An Improved Approximation Algorithm for Maximum Weight 3-Path Packing

TL;DR

This work advances the MW3PP problem by achieving a 10/17-approximation on complete graphs with n divisible by 3, by combining three complementary strategies: a 7/12-guaranteed Alg.1 based on a large matching, a second approach Alg.2 anchored on a smaller matching with cost-based contraction, and a third Alg.3 leveraging a 2-star packing inside the contracted graph. A novel charging method provides the core local-structure proof technique, enabling tighter bounds and extending the analysis across the three algorithms. The final result emerges from a careful trade-off among the three methods, formalized via a linear program and its dual, with a feasible dual achieving the 10/17 bound. The findings contribute to the broader set-packing/3-path optimization literature, offering a robust framework that could extend to related MWkPP/MWkCP problems and inspire improvements in 2-star packing strategies.

Abstract

Given a complete graph with $n$ vertices and non-negative edge weights, where $n$ is divisible by 3, the maximum weight 3-path packing problem is to find a set of $n/3$ vertex-disjoint 3-paths such that the total weight of the 3-paths in the packing is maximized. This problem is closely related to the classic maximum weight matching problem. In this paper, we propose a $10/17$-approximation algorithm, improving the best-known $7/12$-approximation algorithm (ESA 2015). Our result is obtained by making a trade-off among three algorithms. The first is based on the maximum weight matching of size $n/2$, the second is based on the maximum weight matching of size $n/3$, and the last is based on an approximation algorithm for star packing. Our first algorithm is the same as the previous $7/12$-approximation algorithm, but we propose a new analysis method -- a charging method -- for this problem, which is not only essential to analyze our second algorithm but also may be extended to analyze algorithms for some related problems.

Figures (6)

• Figure 1: An illustration of Alg.1: In (a), each red edge has a weight of 1, each omitted edge has a weight of 0, and $M^*_{n/2}$ contains three red edges; In (b), each edge has a cost of -1 by definition and $M^{**}_{n/6}$ contains one blue edge; In (c), Alg.1 outputs two 3-paths with a weight of $2$.
• Figure 2: An illustration of the charging rule, where the edges in $M^*_{n/2}$ are represented by black edges and the edges in $E(P^*)$ are represented by red edges: In (a), we have a 3-path $xyz$ with $xy,yz\in E_1$, $e_x$, $e_y$ and $e_z$ are all charged by $\frac{2}{3}$ point by $xy$ and $yz$ in total; In (b), we have a 3-path $xyz$ with $xy\in E_2$, $e_x$ is charged by $\frac{2}{3}$ point by $xy$, and $e_y$ is charged by $\frac{1}{3}$ point by $xy$.
• Figure 3: An illustration of a cycle $C\in C_{E'}$ and a 3-path $xyz\in P^*$, where $xy\in E(C)$, $e_z$ is an end edge of a path $P\in P_{E'}$, the edges in $M^{*}_{n/2}$ are colored blue, the edges in $E'$ are colored red, and the omitted edges in $E(C)\cup E(P)$ are represented by dotted black edges.
• Figure 4: An illustration of Alg.2: In (a), $M^*_{n/3}$ contains three red edges; In (b), $M^{**}$ contains two blue edges; In (c), $G$ contains three residual vertices, and Alg.2 outputs three 3-paths.
• Figure 5: An illustration of the eight kinds of 3-paths in the optimal solution $P^*$, where the black edges represent the edges in $M^*_{n/3}$, the red edges represent the edges of 3-paths in $P^*$, and the 3-paths on the $i$-th column represent the 3-paths in $P^*_i$.

Theorems & Definitions (44)

• Lemma 1
• proof
• Lemma 2: DBLP:journals/dam/Bar-NoyPRV18, cf. theorem 1
• Lemma 3: DBLP:journals/dam/Bar-NoyPRV18
• proof
• Lemma 4
• proof
• Lemma 5
• proof
• Lemma 6