A Faster Parameterized Algorithm for Temporal Matching
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A temporal graph is a sequence of graphs (called layers) over the same vertex set—describing a graph topology which is subject to discrete changes over time. A Δ-temporal matching M is a set of time edges (e,t) (an edge e paired up with a point in time t) such that for all distinct time edges (e,t),(e',t') ∈ M we have that e and e' do not share an endpoint, or the time-labels t and t' are at least Δ time units apart. Mertzios et al. [STACS '20] provided a 2^O(Δν)· |𝒢|^O(1)-time algorithm to compute the maximum size of Δ-temporal matching in a temporal graph 𝒢, where |𝒢| denotes the size of 𝒢, and ν is the Δ-vertex cover number of 𝒢. The Δ-vertex cover number is the minimum number of vertices which are needed to hit (or cover) all edges in any Δ consecutive layers of the temporal graph. We show an improved algorithm to compute a Δ-temporal matching of maximum size with a running time of Δ^O(ν)· |𝒢| and hence provide an exponential speedup in terms of Δ.
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