Approximability results for the p-centdian and the converse centdian problems
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Given an undirected graph G=(V,E,l) with a nonnegative edge length function l, and two integers p, λ, 0<p<|V|, 0≤λ≤ 1, let V^' be a subset of V with |V^'|=p. For each vertex v ∈ V, we let d(v,V^') denote the shortest distance from v to V^'. An eccentricity _C(V^') of V^' denotes the maximum distance of d(v,V^') for all v ∈ V. A median-distance _M(V^') of V^' denotes the total distance of d(v,V^') for all v ∈ V. The p-centdian problem is to find a vertex set V^' of V with |V^'|=p, such that λ_C(V^')+(1-λ) _M(V^') is minimized. The vertex set V^' is called as the centdian set and λ_C(V^')+(1-λ) _M(V^') is called as the centdian-distance. If we converse the two criteria, that is given the bound U of the centdian-distance and the objective function is to minimize the cardinality of the centdian set, this problem is called as the converse centdian problem. In this paper, we prove the p-centdian problem is NP-Complete even when the centdian-distance is _C(V^')+_M(V^'). Then we design the first non-trivial brute force exact algorithms for the p-centdian problem and the converse centdian problem, respectively. Finally, we design a (1+ϵ)-approximation (respectively, (1+1/ϵ)(ln|V|+1)-approximation) algorithm for the p-centdian problem (respectively, converse centdian problem) satisfying the cardinality of centdian set is less than or equal to (1+1/ϵ)(ln|V|+1)p (respectively, (1+ϵ)U), in which ϵ>0.
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