K-median Algorithms: Theory in Practice

We define the distance metric as dij for i ∈ {1, . . . , n}, j ∈ {1, . . . , n}, such that dij is the distance between points i and j in the metric space X. Kariv and Hakim [1] proved that finding such k medians in a network is an NP-hard problem by reducing the dominating set problem to it. A simple bruteforce algorithm would examine every possible size-k subset in F , compute the closest facility in this set for every client, and return the best set overall. This brute-force algorithm would run in O (( nf k ) nck ) time, where |F | = nf , |C| = nc. Thus, academic research into this problem has focused primarily on producing good approximation algorithms. For a given algorithm, the approximation ratio is defined as the provably worst possible ratio between the cost of the algorithm’s output and the optimal cost. However, for most problem instances, we do not know the actual optimal cost, and we thus compute the approximation ratio as the total cost returned by the algorithm divided by the optimal value of the relaxed linear program discussed in section 1.2. Jain et al. [2] proved that the k-median problem is 1 + 2e ≈ “1.736”-hard to approximate in a metric space. We note that, throughout this paper, all of our distance metrics satisfy the properties of a metric space.