Abstract
In this paper, we study the NP-complete colorful variant of the classical Matching problem, namely, the Rainbow Matching problem. Given an edge-colored graph G and a positive integer k, this problem asks whether there exists a matching of size at least k such that all the edges in the matching have distinct colors. We first develop a deterministic algorithm that solves Rainbow Matching on paths in time $$\mathcal{O}^\star \left( \left( \frac{1+\sqrt{5}}{2}\right) ^k\right) $$ and polynomial space. This algorithm is based on a curious combination of the method of bounded search trees and a “divide-and-conquer-like” approach, where the branching process is guided by the maintenance of an auxiliary bipartite graph where one side captures “divided-and-conquered” pieces of the path. Our second result is a randomized algorithm that solves Rainbow Matching on general graphs in time $$\mathcal {O} ^\star (2^k)$$ and polynomial-space. Here, we show how a result by Bjorklund et al. (J Comput Syst Sci 87:119–139, 2017) can be invoked as a black box, wrapped by a probability-based analysis tailored to our problem. We also complement our two main results by designing kernels for Rainbow Matching on general and bounded-degree graphs.
Highlights
The classical notion of matching has been extensively studied for several decades in the area of Combinatorial Optimization [6, 14]
In the Maximum Matching problem, the objective is to find a matching of maximum size
Similar to the case of μ(P, S ∪ {uv}), k) = 0, we have reduced the problem into checking whether or not, there is a matching in B(S ∪ {uv})) that saturates S ∪ {uv}. 71:6 Parameterized Algorithms and Kernels for Rainbow Matching
Summary
The classical notion of matching has been extensively studied for several decades in the area of Combinatorial Optimization [6, 14]. Close to three decades later, Le and Pfender [13] revisited the computational complexity of this problem They showed that the Rainbow Matching problem is NP-complete even on (edge-colored) paths, complete graphs, P8-free trees in which every color is used at most twice, P5-free linear forests in which every color is used at most twice, and P4-free bipartite graphs in which every color is used at most twice. A parameterized problem Π is said to be FPT if there is an algorithm that solves it in time f (k) · |I|O(1), where |I| is the size of the input and f is a function that depends only on k. Such an algorithm is called a parameterized algorithm. The notation O (·) is used to hide factors polynomial in the input size
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