Matching with Multiple Bottlenecks: Parameterized Complexity and Approximation

2026-10-05

AI Summary

This study addresses the multi-bottleneck matching problem, which involves determining and optimizing perfect matchings where edge weights are vectors and costs are defined as the sum of component-wise maxima. From a parameterized complexity perspective and motivated by scheduling applications, the theoretical analysis integrates reduction techniques with combinatorial optimization methods. The primary contributions include establishing fixed-parameter tractability (FPT) when jointly parameterized by k and Z, while revealing W[1]-hardness under single-parameter settings. Furthermore, this work proposes an efficient approximation scheme based on k and establishes a super-logarithmic lower bound for approximability. Overall, this research systematically completes the complexity classification framework for multi-bottleneck matching.

Abstract

We consider a matching problem in which the cost of each edge is a vector with $k$ components. The cost of a matching is the sum of the bottlenecks over all components, and we ask whether there is a perfect matching of cost at most some value $Z$. This type of matching has applications in heavily synchronized job-shop scheduling problems and in reconfiguration problems, where movement is restricted to a single direction per step. In this paper, we analyze the problem from a parameterized complexity perspective and provide various results including FPT-membership for parameters $k$ and $Z$ combined, as well as W[P]-membership and W[SAT]-hardness for each of the two parameters individually. The reduction also implies para-NP-hardness parameterized by either maximum degree or treewidth. We further show hardness of approximation within a super-logarithmic factor for the optimization variant and provide a $k/d$-approximation algorithm for any constant $d \leq k$ as well as an efficient approximation scheme parameterized by $k$. With parameter $Z$, we show that no FPT-time $F(Z)$-approximation algorithm is possible for any computable function $F$, unless W[1] = FPT.

Problem

• multi-bottleneck matching
• perfect matching
• parameterized complexity
• approximation hardness
• job-shop scheduling

Innovation

• Parameterized Complexity
• Multiple Bottlenecks Matching
• Approximation Algorithms
• W-hardness
• Efficient Parameterized Approximation Scheme