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Separating $k$-Median from the Supplier Version

Aditya Anand, Euiwoong Lee

TL;DR

This work separates the original and supplier versions of the $k$-Median problem by presenting both a parameterized approximation algorithm and a Unique Games Conjecture-based hardness. The algorithm achieves a $1.546$ approximation in time $f(k,\varepsilon)\cdot n^{O(1)}$ by leveraging coresets, a constrained LP relaxation with per-cluster choices, and a probabilistic rounding that mixes leaders and cluster centers. The hardness result shows a $1.416-\varepsilon$ inapproximability under UG via a fine-grained Max-$k$-Coverage construction, using a dictatorship test and Fourier-analytic techniques on d-uniform hypergraphs, which aligns closely with the algorithmic bound and highlights an LP vs SDP gap driving the limits. Overall, the paper advances understanding of both algorithmic and hardness frontiers for k-Median, clarifying how allowing centers to be opened anywhere affects approximability and offering a precise target for future improvements.

Abstract

Given a metric space $(V, d)$ along with an integer $k$, the $k$-Median problem asks to open $k$ centers $C \subseteq V$ to minimize $\sum_{v \in V} d(v, C)$, where $d(v, C) := \min_{c \in C} d(v, c)$. While the best-known approximation ratio of $2.613$ holds for the more general supplier version where an additional set $F \subseteq V$ is given with the restriction $C \subseteq F$, the best known hardness for these two versions are $1+1/e \approx 1.36$ and $1+2/e \approx 1.73$ respectively, using the same reduction from Max $k$-Coverage. We prove the following two results separating them. First, we show a $1.546$-parameterized approximation algorithm that runs in time $f(k) n^{O(1)}$. Since $1+2/e$ is proved to be the optimal approximation ratio for the supplier version in the parameterized setting, this result separates the original $k$-Median from the supplier version. Next, we prove a $1.416$-hardness for polynomial-time algorithms assuming the Unique Games Conjecture. This is achieved via a new fine-grained hardness of Max-$k$-Coverage for small set sizes. Our upper bound and lower bound are derived from almost the same expression, with the only difference coming from the well-known separation between the powers of LP and SDP on (hypergraph) vertex cover.

Separating $k$-Median from the Supplier Version

TL;DR

This work separates the original and supplier versions of the -Median problem by presenting both a parameterized approximation algorithm and a Unique Games Conjecture-based hardness. The algorithm achieves a approximation in time by leveraging coresets, a constrained LP relaxation with per-cluster choices, and a probabilistic rounding that mixes leaders and cluster centers. The hardness result shows a inapproximability under UG via a fine-grained Max--Coverage construction, using a dictatorship test and Fourier-analytic techniques on d-uniform hypergraphs, which aligns closely with the algorithmic bound and highlights an LP vs SDP gap driving the limits. Overall, the paper advances understanding of both algorithmic and hardness frontiers for k-Median, clarifying how allowing centers to be opened anywhere affects approximability and offering a precise target for future improvements.

Abstract

Given a metric space along with an integer , the -Median problem asks to open centers to minimize , where . While the best-known approximation ratio of holds for the more general supplier version where an additional set is given with the restriction , the best known hardness for these two versions are and respectively, using the same reduction from Max -Coverage. We prove the following two results separating them. First, we show a -parameterized approximation algorithm that runs in time . Since is proved to be the optimal approximation ratio for the supplier version in the parameterized setting, this result separates the original -Median from the supplier version. Next, we prove a -hardness for polynomial-time algorithms assuming the Unique Games Conjecture. This is achieved via a new fine-grained hardness of Max--Coverage for small set sizes. Our upper bound and lower bound are derived from almost the same expression, with the only difference coming from the well-known separation between the powers of LP and SDP on (hypergraph) vertex cover.
Paper Structure (18 sections, 21 theorems, 32 equations, 2 algorithms)

This paper contains 18 sections, 21 theorems, 32 equations, 2 algorithms.

Table of Contents

  1. Introduction
  2. Overview
  3. Hardness.
  4. Algorithms.
  5. Discussion and Future Work.
  6. Approximation algorithm
  7. Algorithm
  8. Analysis: Casting as factor-revealing problems
  9. Simplifying distances
  10. Simplifying LP values and degrees
  11. Final simplifications
  12. Hardness
  13. Proof of \ref{['thm:hvc']}
  14. Dictatorship test
  15. Fourier analysis preliminaries
  16. ...and 3 more sections

Key Result

Theorem 1.1

For any $\epsilon > 0$, there is an algorithm for $k$-Median that runs in time $f(k, \epsilon)\cdot n^{O(1)}$ and guarantees an approximation ratio of where the optimal value is achieved with $p^* := (10 - 6\sqrt{2}) /7\approx 0.22$ and $d^* := 3$.

Theorems & Definitions (40)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 2.1
  • Definition 2.1
  • Theorem 2.2: fl11
  • Lemma 2.3
  • proof
  • Theorem 2.4
  • Lemma 2.4
  • proof
  • ...and 30 more