Maximum Weight 3-Path Packing Problem

• Maximum Weight 3-Path Packing is a combinatorial optimization problem that finds vertex-disjoint 3-paths in weighted graphs to maximize total path weight.
• Algorithmic strategies combine matching-based and local-search methods, achieving approximation ratios of 10/17 and 1.786 respectively.
• The problem underpins practical applications in resource allocation and scheduling while extending to broader k-set packing challenges with significant computational complexity.

The Maximum Weight 3-Path Packing Problem (MW3PP) is a fundamental combinatorial optimization problem defined on an edge-weighted undirected graph $G=(V,E)$, where $|V|=n$ and $n$ is divisible by 3. The objective is to compute a collection of $n/3$ vertex-disjoint 3-paths (simple paths each using three distinct vertices $x$–$y$–$z$, with combined weight $w(xy)+w(yz)$) such that every vertex is included in exactly one 3-path and the total sum of path weights is maximized. MW3PP generalizes matching, relates closely to weighted 3-set packing, and captures critical aspects of resource allocation, scheduling, and subgraph covering in combinatorial optimization.

1. Formal Problem Definition

Let $G=(V,E)$ be a complete graph with $n\equiv 0 \pmod{3}$, where each edge $e\in E$ has a nonnegative weight $w(e)\ge0$. A 3-path, also termed a 2-star in some contexts, is a subgraph induced by a triple of distinct vertices $x, y, z$ with edges $xy$ and $yz$. The weight of a 3-path $xyz$ is $w(xy)+w(yz)$. A perfect 3-path packing $P$ is a set of $n/3$ vertex-disjoint 3-paths covering all vertices. The goal is to find $P^*$ with maximal total weight: $\OPT = w(P^*) = \sum_{xyz\in P^*} [w(xy)+w(yz)]$ MW3PP is NP-hard: the unweighted version is a restriction of 3-dimensional matching, included in Karp's original 21 NP-complete problems (Thiery et al., 2023).

2. Algorithmic Approaches and Approximation Schemes

State-of-the-art algorithms for MW3PP achieve provable approximation ratios by combining methods from matching theory, hypergraph packing, and local search.

2.1 Matching-Based $10/17$-Approximation

The currently best-known combinatorial algorithm, attaining a $10/17$ approximation ratio, is constructed by taking the maximum over three distinct packing strategies. Each subroutine computes a candidate packing (denote $P_1$, $P_2$, $P_3$) and chooses the one of greatest total weight. The algorithm operates as follows (Zhao et al., 16 Dec 2025):

Subroutine	Principle	Key Steps
1: $n/2$-Matching	Matching contraction	Maximum matching of size $n/2$ $\rightarrow$ contraction, cost redefinition, cross-match expansion, greedy completion
2: $n/3$-Matching	Mixed contraction	Maximum matching of size $n/3$ $\rightarrow$ mixed contraction (matched + unmatched), cost function adapts, cross-matching, path expansion
3: Star Packing	2-star approximation	On vertices covered by optimal $n/3$-matching, apply $2/3$-approximation for weighted 2-star packing, convert output to valid 3-path packing

The following pseudocode captures Subroutine 1 (others analogous):

Input: Complete graph G=(V,E), |V|=n (n ≡ 0 mod 3) 1. M ← maximum-weight matching of size n/2 in G 2. Contract each e∈M to a supervertex → obtain G/M 3. For each edge uv in G/M, set c(uv) = w(uv) – min{w(e_u), w(e_v)} 4. Compute M' ← maximum-cost matching of size n/6 in G/M w.r.t. c 5. For each uv∈M', expand to 3-path from e_u, cross-edge uv; pair up remaining endpoints arbitrarily Return: Packing P1 by repeated expansion and matching of residuals

Each subroutine's design is underpinned by a detailed charging scheme (see Section 3) ensuring that at least one of $P_1,P_2,P_3$ achieves at least $10/17$ of optimum.

2.2 Local-Search $1.786$-Approximation (Weighted 3-Set Packing)

An alternative approach realizes MW3PP as an instance of weighted 3-set packing. Vertices correspond to candidate 3-paths; edges in the conflict graph connect pairs of 3-paths sharing a vertex, yielding a $4$-claw-free graph structure (Thiery et al., 2023). The local-search algorithm uses a weight-squared potential

$\Phi(A) = \sum_{v\in A} w_v^2$

for the packing $A$ (collection of vertex-disjoint 3-paths). The procedure searches for local exchanges of up to 39 3-paths that improve $\Phi$. This exchange-based method achieves a $1.786$-approximation guarantee for weighted MW3PP, surpassing previous bounds for this generality.

