[Paper Review] Augmented Sparsifiers for Generalized Hypergraph Cuts.

Austin R. Benson, Jon Kleinberg+1 more|arXiv (Cornell University)|Jul 16, 2020

Complexity and Algorithms in Graphs60 references4 citations

TL;DR

This paper introduces a sparsification framework that approximates generalized hypergraph cuts with submodular cardinality-based splitting functions using sparse directed graphs, enabling faster approximate min $s$-$t$ cut algorithms. By approximating concave functions via piecewise linear curves, it reduces each hyperedge to $O(\varepsilon^{-1}|e|\log|e|)$ edges, significantly improving runtime efficiency over dense reductions.

ABSTRACT

In recent years, hypergraph generalizations of many graph cut problems have been introduced and analyzed as a way to better explore and understand complex systems and datasets characterized by multiway relationships. Recent work has made use of a generalized hypergraph cut function which for a hypergraph $\mathcal{H} = (V,E)$ can be defined by associating each hyperedge $e \in E$ with a splitting function ${\bf w}_e$, which assigns a penalty to each way of separating the nodes of $e$. When each ${\bf w}_e$ is a submodular cardinality-based splitting function, meaning that ${\bf w}_e(S) = g(|S|)$ for some concave function $g$, previous work has shown that a generalized hypergraph cut problem can be reduced to a directed graph cut problem on an augmented node set. However, existing reduction procedures often result in a dense graph, even when the hypergraph is sparse, which leads to slow runtimes for algorithms that run on the reduced graph. We introduce a new framework of sparsifying hypergraph-to-graph reductions, where a hypergraph cut defined by submodular cardinality-based splitting functions is $(1+\varepsilon)$-approximated by a cut on a directed graph. Our techniques are based on approximating concave functions using piecewise linear curves. For $\varepsilon > 0$ we need at most $O(\varepsilon^{-1}|e| \log |e|)$ edges to reduce any hyperedge $e$, which leads to faster runtimes for approximating generalized hypergraph $s$-$t$ cut problems. For the machine learning heuristic of a clique splitting function, our approach requires only $O(|e| \varepsilon^{-1/2} \log \log \frac{1}{\varepsilon})$ edges. This sparsification leads to faster approximate min $s$-$t$ graph cut algorithms for certain classes of co-occurrence graphs. Finally, we apply our sparsification techniques to develop approximation algorithms for minimizing sums of cardinality-based submodular functions.

Motivation & Objective

• To address the inefficiency of existing hypergraph-to-graph reductions that produce dense graphs, slowing down $s$-$t$ cut algorithms.
• To develop a sparsification technique that approximates generalized hypergraph cuts with $(1+\varepsilon)$-accuracy using fewer edges.
• To enable faster approximate min $s$-$t$ cut computation on co-occurrence graphs by reducing hyperedge complexity.
• To provide efficient approximation algorithms for minimizing sums of cardinality-based submodular functions.

Proposed method

• Approximate concave functions using piecewise linear curves to model submodular cardinality-based splitting functions.
• Construct an augmented directed graph where each hyperedge is replaced by a sparse subgraph of size $O(\varepsilon^{-1}|e|\log|e|)$.
• Ensure the resulting graph preserves the hypergraph cut value within a $1+\varepsilon$ factor for all node partitions.
• Apply the sparsification to clique splitting functions, achieving $O(|e|\varepsilon^{-1/2}\log\log\frac{1}{\varepsilon})$ edges per hyperedge.
• Leverage the sparse graph structure to accelerate min $s$-$t$ cut algorithms on co-occurrence graphs.
• Use the sparsified reductions to derive approximation algorithms for minimizing sums of submodular functions.

Experimental results

Research questions

• RQ1 Can we reduce the number of edges in hypergraph-to-graph reductions while preserving $(1+\varepsilon)$-approximate cut values?
• RQ2 What is the minimum number of edges required to sparsify a hyperedge with a submodular cardinality-based splitting function?
• RQ3 How does the sparsification performance vary for specific functions like clique splitting?
• RQ4 Can the sparsified reductions accelerate approximate min $s$-$t$ cut computation in practice?
• RQ5 Can the sparsification framework be extended to minimize sums of submodular functions?

Key findings

• The proposed method reduces each hyperedge to $O(\varepsilon^{-1}|e|\log|e|)$ edges while preserving a $(1+\varepsilon)$-approximation of the generalized hypergraph cut.
• For clique splitting functions, the edge count is reduced to $O(|e|\varepsilon^{-1/2}\log\log\frac{1}{\varepsilon})$, improving over general bounds.
• The sparsification leads to faster approximate min $s$-$t$ cut algorithms on co-occurrence graphs due to reduced graph density.
• The framework enables efficient approximation algorithms for minimizing sums of cardinality-based submodular functions.
• The use of piecewise linear approximations of concave functions enables accurate and scalable cut approximations.