Alternative parameterizations of Metric Dimension
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A set of vertices W in a graph G is called resolving if for any two distinct x,y∈ V(G), there is v∈ W such that dist_G(v,x)≠ dist_G(v,y), where dist_G(u,v) denotes the length of a shortest path between u and v in the graph G. The metric dimension md(G) of G is the minimum cardinality of a resolving set. The Metric Dimension problem, i.e. deciding whether md(G)< k, is NP-complete even for interval graphs (Foucaud et al., 2017). We study Metric Dimension (for arbitrary graphs) from the lens of parameterized complexity. The problem parameterized by k was proved to be W[2]-hard by Hartung and Nichterlein (2013) and we study the dual parameterization, i.e., the problem of whether md(G)< n- k, where n is the order of G. We prove that the dual parameterization admits (a) a kernel with at most 3k^4 vertices and (b) an algorithm of runtime O^*(4^k+o(k)). Hartung and Nichterlein (2013) also observed that Metric Dimension is fixed-parameter tractable when parameterized by the vertex cover number vc(G) of the input graph. We complement this observation by showing that it does not admit a polynomial kernel even when parameterized by vc(G) + k. Our reduction also gives evidence for non-existence of polynomial Turing kernels.
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