3. Analysis via Charging Schemes and Duality

The correctness and tightness of the $10/17$-approximation algorithm (Zhao et al., 16 Dec 2025) are established through an intricate charging scheme, which balances the contribution of selected matching and cross edges against the value of the optimal 3-path packing. Central to this machinery are

• Contracted graphs: Formed by contracting matched edges to supervertices.
• Edge cost functions: $c(uv)=w(uv)-\min\{w(e_u),w(e_v)\}$ following contraction.
• Charging arguments: Assigning portions of the optimum's value to components in the constructed packing, partitioned among edge types.

These ingredients yield for $P_1$ the bound $w(P_1)\geq \frac{7}{12}\OPT$, and via similar reasoning for $P_2$ and $P_3$, a polyhedral analysis shows the maximum of $\{w(P_1),w(P_2),w(P_3)\}$ must satisfy $w(P_i)\geq \frac{10}{17}\OPT$ for some $i$. This is certifiable by exhibiting a dual-feasible solution matching the lower bound.

The local-search analysis for the $1.786$-approximation is based on charging via the so-called "potential-slack" lemma, leveraging weight-squared potential to distribute "slack" among vertices and their claws, and grouping actions into trees or isolated structures in the exchange digraph. Finer lower-bounding of the aggregate slack $S$ underpins the improved constant (Thiery et al., 2023).

4. Computational Complexity

All matching and star-packing subroutines in the $10/17$-approximation run in $O(n^3)$ time (Edmonds' maximum matching, Babenkō–Nederlof's 2-star approximation), with $O(n^2)$ space to store the graph (Zhao et al., 16 Dec 2025).

For local-search, listings of all potential 3-paths takes $O(n^3)$ time per local improvement step; exhaustive search of exchange candidates is mitigated by pruning to feasible independent subsets.

These complexities are currently optimal among general polynomial-time combinatorial algorithms for this class of problems.

5. Connections to General Weighted $k$-Set Packing

MW3PP generalizes to the maximum weight $k$-set packing problem, in which disjoint subsets of size $k$ are selected for maximum total weight. For $k=3$, as shown in (Thiery et al., 2023), the same local-search paradigm (with weight-squared potential and O($k^3$)-size exchanges) achieves an approximation factor of $(3+1-\tau)/2$ with $\tau\approx0.214$, yielding the $1.786$ constant for $k=3$. As $k$ grows, similar analyses approach the bound $(k+1)/2$.

6. Extensions, Open Questions, and Research Directions

Several open problems remain in MW3PP and its relatives (Zhao et al., 16 Dec 2025):

• Whether the $3/4$-barrier for Maximum Weight 4-Path Packing can be broken using similar trade-off or charging-based methodologies.
• Whether improving the $2$-star-packing approximation factor beyond $2/3$ (currently used in subroutine 3) can further enhance the overall guarantee.
• Generalizing the local charging scheme developed here to Maximum Weight 3-Cycle Packing, or to larger values of $k$, such as $k$-path or $k$-cycle packings.
• For local-search methods: whether the factor $1.786$ for 3-path (3-set) packing can be further reduced, e.g., down to the single-claw configuration lower bound of $\sqrt{3}\approx1.732$, perhaps by combining swap strategies more aggressively.

A plausible implication is that the local-charging method—carefully accounting for the redistribution of weight from optimum 3-paths to matching edges—may find application in broader families of constrained-packing problems.

7. Summary Table: Approximation Results

Algorithm or Approach	Guarantee	Reference
Matching + Star-Packing Combination	$10/17$ ($\approx0.588$)	(Zhao et al., 16 Dec 2025)
Local-Search, Weight-Squared Potential (large swap)	$1.786$-approximation	(Thiery et al., 2023)
Previous Matching-Based Method (2015)	$7/12$ ($\approx0.583$)	(Zhao et al., 16 Dec 2025)

These results currently define the best-known performance guarantees for polynomial-time approximation of MW3PP in the weighted setting.