider-free graphs.Source: §4 · p. 10
Running time
- The running time is O(n^(3k/2+1)·poly(k) + n·f(k)); because the algorithms are exponential in k, k must be fixed for polynomial time.Here f(k) is exponential in k.Source: §4 · p. 11
Approximation guarantee
- For every fixed k ≥3, LRE is a polynomial-time ρ_k-approximation, with ρ_k = (3k−1)/4 for odd k and 3k/4 for even k.The guarantee follows when each maximal-arm spider layer is ρ-good and the spider-free case is solved exactly.Source: §4 · pp. 10–11
5 Conclusion
The paper studies approximation algorithms for weighted k-path vertex cover and uses local ratio with layering and reverse extension. It reports a ratio of at most 3k/4, while noting that the algorithms require fixed k.
Approach
- The approach designs maximal-arm spider layers in the Layering and Reverse Extension framework and solves spider-free graphs exactly using dynamic programming based on bounded-width path decompositions.Source: §5 · p. 12
Approximation result
- The algorithm achieves an approximation ratio of at most 3k/4, improving on the previous best ratio k−1/2 and the earlier ratio k.The paper also describes a maximal-arm spider approximation incorporated into the layering framework.Source: §5 · p. 12
Scope and open question
- Both the dynamic programming and LRE algorithms are exponential in k, so their running time is not necessarily polynomial when k is part of the input.The paper identifies improving the ratio to o(k), if possible, as an open question.Source: §5 · p. 12