eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2024-09-23
39:1
39:20
10.4230/LIPIcs.ESA.2024.39
article
List Homomorphisms by Deleting Edges and Vertices: Tight Complexity Bounds for Bounded-Treewidth Graphs
Can Esmer, Barış
1
2
https://orcid.org/0000-0001-5694-1465
Focke, Jacob
1
https://orcid.org/0000-0002-6895-755X
Marx, Dániel
1
https://orcid.org/0000-0002-5686-8314
Rzążewski, Paweł
3
4
https://orcid.org/0000-0001-7696-3848
CISPA Helmholtz Center for Information Security, Saarbrücken, Germany
Saarbrücken Graduate School of Computer Science, Saarland Informatics Campus, Germany
Warsaw University of Technology, Poland
University of Warsaw, Poland
The goal of this paper is to investigate a family of optimization problems arising from list homomorphisms, and to understand what the best possible algorithms are if we restrict the problem to bounded-treewidth graphs. Given graphs G, H, and lists L(v) ⊆ V(H) for every v ∈ V(G), a list homomorphism from (G,L) to H is a function f:V(G) → V(H) that preserves the edges (i.e., uv ∈ E(G) implies f(u)f(v) ∈ E(H)) and respects the lists (i.e., f(v) ∈ L(v)). The graph H may have loops. For a fixed H, the input of the optimization problem LHomVD(H) is a graph G with lists L(v), and the task is to find a set X of vertices having minimum size such that (G-X,L) has a list homomorphism to H. We define analogously the edge-deletion variant LHomED(H), where we have to delete as few edges as possible from G to obtain a graph that has a list homomorphism. This expressive family of problems includes members that are essentially equivalent to fundamental problems such as Vertex Cover, Max Cut, Odd Cycle Transversal, and Edge/Vertex Multiway Cut.
For both variants, we first characterize those graphs H that make the problem polynomial-time solvable and show that the problem is NP-hard for every other fixed H. Second, as our main result, we determine for every graph H for which the problem is NP-hard, the smallest possible constant c_H such that the problem can be solved in time c^t_H⋅ n^{𝒪(1)} if a tree decomposition of G having width t is given in the input. Let i(H) be the maximum size of a set of vertices in H that have pairwise incomparable neighborhoods. For the vertex-deletion variant LHomVD(H), we show that the smallest possible constant is i(H)+1 for every H:
- Given a tree decomposition of width t of G, LHomVD(H) can be solved in time (i(H)+1)^t⋅ n^{𝒪(1)}.
- For any ε > 0 and H, an (i(H)+1-ε)^t⋅ n^{𝒪(1)} algorithm would violate the Strong Exponential-Time Hypothesis (SETH).
The situation is more complex for the edge-deletion version. For every H, one can solve LHomED(H) in time i(H)^t⋅ n^{𝒪(1)} if a tree decomposition of width t is given. However, the existence of a specific type of decomposition of H shows that there are graphs H where LHomED(H) can be solved significantly more efficiently and the best possible constant can be arbitrarily smaller than i(H). Nevertheless, we determine this best possible constant and (assuming the SETH) prove tight bounds for every fixed H.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol308-esa2024/LIPIcs.ESA.2024.39/LIPIcs.ESA.2024.39.pdf
Graph Homomorphism
List Homomorphism
Vertex Deletion
Edge Deletion
Multiway Cut
Parameterized Complexity
Tight Bounds
Treewidth
SETH