Partial Optimality in Cubic Correlation Clustering for General Graphs

TL;DR

This work addresses partial optimality in cubic correlation clustering on general graphs by extending prior complete-graph results. It develops improving-map based cut and join conditions, and provides exact and heuristic algorithms that reduce CP3 instances to min-cut / max-flow problems for efficient verification. Empirical evaluations on partition and geometric data demonstrate substantial variable fixing primarily via cut-based conditions, and establish scalability on sparse graphs while outlining practical deployment considerations. The study also clarifies the relationship between cubic clustering and cubic multicut, offering a useful theoretical-aided reduction framework. Overall, the approach yields practical reductions that can accelerate solving or approximating CP3 on large, incomplete graphs with meaningful implications for clustering and high-order graph partitioning tasks.

Abstract

The higher-order correlation clustering problem for a graph $G$ and costs associated with cliques of $G$ consists in finding a clustering of $G$ so as to minimize the sum of the costs of those cliques whose nodes all belong to the same cluster. To tackle this NP-hard problem in practice, local search heuristics have been proposed and studied in the context of applications. Here, we establish partial optimality conditions for cubic correlation clustering, i.e., for the special case of at most 3-cliques. We define and implement algorithms for deciding these conditions and examine their effectiveness numerically, on two data sets.

Figures (10)

• Figure 1: In order to examine the effectiveness of partial optimality conditions empirically, we implement algorithms for deciding these conditions and measure the fraction of fixed variables with respect to a parameter controlling the noise of the problem, for (a) synthetic instances with four clusters and noisy costs, and (b) instances for the task of finding equilateral triangles in a noisy point cloud.
• Figure 2: We report above for the partition data set the percentage of fixed variables and runtime after applying all conditions jointly, as described in Section \ref{['section:mixing-conditions']}, with respect to $\alpha$ (Rows 1--2) and with respect to $\vert V\vert$ (Rows 3--4). From left to right, the sparsity of the graph decreases as written in the column title, $(p_E, \lvert V\rvert, \bar{E})$ (Rows 1--2) and $p_E$ (Rows 3--4).
• Figure 3: We report above for the partition data set the percentage of fixed variables and runtime after applying Proposition \ref{['lemma:persistency-subset-separation']} (Rows 1--2), Proposition \ref{['proposition:edge-cut-persistency']} (Rows 3--4) and Corollary \ref{['corollary:subset-join-all-pairs-at-once']} (Rows 5--6), separately. From left to right, the sparsity of the graph decreases as written in the column title $(p_E, \vert V\vert, \bar{E})$.
• Figure 4: We report above for the partition data set the percentage of fixed triples and runtime after applying Proposition \ref{['lemma:persistency-triplet-cut']}. From left to right, the sparsity of the graph decreases as written in the column title $(p_E, \vert V\vert, \bar{E})$.
• Figure 5: We report above for the geometric data set the percentage of fixed variables and runtime after applying all conditions jointly, as described in Section \ref{['section:mixing-conditions']}, with respect to $\sigma$ (Rows 1--2) and with respect to $\vert V\vert$ (Rows 3--4). From left to right, the sparsity of the graph decreases as written in the column title, $(k, \vert V\vert, \bar{E})$ (Rows 1--2) and $k$ (Rows 3--4).

Theorems & Definitions (29)

• Definition 1
• Proposition 2
• Definition 3
• Proposition 4
• Corollary 5
• Definition 6
• Lemma 7
• Definition 8
• Lemma 9
• Proposition 10