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The most well known and ubiquitous clustering problem encountered in nearly every branch of science is undoubtedly k-means: given a set of data points and a parameter k, select k centres and partition the data points into k clusters around these centres so that the sum of squares of distances of the points to their cluster centre is minimized. Typically these data points lie in Euclidean space R d for some d ≥ 2.k-means and the first algorithms for it were introduced in the 1950's. Over the last six decades, hundreds of papers have studied this problem and different algorithms have been proposed for it. The most commonly used algorithm in practice is known as Lloyd-Forgy, which is also referred to as "the" k-means algorithm, and various extensions of it often work very well in practice. However, they may produce solutions whose cost is arbitrarily large compared to the optimum solution. Kanungo et al. [2004] analyzed a very simple local search heuristic to get a polynomial-time algorithm with approximation ratio 9 + for any fixed > 0 for k-means in Euclidean space.Finding an algorithm with a better worst-case approximation guarantee has remained one of the biggest open questions in this area, in particular whether one can get a true PTAS for fixed dimension Euclidean space. We settle this problem by showing that a simple local search algorithm provides a PTAS for k-means for R d for any fixed d.More precisely, for any error parameter > 0, the local search algorithm that considers swaps of up to ρ = d O(d) · −O(d/ ) centres at a time will produce a solution using exactly k centres whose cost is at most a (1 + )-factor greater than the optimum solution. Our analysis extends very easily to the more general settings where we want to minimize the sum of q'th powers of the distances between data points and their cluster centres (instead of sum of squares of distances as in k-means) for any fixed q ≥ 1 and where the metric may not be Euclidean but still has fixed doubling dimension.Finally, our techniques also extend to other classic clustering problems. We provide the first demonstration that local search yields a PTAS for uncapacitated facility location and the generalization of k-median to the setting with non-uniform opening costs in doubling metrics.

We develop polynomial-size LP-relaxations for orienteering and the regret-bounded vehicle routing problem (RVRP) and devise suitable LP-rounding algorithms that lead to various new insights and approximation results for these problems. In orienteering, the goal is to find a maximum-reward r-rooted path, possibly ending at a specified node, of length at most some given budget B. In RVRP, the goal is to find the minimum number of r-rooted paths of regret at most a given bound R that cover all nodes, where the regret of an r-v path is its length − c rv .For rooted orienteering, we introduce a natural bidirected LP-relaxation and obtain a simple 3-approximation algorithm via LP-rounding. This is the first LP-based guarantee for this problem. We also show that point-to-point (P2P) orienteering can be reduced to a regret-version of rooted orienteering at the expense of a factor-2 loss in approximation. For RVRP, we propose two compact LPs that lead to significant improvements, in both approximation ratio and running time, over the approach in [14]. One of these is a natural modification of the LP for rooted orienteering; the other is an unconventional formulation that is motivated by certain structural properties of an RVRP-solution, which leads to a 15-approximation algorithm for RVRP.

We consider the mobile facility location (MFL) problem. We are given a set of facilities and clients located in a common metric space G = (V, c). The goal is to move each facility from its initial location to a destination (in V ) and assign each client to the destination of some facility so as to minimize the sum of the movementcosts of the facilities and the client-assignment costs. This abstracts facility-location settings where one has the flexibility of moving facilities from their current locations to other destinations so as to serve clients more efficiently by reducing their assignment costs.We give the first local-search based approximation algorithm for this problem and achieve the best-known approximation guarantee. Our main result is (3 + )-approximation for this problem for any constant > 0 using local search. The previous best guarantee for MFL was an 8-approximation algorithm due to Friggstad and Salavatipour [12] based on LP-rounding. Our guarantee matches the best-known approximation guarantee for the k-median problem. Since there is an approximation-preserving reduction from the k-median problem to MFL, any improvement of our result would imply an analogous improvement for the k-median problem. Furthermore, our analysis is tight (up to o(1) factors) since the tight example for the local-search based 3-approximation algorithm for k-median can be easily adapted to show that our local-search algorithm has a tight approximation ratio of 3. Our results extend to the weighted generalization wherein each facility i has a non-negative weight w i and the movement cost for i is w i times the distance traveled by i.In contrast to the k-median problem, the local search procedure that moves, at each step, a constant number of facilities (to chosen destinations) and assigns each client to the nearest destination, is known to * {sahmadian,zfriggstad,cswamy}@math.uwaterloo.ca. Dept. of Combinatorics and Optimization, Univ. Waterloo, Waterloo, ON N2L 3G1. Supported in part by NSERC grant 327620-09. The second and third authors are also supported by the third author's Ontario Early Researcher Award.have an unbounded locality gap. Our local-search algorithm is a natural and simple variant, where we only select the destinations of the facilities in each step and optimally rematch the facilities to these destinations (which might entail moving all facilities). One of the chief novelties in the analysis is that in order to generate a suitable collection of local-search moves whose resulting inequalities yield the desired bound on the cost of a local-optimum, we define a treelike structure that (loosely speaking) functions as a "recursion tree", using which we spawn off local-search moves by exploring this tree to a constant depth.

We consider vehicle-routing problems (VRPs) that incorporate the notion of regret of a client, which is a measure of the waiting time of a client relative to its shortest-path distance from the depot. Formally, we consider both the additive and multiplicative versions of, what we call, the regret-bounded vehicle routing problem (RVRP). In these problems, we are given an undirected complete graph G = ({r} ∪ V, E) on n nodes with a distinguished root (depot) node r, edge costs {cuv} that form a metric, and a regret bound R. Given a path P rooted at r and a node v ∈ P , let cP (v) be the distance from r to v along P . The goal is to find the fewest number of paths rooted at r that cover all the nodes so that for every node v covered by (say) path P : (i) its additive regret cP (v) − crv, with respect to P is at most R in additive-RVRP; or (ii) its multiplicative regret, cP (v)/crv, with respect to P is at most R in multiplicative-RVRP.Our main result is the first constant-factor approximation algorithm for additive-RVRP. This is a substantial improvement over the previous-best O(log n)-approximation. Additive-RVRP turns out be a rather central vehicle-routing problem, whose study reveals insights into a variety of other regret-related problems as well as the classical distance-constrained VRP (DVRP), enabling us to obtain guarantees for these various problems by leveraging our algorithm for additive-RVRP and the underlying techniques. We obtain approximation ratios of O log( Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than the author(s) must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. bound D via reductions to additive-RVRP; the latter improves upon the previous-best approximation for DVRP. A noteworthy aspect of our results is that they are obtained by devising rounding techniques for a natural configuration-style LP. This furthers our understanding of LP-relaxations for VRPs and enriches the toolkit of techniques that have been utilized for configuration LPs.