FPT Approximation for Constrained Metric k-Median/Means

Abstract

The Metric k-median problem over a metric space (𝒳, d) is defined as follows: given a set L ⊆ 𝒳 of facility locations and a set C ⊆ 𝒳 of clients, open a set F ⊆ L of k facilities such that the total service cost, defined as Φ(F, C) := ∑_{x ∈ C} min_{f ∈ F} d(x, f), is minimised. The metric k-means problem is defined similarly using squared distances (i.e., d²(., .) instead of d(., .)). In many applications there are additional constraints that any solution needs to satisfy. For example, to balance the load among the facilities in resource allocation problems, a capacity u is imposed on every facility. That is, no more than u clients can be assigned to any facility. This problem is known as the capacitated k-means/k-median problem. Likewise, various other applications have different constraints, which give rise to different constrained versions of the problem such as r-gather, fault-tolerant, outlier k-means/k-median problem. Surprisingly, for many of these constrained problems, no constant-approximation algorithm is known. Moreover, the unconstrained problem itself is known [Marek Adamczyk et al., 2019] to be W[2]-hard when parameterized by k. We give FPT algorithms with constant approximation guarantee for a range of constrained k-median/means problems. For some of the constrained problems, ours is the first constant factor approximation algorithm whereas for others, we improve or match the approximation guarantee of previous works. We work within the unified framework of Ding and Xu [Ding and Xu, 2015] that allows us to simultaneously obtain algorithms for a range of constrained problems. In particular, we obtain a (3+ε)-approximation and (9+ε)-approximation for the constrained versions of the k-median and k-means problem respectively in FPT time. In many practical settings of the k-median/means problem, one is allowed to open a facility at any client location, i.e., C ⊆ L. For this special case, our algorithm gives a (2+ε)-approximation and (4+ε)-approximation for the constrained versions of k-median and k-means problem respectively in FPT time. Since our algorithm is based on simple sampling technique, it can also be converted to a constant-pass log-space streaming algorithm. In particular, here are some of the main highlights of this work:  
1) For the uniform capacitated k-median/means problems our results matches previously known results of Addad et al. [Vincent Cohen-Addad and Jason Li, 2019]. 
2) For the r-gather k-median/means problem (clustering with lower bound on the size of clusters), our FPT approximation bounds are better than what was previously known. 
3) Our approximation bounds for the fault-tolerant, outlier, and uncertain versions is better than all previously known results, albeit in FPT time. 
4) For certain constrained settings such as chromatic, l-diversity, and semi-supervised k-median/means, we obtain the first constant factor approximation algorithms to the best of our knowledge. 
5) Since our algorithms are based on a simple sampling based approach, we also obtain constant-pass log-space streaming algorithms for most of the above-mentioned problems.