Abstract
We design FPT-algorithms for the following problems. The first is List Digraph Homomorphism: given two digraphs G and H, with order n and h, respectively, and a list of allowed vertices of H for every vertex of G, does there exist a homomorphism from G to H respecting the list constraints? Let ℓ be the number of edges of G mapped to non-loop edges of H. The second problem is Min-Max Multiway Cut: given a graph G, an integer ℓ≥0, and a set T of r terminals, can we partition V(G) into r parts such that each part contains one terminal and there are at most ℓ edges with only one endpoint in this part? We solve both problems in time 2O(ℓ⋅logh+ℓ2⋅logℓ)⋅n4⋅logn and 2O((ℓr)2logℓr)⋅n4⋅logn, respectively, via a reduction to a new problem called List Allocation, which we solve adapting the randomized contractions technique of Chitnis et al. (2012) [4].
Highlights
The Multiway Cut problem asks, given a graph G, a set of r terminals T, and a non-negative integer, whether it is possible to partition V (G) into r parts such that each part contains exactly one of the terminals of T and there are at most edges between different parts
In this paper we use it in order to design FPT-algorithms for parameterizations of two problems that do not seem to be directly related to each other: the Min-Max-Multiway Cut problem [32] and the List Digraph Homomorphism problem
We introduce a new parameterization of List Digraph Homomorphism where the parameter is, apart from h = |V (H)|, the number of “crossing edges”, i.e., the edges of G whose endpoints are mapped to different vertices of H
Summary
The Multiway Cut problem asks, given a graph G, a set of r terminals T , and a non-negative integer , whether it is possible to partition V (G) into r parts such that each part contains exactly one of the terminals of T and there are at most edges between different parts (i.e., at most crossing edges). The existence of an FPT-algorithm for Multiway Cut (when parameterized by ), i.e., an f ( ) · nO(1)-step algorithm, had been a long-standing open problem This question was answered positively by Marx in [26] with the use of the important separators technique which was used for the design of FPT-algorithms for several other problems such as Directed Multiway Cut [4], Vertex Multicut, and Edge Multicut [28]. This technique has been extended to the powerful framework of randomized contractions technique, introduced in [5]. In this paper we use it in order to design FPT-algorithms for parameterizations of two problems that do not seem to be directly related to each other: the Min-Max-Multiway Cut problem [32] and the List Digraph Homomorphism problem
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
