---
title: Maximum Weight 3-Path Packing Problem
url: https://www.emergentmind.com/topics/maximum-weight-3-path-packing-problem
type: topic
---

# Maximum Weight 3-Path Packing Problem

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 [2301.07537].

## 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 [2512.14457]:

| 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):

```python
# Subroutine 1: Matching of size n/2 (7/12-approximation framework)
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 [2301.07537]. 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 [2512.14457] 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 [2301.07537].

## 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 [2512.14457].

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 [2301.07537], 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 [2512.14457]:

- 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                 | $10/17$ ($\approx0.588$) | [2512.14457]          |
| Local-Search, Weight-Squared Potential (large swap) | $1.786$-approximation    | [2301.07537]          |
| Previous Matching-Based Method (2015)               | $7/12$ ($\approx0.583$)  | [2512.14457]          |

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

Source: https://www.emergentmind.com/topics/maximum-weight-3-path-packing-problem