[Paper Review] Approximation algorithms for stochastic clustering

David G. Harris, Li Shi+3 more|arXiv (Cornell University)|Sep 7, 2018

Data Management and Algorithms3 citations

TL;DR

This paper introduces approximation algorithms for stochastic clustering using k-lotteries—randomized distributions over k-center solutions—to achieve better per-point service guarantees than deterministic methods. It presents a (1, k+2)-determinization algorithm that ensures expected connection cost ≤ (1+2/e)T while maintaining worst-case radius ≤ 3T, improving fairness and long-term user satisfaction without sacrificing global approximation guarantees.

ABSTRACT

We consider stochastic settings for clustering, and develop provably-good approximation algorithms for a number of these notions. These algorithms yield better approximation ratios compared to the usual deterministic clustering setting. Additionally, they offer a number of advantages including clustering which is fairer and has better long-term behavior for each user. In particular, they ensure that *every user* is guaranteed to get good service (on average). We also complement some of these with impossibility results.

Motivation & Objective

• To develop approximation algorithms for stochastic clustering that improve per-point service quality beyond deterministic k-center and k-median solutions.
• To design k-lotteries—probability distributions over k-center solutions—that ensure every client has a good expected connection cost while respecting global approximation guarantees.
• To provide provable fairness and long-term behavioral advantages in clustering by ensuring each user is guaranteed good average service.
• To establish theoretical limits via impossibility results, showing that certain improvements are unattainable under standard assumptions.
• To bridge the gap between global approximation ratios and individual user experience in clustering, particularly in settings where worst-case performance harms fairness.

Proposed method

• Proposes a stochastic framework where the solution is a k-lottery—i.e., a probability distribution over k-element subsets of facilities—rather than a single deterministic set.
• Introduces the concept of (α, β)-determinization, where α controls the number of facilities used (α ≤ k) and β bounds the worst-case connection cost (d(j,S) ≤ βT) with probability 1.
• Develops a greedy algorithm that selects facilities iteratively based on minimizing t_j, the target expected connection cost for client j, to achieve a (1, k+2)-determinization.
• Uses probabilistic and combinatorial arguments, including the pigeonhole principle and triangle inequality, to prove that the algorithm cannot exceed k+1 facilities without violating expected cost constraints.
• Employs a contradiction argument based on expected values: if the algorithm used k+2 facilities, the expected sum of normalized connection costs would exceed k+1, contradicting the feasibility of the target t_j values.
• Applies the framework to k-center, k-median, and k-supplier problems, showing that stochastic solutions can outperform deterministic ones in terms of expected cost while preserving worst-case bounds.

Experimental results

Research questions

• RQ1 Can stochastic clustering with k-lotteries achieve better expected connection costs per client than deterministic k-center or k-median algorithms?
• RQ2 Is it possible to maintain a worst-case radius bound of 3T for k-supplier while improving the expected connection cost below the deterministic lower bound of 1+2/e ≈ 1.736T?
• RQ3 What are the theoretical limits of improving per-point guarantees in stochastic clustering under global approximation constraints?
• RQ4 Can a (1, β)-determinization be achieved with β < k+2 for general k-center instances?
• RQ5 How does the use of k-lotteries affect fairness and long-term user satisfaction compared to deterministic clustering?

Key findings

• The paper presents a (1, k+2)-determinization algorithm that guarantees worst-case connection cost ≤ 3T with probability one, while ensuring E[d(j,S)] ≤ (1+2/e)T ≈ 1.736T for all clients j.
• The algorithm runs in O(|F||C|) time and is proven correct via contradiction, showing that using more than k+1 facilities leads to a violation of expected cost constraints.
• It is shown that a (1, k+1)-determinization is not always possible, establishing a tight lower bound on β for the α=1 case.
• The framework improves fairness and long-term behavior by ensuring every user receives good average service, even when deterministic algorithms produce worst-case outcomes.
• The results demonstrate that stochastic solutions can circumvent the inherent limitations of deterministic approximation algorithms in center-type clustering problems.
• Impossibility results confirm that certain improvements in expected cost cannot be achieved without violating worst-case bounds, highlighting the optimality of the proposed bounds.