Augmented Sparsifiers for Generalized Hypergraph Cuts
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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 ℋ = (V,E) can be defined by associating each hyperedge e ∈ E with a splitting function w_e, which assigns a penalty to each way of separating the nodes of e. When each w_e is a submodular cardinality-based splitting function, meaning that 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+ε)-approximated by a cut on a directed graph. Our techniques are based on approximating concave functions using piecewise linear curves. For ε > 0 we need at most O(ε^-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| ε^-1/2loglog1/ε) 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.
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