eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2023-11-28
53:1
53:17
10.4230/LIPIcs.ISAAC.2023.53
article
On the Complexity of the Eigenvalue Deletion Problem
Misra, Neeldhara
1
https://orcid.org/0000-0003-1727-5388
Mittal, Harshil
1
Saurabh, Saket
2
3
https://orcid.org/0000-0001-7847-6402
Thakkar, Dhara
1
https://orcid.org/0000-0002-4234-0105
Indian Institute of Technology, Gandhinagar, India
Institute of Mathematical Sciences, Chennai, India
University of Bergen, Norway
For any fixed positive integer r and a given budget k, the r-Eigenvalue Vertex Deletion (r-EVD) problem asks if a graph G admits a subset S of at most k vertices such that the adjacency matrix of G⧵S has at most r distinct eigenvalues. The edge deletion, edge addition, and edge editing variants are defined analogously. For r = 1, r-EVD is equivalent to the Vertex Cover problem. For r = 2, it turns out that r-EVD amounts to removing a subset S of at most k vertices so that G⧵ S is a cluster graph where all connected components have the same size.
We show that r-EVD is NP-complete even on bipartite graphs with maximum degree four for every fixed r > 2, and FPT when parameterized by the solution size and the maximum degree of the graph.
We also establish several results for the special case when r = 2. For the vertex deletion variant, we show that 2-EVD is NP-complete even on triangle-free and 3d-regular graphs for any d ≥ 2, and also NP-complete on d-regular graphs for any d ≥ 8. The edge deletion, addition, and editing variants are all NP-complete for r = 2. The edge deletion problem admits a polynomial time algorithm if the input is a cluster graph, while - in contrast - the edge addition variant is hard even when the input is a cluster graph. We show that the edge addition variant has a quadratic kernel. The edge deletion and vertex deletion variants admit a single-exponential FPT algorithm when parameterized by the solution size alone.
Our main contribution is to develop the complexity landscape for the problem of modifying a graph with the aim of reducing the number of distinct eigenvalues in the spectrum of its adjacency matrix. It turns out that this captures, apart from Vertex Cover, also a natural variation of the problem of modifying to a cluster graph as a special case, which we believe may be of independent interest.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol283-isaac2023/LIPIcs.ISAAC.2023.53/LIPIcs.ISAAC.2023.53.pdf
Graph Modification
Rank Reduction
Eigenvalues