Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Colorful Priority $k$-Supplier

Chandra Chekuri, Junkai Song

TL;DR

This paper builds upon the work of [Bajpai et al., 2022] and improves their $9-approximation Priority $k$-Supplier with Outliers problem to a $1+3\sqrt{3}\approx 6.196$-approximation, and presents the first set of approximation algorithms for the Priority Colorful $k$-Supplier problem.

Abstract

In the Priority $k$-Supplier problem the input consists of a metric space $(F \cup C, d)$ over set of facilities $F$ and a set of clients $C$, an integer $k > 0$, and a non-negative radius $r_v$ for each client $v \in C$. The goal is to select $k$ facilities $S \subseteq F$ to minimize $\max_{v \in C} \frac{d(v,S)}{r_v}$ where $d(v,S)$ is the distance of $v$ to the closes facility in $S$. This problem generalizes the well-studied $k$-Center and $k$-Supplier problems, and admits a $3$-approximation [Plesník, 1987, Bajpai et al., 2022. In this paper we consider two outlier versions. The Priority $k$-Supplier with Outliers problem [Bajpai et al., 2022] allows a specified number of outliers to be uncovered, and the Priority Colorful $k$-Supplier problem is a further generalization where clients are partitioned into $c$ colors and each color class allows a specified number of outliers. These problems are partly motivated by recent interest in fairness in clustering and other optimization problems involving algorithmic decision making. We build upon the work of [Bajpai et al., 2022] and improve their $9$-approximation Priority $k$-Supplier with Outliers problem to a $1+3\sqrt{3}\approx 6.196$-approximation. For the Priority Colorful $k$-Supplier problem, we present the first set of approximation algorithms. For the general case with $c$ colors, we achieve a $17$-pseudo-approximation using $k+2c-1$ centers. For the setting of $c=2$, we obtain a $7$-approximation in random polynomial time, and a $2+\sqrt{5}\approx 4.236$-pseudo-approximation using $k+1$ centers.

Colorful Priority $k$-Supplier

TL;DR

This paper builds upon the work of [Bajpai et al., 2022] and improves their k1+3\sqrt{3}\approx 6.196k$-Supplier problem.

Abstract

In the Priority -Supplier problem the input consists of a metric space over set of facilities and a set of clients , an integer , and a non-negative radius for each client . The goal is to select facilities to minimize where is the distance of to the closes facility in . This problem generalizes the well-studied -Center and -Supplier problems, and admits a -approximation [Plesník, 1987, Bajpai et al., 2022. In this paper we consider two outlier versions. The Priority -Supplier with Outliers problem [Bajpai et al., 2022] allows a specified number of outliers to be uncovered, and the Priority Colorful -Supplier problem is a further generalization where clients are partitioned into colors and each color class allows a specified number of outliers. These problems are partly motivated by recent interest in fairness in clustering and other optimization problems involving algorithmic decision making. We build upon the work of [Bajpai et al., 2022] and improve their -approximation Priority -Supplier with Outliers problem to a -approximation. For the Priority Colorful -Supplier problem, we present the first set of approximation algorithms. For the general case with colors, we achieve a -pseudo-approximation using centers. For the setting of , we obtain a -approximation in random polynomial time, and a -pseudo-approximation using centers.
Paper Structure (10 sections, 17 theorems, 43 equations, 2 figures, 1 table, 5 algorithms)

This paper contains 10 sections, 17 theorems, 43 equations, 2 figures, 1 table, 5 algorithms.

Table of Contents

  1. Introduction
  2. Results and ideas:
  3. Organization:
  4. Preliminaries
  5. Priority $k$-Supplier with Outliers
  6. Priority Knapsack Supplier with Outliers
  7. DAG to Forest
  8. Round-or-cut
  9. Priority Colorful $k$-Supplier
  10. Conclusions

Key Result

Lemma 1

The output of Algorithm alg:filter has the following properties: (a) $\{D(v):v\in R\}$ is a partition of $C$. (b) $\forall u,v\in R, d(u,v)>r_u+r_v$. (c) $\forall v\in R,u\in D(v),d(u,v)\leq r_u+r_v$ and $\textsf{cov}\xspace(v)\geq \textsf{cov}\xspace(u)$.

Figures (2)

  • Figure 1: Illustration of 2 radii
  • Figure 2: Illustration for the case of $3$ distinct radii

Theorems & Definitions (45)

  • Definition 1: Priority $k$-Supplier with Outliers (P$k$SO)
  • Lemma 1
  • Theorem 2
  • proof
  • Definition 2: Weighted $k$-Path Packing (W$k$PP)
  • Definition 3: contact DAG
  • Lemma 3
  • proof
  • Theorem 4
  • proof
  • ...and 35 more