[Paper Review] Interpolating between $k$-Median and $k$-Center: Approximation Algorithms for Ordered $k$-Median

Deeparnab Chakrabarty, Chaitanya Swamy|arXiv (Cornell University)|Nov 23, 2017

Facility Location and Emergency Management11 citations

TL;DR

This paper presents an (18+ε)-approximation algorithm for the ordered k-median problem, a generalization of k-median and k-center that minimizes a weighted sum of ordered assignment costs. It leverages Lagrangian relaxation, primal-dual methods, and an enumeration technique to achieve this guarantee, with a tighter (8.5+ε)-approximation for the special case of {0,1} weights via a novel reduction to standard k-median.

ABSTRACT

We consider a generalization of $k$-median and $k$-center, called the {\\em ordered $k$-median} problem. In this problem, we are given a metric space $(\\mathcal{D},\\{c_{ij}\\})$ with $n=|\\mathcal{D}|$ points, and a non-increasing weight vector $w\\in\\mathbb{R}_+^n$, and the goal is to open $k$ centers and assign each point each point $j\\in\\mathcal{D}$ to a center so as to minimize $w_1\\cdot\ ext{(largest assignment cost)}+w_2\\cdot\ ext{(second-largest assignment cost)}+\\ldots+w_n\\cdot\ ext{($n$-th largest assignment cost)}$. We give an $(18+\\epsilon)$-approximation algorithm for this problem. Our algorithms utilize Lagrangian relaxation and the primal-dual schema, combined with an enumeration procedure of Aouad and Segev. For the special case of $\\{0,1\\}$-weights, which models the problem of minimizing the $\\ell$ largest assignment costs that is interesting in and of by itself, we provide a novel reduction to the (standard) $k$-median problem showing that LP-relative guarantees for $k$-median translate to guarantees for the ordered $k$-median problem; this yields a nice and clean $(8.5+\\epsilon)$-approximation algorithm for $\\{0,1\\}$ weights.

Motivation & Objective

• To develop a constant-factor approximation algorithm for the ordered k-median problem, which generalizes both k-median and k-center by minimizing a weighted sum of ordered assignment costs.
• To address the limited prior work on approximation algorithms for ordered k-median, especially in general metrics.
• To provide a clean (8.5+ε)-approximation for the special case of {0,1} weights, where the goal is to minimize the ℓ largest assignment costs.
• To extend techniques from the {0,1} case to the general weighted setting using novel LP relaxations and enumeration.
• To achieve improved approximation guarantees compared to prior work, including a better bound than the concurrent result of Byrka et al. (38+ε) for the {0,1} case.

Proposed method

• Use a novel LP relaxation for the {0,1}-weight case that avoids the Ω(ℓ) integrality gap of the natural relaxation.
• Apply a primal-dual framework combined with Lagrangian relaxation to handle the ordered cost structure.
• Employ an enumeration procedure inspired by Aouad and Segev to explore candidate solutions based on estimated optimal values.
• For the general weighted case, use a two-stage approach: first estimate the largest cost M and the weight vector w^est, then solve a modified LP to select centers.
• Introduce a function g_{M,w^est}(9; c_j) to bound the contribution of each client’s cost in the final solution relative to the estimated optimal.
• Combine theoretical guarantees from multiple lemmas and claims to bound the total cost within (18+O(ε)) times the optimal.

Experimental results

Research questions

• RQ1 Can a constant-factor approximation be achieved for the general ordered k-median problem, which interpolates between k-median and k-center?
• RQ2 What is the best possible approximation ratio for the {0,1}-weight variant of ordered k-median, where the goal is to minimize the ℓ largest assignment costs?
• RQ3 How can LP-based techniques like primal-dual and Lagrangian relaxation be adapted to handle the non-separable, ordered cost function?
• RQ4 Can a reduction from the {0,1}-weight ordered k-median problem to standard k-median preserve approximation guarantees?
• RQ5 What is the trade-off between approximation ratio and algorithmic complexity in the general weighted setting?

Key findings

• The paper achieves an (18+ε)-approximation for the general ordered k-median problem using a novel combination of primal-dual, Lagrangian relaxation, and enumeration.
• For the {0,1}-weight case, the authors obtain a (8.5+ε)-approximation by reducing the problem to standard k-median, leveraging LP-relative guarantees.
• The algorithm uses a carefully constructed LP relaxation that avoids the Ω(ℓ) integrality gap of the natural relaxation for the ℓ-centrum problem.
• The analysis shows that the solution cost is bounded by 9 times the estimated optimal cost plus lower-order terms, which are then amplified and combined with other bounds.
• The final approximation ratio of (18+O(ε)) is derived by combining the estimated cost bound with the optimal solution cost and error terms from weight estimation.
• The approach improves upon the concurrent result of Byrka et al. (38+ε) for the {0,1} case, using simpler techniques and a better approximation factor.