On the Complexity of the Eigenvalue Deletion Problem
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.
Graph Modification
Rank Reduction
Eigenvalues
Theory of computation~Design and analysis of algorithms
53:1-53:17
Regular Paper
https://arxiv.org/abs/2310.00600
We thank Daniel Lokshtanov for helpful discussions.
Neeldhara
Misra
Neeldhara Misra
Indian Institute of Technology, Gandhinagar, India
https://www.neeldhara.com
https://orcid.org/0000-0003-1727-5388
Supported by DST-SERB and IIT Gandhinagar.
Harshil
Mittal
Harshil Mittal
Indian Institute of Technology, Gandhinagar, India
Supported by IIT Gandhinagar.
Saket
Saurabh
Saket Saurabh
Institute of Mathematical Sciences, Chennai, India
University of Bergen, Norway
https://sites.google.com/view/sakethome
https://orcid.org/0000-0001-7847-6402
Supported by ERC, the University of Bergen, and IMSc.
Dhara
Thakkar
Dhara Thakkar
Indian Institute of Technology, Gandhinagar, India
https://sites.google.com/iitgn.ac.in/dharathakkar
https://orcid.org/0000-0002-4234-0105
Supported by CSIR-UGC NET JRF Fellowship.
10.4230/LIPIcs.ISAAC.2023.53
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Neeldhara Misra, Harshil Mittal, Saket Saurabh, and Dhara Thakkar
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