Parameterized Complexity of Maximum Edge Colorable Subgraph

• Published in: Algorithmica Vol. 84; no. 10; pp. 3075 - 3100
• Main Authors: Agrawal, Akanksha, Kundu, Madhumita, Sahu, Abhishek, Saurabh, Saket, Tale, Prafullkumar
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.10.2022; Springer Nature B.V
• Subjects: Algorithm Analysis and Problem Complexity; Algorithms; Apexes; Color coding; Complexity; Computer Science; Computer Systems Organization and Communication Networks; Data Structures and Information Theory; Graph coloring; Graph theory; Integer programming; Kernels; Linear programming; Matching; Mathematics of Computing; Parameterization; Theory of Computation; Kernel Lower Bound; Edge Coloring; FPT Algorithms; Kernelization
• ISSN: 0178-4617; 1432-0541
• DOI: 10.1007/s00453-022-01003-0

A graph H is p -edge colorable if there is a coloring ψ : E ( H ) → { 1 , 2 , ⋯ , p } , such that for distinct u v , v w ∈ E ( H ) , we have ψ ( u v ) ≠ ψ ( v w ) . The Maximum Edge-Colorable Subgraph problem takes as input a graph G and integers l and p , and the objective is to find a subgraph H of G and a p -edge-coloring of H , such that | E ( H ) | ≥ l . We study the above problem from the viewpoint of Parameterized Complexity. We obtain FPT algorithms when parameterized by: (1) the vertex cover number of G , by using Integer Linear Programming , and (2) l , a randomized algorithm via a reduction to Rainbow Matching , and a deterministic algorithm by using color coding, and divide and color. With respect to the parameters p + k , where k is one of the following: (1) the solution size, l , (2) the vertex cover number of G , and (3) l - mm ( G ) , where mm ( G ) is the size of a maximum matching in G ; we show that the (decision version of the) problem admits a kernel with O ( k · p ) vertices. Furthermore, we show that there is no kernel of size O ( k 1 - ϵ · f ( p ) ) , for any ϵ > 0 and computable function f , unless NP ⊆ coNP / poly .