[Paper Review] Constant factor FPT approximation for capacitated k-median

Marek Adamczyk, Jarosław Byrka+3 more|arXiv (Cornell University)|Sep 16, 2018

Computational Geometry and Mesh Generation15 references8 citations

TL;DR

This paper presents the first FPT-time (7 + ε)-approximation algorithm for the capacitated k-median problem that respects both the cardinality (k facilities) and capacity constraints. It combines metric tree embedding, clustering into ℓ = O(k log n/ε) centers, and a parameterized dynamic programming approach on ℓ-centered instances to achieve a constant-factor approximation in time 2^{O(k log k)}n^{O(1}).

ABSTRACT

Capacitated k-median is one of the few outstanding optimization problems for which the existence of a polynomial time constant factor approximation algorithm remains an open problem. In a series of recent papers algorithms producing solutions violating either the number of facilities or the capacity by a multiplicative factor were obtained. However, to produce solutions without violations appears to be hard and potentially requires different algorithmic techniques. Notably, if parameterized by the number of facilities $k$, the problem is also $W[2]$ hard, making the existence of an exact FPT algorithm unlikely. In this work we provide an FPT-time constant factor approximation algorithm preserving both cardinality and capacity of the facilities. The algorithm runs in time $2^{\mathcal{O}(k\log k)}n^{\mathcal{O}(1)}$ and achieves an approximation ratio of $7+\varepsilon$.

Motivation & Objective

• To develop a fixed-parameter tractable (FPT) approximation algorithm for the capacitated k-median problem that respects both the number of facilities k and their capacities.
• To overcome the longstanding open problem of achieving a polynomial-time constant-factor approximation without violating capacity or cardinality constraints.
• To bridge approximation algorithms and parameterized complexity by designing an FPT algorithm with a constant approximation ratio.
• To explore whether better approximation ratios (e.g., (1+ε)) are achievable in FPT time for this problem.

Proposed method

• Uses probabilistic tree embedding to reduce the metric to a tree structure with O(log k) expected distortion.
• Clusters clients into ℓ = O(k log n / ε) centers, forming an ℓ-centered instance.
• Guesses the distribution of the k facilities across these ℓ centers to reduce the problem to a structured instance.
• Applies dynamic programming on the tree-embedded instance, computing D(t, k′, b) for subtrees t, facility counts k′, and client balance b.
• Rounds distances and reduces the problem to linear programming over a totally unimodular matrix to ensure integral solutions.
• Combines the FPT algorithm for uncapacitated k-median with the tree-embedding framework to maintain approximation guarantees.

Experimental results

Research questions

• RQ1 Can a constant-factor FPT approximation be achieved for capacitated k-median without violating capacity or facility count constraints?
• RQ2 Is the integrality gap barrier of the standard LP relaxation surmountable via FPT techniques and structured instance reduction?
• RQ3 Can the (7+ε)-approximation ratio be improved to (1+ε) in FPT time using alternative embedding or rounding techniques?
• RQ4 Is it possible to design a polynomial-time constant-factor approximation for capacitated k-median by exploiting the ℓ-centered instance structure?

Key findings

• The paper presents a (7 + ε)-approximation algorithm for capacitated k-median that runs in time 2^{O(k log k)}n^{O(1)} and respects both k and capacity constraints.
• The algorithm achieves this by combining tree embedding with a novel FPT dynamic programming approach on ℓ-centered instances.
• The tree embedding step ensures that the expected cost of the optimal solution increases by at most O(log k) factor, enabling a polynomial-time exact solution on trees.
• The dynamic programming formulation D(t, k′, b) computes the minimum cost of opening k′ facilities in a subtree t and routing b clients across the edge to its parent, with cost counted at the top endpoint.
• The framework allows for a clean reduction to totally unimodular LP, ensuring integral solutions without capacity violations.
• The result improves upon the best-known polynomial-time approximation ratio of O(log k) and provides the first FPT algorithm with a constant approximation ratio for this problem.