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Edge-Colored Clustering in Hypergraphs: Beyond Minimizing Unsatisfied Edges

Alex Crane, Thomas Stanley, Blair D. Sullivan, Nate Veldt

TL;DR

This work advances edge-colored clustering by (i) delivering the first approximation for MaxECC on hypergraphs with factor $(1/(r+1))(2/e)^r$ and refining graph MaxECC to $154/405\approx0.38$ via LP-rounding and color-priority techniques, (ii) introducing generalized $\ell_p$-norm MinECC and fair/balanced variants—Color-Fair MinECC and Protected-Color MinECC—with hardness, approximation, and FPT results, and (iii) validating methods experimentally to show practical gains in fairness-aware ECC while preserving standard clustering quality. The approach blends LP relaxations, randomized rounding, and a careful analysis of independent events and color orderings, connecting ECC to conflict-graph and vertex-cover perspectives. Collectively, the results expand ECC tools for balanced clustering, fairness constraints, and protected-interaction considerations, with implications for team formation, temporal clustering, and other multiway interaction tasks. Overall, the paper highlights both improved theoretical guarantees and practical algorithms that balance accuracy with fairness in edge-colored hypergraph clustering.

Abstract

We consider a framework for clustering edge-colored hypergraphs, where the goal is to cluster (equivalently, to color) objects based on the primary type of multiway interactions they participate in. One well-studied objective is to color nodes to minimize the number of unsatisfied hyperedges -- those containing one or more nodes whose color does not match the hyperedge color. We motivate and present advances for several directions that extend beyond this minimization problem. We first provide new algorithms for maximizing satisfied edges, which is the same at optimality but is much more challenging to approximate, with all prior work restricted to graphs. We develop the first approximation algorithm for hypergraphs, and then refine it to improve the best-known approximation factor for graphs. We then introduce new objective functions that incorporate notions of balance and fairness, and provide new hardness results, approximations, and fixed-parameter tractability results.

Edge-Colored Clustering in Hypergraphs: Beyond Minimizing Unsatisfied Edges

TL;DR

This work advances edge-colored clustering by (i) delivering the first approximation for MaxECC on hypergraphs with factor and refining graph MaxECC to via LP-rounding and color-priority techniques, (ii) introducing generalized -norm MinECC and fair/balanced variants—Color-Fair MinECC and Protected-Color MinECC—with hardness, approximation, and FPT results, and (iii) validating methods experimentally to show practical gains in fairness-aware ECC while preserving standard clustering quality. The approach blends LP relaxations, randomized rounding, and a careful analysis of independent events and color orderings, connecting ECC to conflict-graph and vertex-cover perspectives. Collectively, the results expand ECC tools for balanced clustering, fairness constraints, and protected-interaction considerations, with implications for team formation, temporal clustering, and other multiway interaction tasks. Overall, the paper highlights both improved theoretical guarantees and practical algorithms that balance accuracy with fairness in edge-colored hypergraph clustering.

Abstract

We consider a framework for clustering edge-colored hypergraphs, where the goal is to cluster (equivalently, to color) objects based on the primary type of multiway interactions they participate in. One well-studied objective is to color nodes to minimize the number of unsatisfied hyperedges -- those containing one or more nodes whose color does not match the hyperedge color. We motivate and present advances for several directions that extend beyond this minimization problem. We first provide new algorithms for maximizing satisfied edges, which is the same at optimality but is much more challenging to approximate, with all prior work restricted to graphs. We develop the first approximation algorithm for hypergraphs, and then refine it to improve the best-known approximation factor for graphs. We then introduce new objective functions that incorporate notions of balance and fairness, and provide new hardness results, approximations, and fixed-parameter tractability results.

Paper Structure

This paper contains 16 sections, 24 theorems, 93 equations, 3 figures, 1 table, 2 algorithms.

Table of Contents

  1. Introduction
  2. Improved MaxECC Algorithms
  3. Technical preliminaries for LP rounding algorithms
  4. Hypergraph MaxECC Algorithm
  5. Graph MaxECC Algorithm
  6. Alternative ECC Objectives
  7. Generalized Norm Objectives
  8. Color-fair ECC variants
  9. Protected-color ECC
  10. Experiments
  11. Conclusions
  12. MaxECC Proofs
  13. Step 1: Proving minimizers exist in $\mathcal{S}$.
  14. Step 2: Editing a vector from $\mathcal{S}$ to $\mathcal{I}$.
  15. Full proof of
  16. ...and 1 more sections

Key Result

Theorem 2.1

Algorithm alg:hyper_maxecc is a $\left(\frac{1}{r+1} \left(\frac{2}{e}\right)^r\right)$-approximation algorithm for MaxECC in hypergraphs with rank $r$.

Figures (3)

  • Figure 1: The node coloring on the left satisfies 4 edges (2 blue, 2 red, 0 green). The coloring on the right only satisfies 3, but each color has a satisfied edge.
  • Figure 2: Results running Protected-Color MinECC on multiple datasets with a varying constraint on the number of unsatisfied edges for the color with the median amount of edges. Figure \ref{['fig:pcecc_experiments_constraint']} is the percent of constraint violation of the protected color. Line $y=x$ represents the constraint imposed by the problem definition, and $y=2x$ is the theoretical limit for our bicriteria approximation algorithm. Figure \ref{['fig:pcecc_experiments_runtime']} is the runtime in seconds for solving the linear program (note that the rounding step has a negligible runtime). Figure \ref{['fig:pcecc_experiments_pc_approx']} is the approximation upper bound for the protected color objective. Figure \ref{['fig:pcecc_experiments_st_approx']} is the approximation upper bound for the standard MinECC objective. Approximation upper bounds are given by the objective value for the algorithm divided by the lower bound for the objective determined by the LP relaxation.
  • Figure 3: The construction given by \ref{['thm:cfminecc-NPhard']} for the CNF formula on three clauses $C_1 = (x_1 \lor x_2 \lor x_3)$, $C_2 = (x_1 \lor \neg x_2 \lor \neg x_3)$, and $C_3 = (\neg x_1 \lor \neg x_2)$. Notice that the vertices are partitioned visually into four horizontal layers. The top layer contains conflict vertices, the second from top contains free vertices, the second from bottom bridge vertices, and the bottom spare vertices. Every edge containing a conflict vertex is a conflict edge, every edge containing a bridge vertex is a bridge edge, and all other edges are spare edges. The colors $c_1, c_2$ and $c_3$ correspond to the clauses $C_1, C_2$, and $C_3$. The color $c_3'$ is the spare color associated with $C_3$. The remaining colors are bridge colors. We refer to the proof of \ref{['thm:cfminecc-NPhard']} for a formal description of the construction and the accompanying analysis.

Theorems & Definitions (43)

  • Theorem 2.1
  • proof
  • Definition 2.2
  • Lemma 2.2
  • Lemma 2.2
  • Lemma 2.2
  • Lemma 2.2
  • Theorem 2.3
  • proof
  • Theorem 3.1
  • ...and 33 more