[Paper Review] The Guided Team-Partitioning Problem: Definition, Complexity, and Algorithm

Sanaz Bahargam, Theodoros Lappas+1 more|arXiv (Cornell University)|Apr 30, 2019

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TL;DR

This paper introduces the Guided Team-Partitioning (GTP) problem, which partitions a population into teams such that each team's centroid is as close as possible to a given target vector, enabling controlled, fair team composition. The authors prove GTP is NP-hard and propose efficient heuristic algorithms that outperform baselines in both solution quality and runtime on real and synthetic datasets.

ABSTRACT

A long line of literature has focused on the problem of selecting a team of individuals from a large pool of candidates, such that certain constraints are respected, and a given objective function is maximized. Even though extant research has successfully considered diverse families of objective functions and constraints, one of the most common limitations is the focus on the single-team paradigm. Despite its well-documented applications in multiple domains, this paradigm is not appropriate when the team-builder needs to partition the entire population into multiple teams. Team-partitioning tasks are very common in an educational setting, in which the teacher has to partition the students in her class into teams for collaborative projects. The task also emerges in the context of organizations, when managers need to partition the workforce into teams with specific properties to tackle relevant projects. In this work, we extend the team formation literature by introducing the Guided Team-Partitioning (GTP) problem, which asks for the partitioning of a population into teams such that the centroid of each team is as close as possible to a given target vector. As we describe in detail in our work, this formulation allows the team-builder to control the composition of the produced teams and has natural applications in practical settings. Algorithms for the GTP need to simultaneously consider the composition of multiple non-overlapping teams that compete for the same population of candidates. This makes the problem considerably more challenging than formulations that focus on the optimization of a single team. In fact, we prove that GTP is NP-hard to solve and even to approximate. The complexity of the problem motivates us to consider efficient algorithmic heuristics, which we evaluate via experiments on both real and synthetic datasets.

Motivation & Objective

• To address the gap in team formation research that focuses only on single-team optimization, especially in settings requiring partitioning a full population into multiple non-overlapping teams.
• To formalize the Guided Team-Partitioning (GTP) problem, where team centroids must be guided toward specific target vectors to ensure balanced or desired team compositions.
• To analyze the computational complexity of GTP and prove it is NP-hard to solve and approximate.
• To design and evaluate efficient heuristic algorithms for GTP that scale well and produce high-quality solutions.
• To demonstrate the practical utility of GTP in educational and organizational settings through experiments on real and synthetic data.

Proposed method

• Formulate GTP as a constrained optimization problem where the objective is to minimize the sum of squared distances between team centroids and target vectors.
• Propose the GuidedSplit algorithm, a heuristic that iteratively assigns individuals to teams by balancing centroid proximity and team size constraints.
• Use a convex optimization subroutine (ConvexOpt) to refine team centroids during the partitioning process, improving alignment with target vectors.
• Integrate a Hungarian algorithm-based matching step to assign cluster centers to target vectors when needed, ensuring optimal target assignment.
• Design a greedy variant (BTF_Greedy) and a baseline (BTF_CVX) for comparison, both using iterative refinement with target constraints.
• Evaluate performance using synthetic and real-world datasets, measuring solution quality via centroid-target distance and runtime efficiency.

Experimental results

Research questions

• RQ1 Is the Guided Team-Partitioning problem NP-hard, and can it be approximated within a constant factor?
• RQ2 Can heuristic algorithms be designed that effectively balance team composition toward specified target vectors while maintaining computational efficiency?
• RQ3 How do the proposed algorithms compare to existing clustering and team formation baselines in terms of solution quality and runtime?
• RQ4 What impact do parameters such as the number of clusters and the number of points removed per iteration have on algorithm performance?
• RQ5 In what real-world scenarios does guided partitioning significantly outperform standard clustering or team formation methods?

Key findings

• The GTP problem is proven to be NP-hard to solve and even to approximate within any constant factor, establishing its computational intractability.
• The proposed GuidedSplit algorithm achieves significantly better solution quality than k-means, k_targets, Random, and baseline methods in minimizing centroid-target distance.
• GuidedSplit runs in comparable time to other heuristics, with runtime scaling sub-linearly with population size when clusters are balanced, and increasing with the number of clusters due to point removal overhead.
• The k_means baseline exhibits high runtime and poor performance on large datasets due to highly unbalanced cluster formation, especially at n = 5000.
• When ℓ = 0 (no convex optimization), BTF_CVX and BTF_Greedy are slower than others due to running the ConvexOpt or Greedy step n times per iteration.
• The k_targets baseline performs better than k_means in runtime due to more balanced initial cluster formation when using predefined means as seeds.