Graph sparsification is a fundamental tool in theoretical computer science that helps to reduce the size of a graph without losing key properties. Although many sparsification methods have been introduced, hypergraph separation and cut problems have become highly relevant due to their widespread application and theoretical challenges. Hypergraphs offer more accurate modeling of complex real-world scenarios than normal graphs, and the transition from graphs to hypergraphs has led to the development of new algorithms and theoretical frameworks to address the unique complexities of hypergraphs. This highlights the critical importance of these problems in both theory and practice.
Existing research has explored various approaches to address the challenges in graph sparsification. One major problem is the mimicking problem, which aims to find a graph that preserves the minimum cut sizes between any of the two subsets of vertices called terminals, with a mimicking network of O(τ³) edges, where τ is the number of edges incident to terminals. Further, the connectivity-c mimicking problem is developed to preserve minimum cut sizes of at most c, showing a graph with O(kc^3) edges, where k is the number of terminals. Another important variant is the multicut-mimicking problem, for which a method was introduced to obtain a multicut-mimicking network by contracting edges, however, a constrained version of the multicut-mimicking problem remains an open challenge, even for graphs.
Researchers from the Department of Computer Science and Engineering, POSTECH, Korea have proposed a new approach to address the multicut-mimicking network problem for hypergraphs. They introduced a multicut-mimicking network that preserves the minimum multicut values of any set of terminal pairs with a value at most c. This extends the connectivity-c mimicking network concept introduced earlier to the more complex domain of hypergraphs. The researchers have developed new notions and algorithms to effectively handle the unique challenges posed by hypergraph structures while building on previous methodologies, allowing the construction of smaller and more efficient networks.
The proposed method to compute a minimal multicut-mimicking network for hypergraphs builds upon the design of an algorithm to find a connectivity-c mimicking network for hypergraphs using expander decomposition. It utilizes the expander decomposition technique, introducing the concept of a ϕ-expander hypergraph. Moreover, the algorithm uses a recursive approach using a submodule called MimickingExpander, which computes a small multicut-mimicking network based on the expander decomposition. This helps the method to achieve a significantly smaller solution, effectively addressing the challenges posed by hypergraph structures in multicut-mimicking network computation.
The researchers focused on “vertex sparsifiers for multiway connectivity” with a parameter c > 0. The instance (G, T, c) consists of an undirected hypergraph G, a terminal set T ⊆ V(G), and a parameter c. The goal is to construct a hypergraph that preserves the minimum multicut values over T where the value is at most c. This represents the first result for the multicut-mimicking network problem that adapts the parameter c, even for graphs. Previously, the best-known multicut-mimicking network had a quasipolynomial size in T, specifically |∂T|^O(log |∂T|). Introducing parameter c, a multicut-mimicking network for a given instance can exist with a size linear in |T|. This uses a near-linear time framework to find a mimicking network using expander decomposition.
In conclusion, the researchers have demonstrated that for a hypergraph instance (G, T, c) with more than |T|cO(r log c) hyperedges, a smaller “multicut-mimicking” network can be created by contracting a hyperedge. An efficient algorithm is introduced in this paper for this purpose. This extends the current research on mimicking networks by introducing a parameter c and handling the complexities of hypergraphs. This has led to a significant advancement in graph sparsification, especially for hypergraph separation and cut problems, which have important theoretical and practical applications. In the future, the focus should be on reducing the time complexity or size of the “multicut-mimicking” network, such as exploring whether a network of size |T|cO(log (rc)) is achievable.
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