Minimum Consistent Subset in Trees and Interval Graphs
arxiv(2024)
摘要
In the Minimum Consistent Subset (MCS) problem, we are presented with a
connected simple undirected graph G=(V,E), consisting of a vertex set V of
size n and an edge set E. Each vertex in V is assigned a color from the
set {1,2,…, c}. The objective is to determine a subset V' ⊆
V with minimum possible cardinality, such that for every vertex v ∈ V, at
least one of its nearest neighbors in V' (measured in terms of the hop
distance) shares the same color as v. The decision problem, indicating
whether there exists a subset V' of cardinality at most l for some positive
integer l, is known to be NP-complete even for planar graphs.
In this paper, we establish that the MCS problem for trees, when the number
of colors c is considered an input parameter, is NP-complete. We propose a
fixed-parameter tractable (FPT) algorithm for MCS on trees running in
O(2^6cn^6) time, significantly improving the currently best-known algorithm
whose running time is O(2^4cn^2c+3).
In an effort to comprehensively understand the computational complexity of
the MCS problem across different graph classes, we extend our investigation to
interval graphs. We show that it remains NP-complete for interval graphs, thus
enriching graph classes where MCS remains intractable.
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