Constant-factor approximation for ordered k-median

Abstract

We study the Ordered k-Median problem, in which the solution is evaluated by first sorting the client connection costs and then multiplying them with a predefined non-increasing weight vector (higher connection costs are taken with larger weights). Since the 1990s, this problem has been studied extensively in the discrete optimization and operations research communities and has emerged as a framework unifying many fundamental clustering and location problems such as k-Median and k-Center. This generality, however, renders the problem intriguing from the algorithmic perspective and obtaining non-trivial approximation algorithms was an open problem even for simple topologies such as trees. Recently, Aouad and Segev were able to obtain an O(log n) approximation algorithm for Ordered k-Median using a sophisticated local-search approach and the concept of surrogate models thereby extending the result by Tamir (2001) for the case of a rectangular weight vector, also known as k-Facility p-Centrum. The existence of a constant-factor approximation algorithm, however, remained open even for the rectangular case.In this paper, we provide an LP-rounding constant-factor approximation algorithm for the Ordered k-Median problem.We obtain this result by revealing an interesting connection to the classic k-Median problem. We first provide a new analysis of the rounding process by Charikar and Li (2012) for k-Median, when applied to a fractional solution obtained from solving an LP relaxation over a non-metric, truncated cost vector, resulting in an elegant 15-approximation for the rectangular case. In our analysis, the connection cost of a single client is partly charged to a deterministic budget related to a combinatorial bound based on guessing, and partly to a budget whose expected value is bounded with respect to the fractional LP-solution. This approach allows us to limit the problematic effect of the variance of individual client connection costs on the value of the ranking-based objective function of Ordered k-Median. Next, we analyze objective-oblivious clustering, which allows us to handle multiple rectangles in the weight vector and obtain a constant-factor approximation for the case of O(1) rectangles. Then, we show that a simple weight bucketing can be applied to the general case resulting in O(log n) rectangles and hence in a constant-factor approximation in quasi-polynomial time. Finally, with a more involved argument, we show that also the clever distance bucketing by Aouad and Segev can be combined with the objective-oblivious version of our LP-rounding for the rectangular case, and that it results in a true, polynomial time, constant-factor approximation algorithm.

Approximation algorithms for minimum norm and ordered optimization problems

“…The theorem implies that (x, y) is an optimal solution to (OCl-P t ), then we obtain an O(1)-approximation for ordered k-median. We remark that Byrka et al [13] show that a randomized rounding procedure of Charikar and Li [19] for the standard k-median LP has the property that it produces an assignment-cost vector c satisfying Exp[ j∈D ( c j − 19ρ) + ] ≤ j,i (c ij − ρ) + x ij for every ρ ∈ R + ; that is, it gives a thresholdoblivious rounding for ℓ-centrum. Since cost(w; v) is a nonnegative linear combination of cost(ℓ; v) terms, this also gives a randomized weight-oblivious rounding for ordered k-median.…”

“…However, it is not at all clear how to use this LP for min-max ordered problems with multiple weight functions. The algorithms of Byrka et al [13] are randomized which bound the expected cost of the ordered k-median; with multiple weights, this won't help solve the min-max problem unless one can argue very sharp concentration properties of the algorithm. The same is true for our load-balancing algorithm.…”

“…For the special case of the (single) ordered k-median problem we can obtain a much better approximation factor. Specifically, this improves upon the factors from [16,13]. Our technique for this, however, is different from LP-rounding.…”

“…In k-clustering, the Top-ℓ optimization problem is called the ℓ-centrum problem, and the orderednorm minimization problem is called the ordered k-median problem. Only recently, a 38-factor [13] and 18 + ε-factor [16] approximation algorithm was given for the ordered k-median problem. We give a much improved (5 + ε)-factor approximation algorithm for the ordered k-median problem (Theorem 9.3).…”

“…From an approximation algorithms point of view, Tamir [42] gives the first O(log n)-approximation for the ℓ-centrum problem, and Aouad and Segev [5] give the first O(log n)approximations for the ordered k-median problem. Very recently, Byrka, Sornat, and Spoerhase [13] and our earlier paper [16] give the first constant-factor approximations for the ℓ-centrum and ordered k-median problems. Another recent relevant work is of Alamdari and Shmoys [2] who consider the k-centridian problem where the objective is a weighted average of the k-center and the k-median objective (a special case of the ordered k-median problem); [2] give a constant-factor approximation algorithm for this problem.…”