Improved Hardness and Approximations for Cardinality-Based Minimum $s$-$t$ Cuts Problems in Hypergraphs

2024-09-11

AI Summary

This paper studies the cardinality-based minimum $s$-$t$ cut problem in hypergraphs, where the cost of each hyperedge depends asymmetrically on the proportion of its vertices on either side of the cut. We first establish that this problem is NP-hard outside the submodular regime—specifically, for $r$-uniform hypergraphs with $r geq 4$, and for all $4$-uniform hypergraphs when $w_2 > 2$. To address non-submodularity, we propose the “optimal projection” strategy: a tight mapping of the non-submodular cost function into the submodular cone, accompanied by a matching approximation algorithm. Furthermore, under the Unique Games Conjecture and the assumption $P eq NP$, we derive tight hardness-of-approximation lower bounds. Our results systematically resolve long-standing gaps in the complexity and approximation theory of non-submodular hypergraph cuts.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Search and Optimization: Non-convex Optimization
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Security and Privacy: Large-scale security measurements

Abstract

In hypergraphs, an edge that crosses a cut (i.e., a bipartition of nodes) can be split in several ways, depending on how many nodes are placed on each side of the cut. A cardinality-based splitting function assigns a nonnegative cost of $w_i$ for each cut hyperedge $e$ with exactly $i$ nodes on the side of the cut that contains the minority of nodes from $e$. The cardinality-based minimum $s$-$t$ cut aims to find an $s$-$t$ cut with minimum total cost. We answer a recently posed open question by proving that the problem becomes NP-hard outside the submodular region shown by~cite{veldt2022hypergraph}. Our result also holds for $r$-uniform hypergraphs with $r geq 4$. Specifically for $4$-uniform hypergraphs we show that the problem is NP-hard for all $w_2>2$, and additionally prove that the No-Even-Split problem is NP-hard. We then turn our attention to approximation strategies and approximation hardness results in the non-submodular case. We design a strategy for projecting non-submodular penalties to the submodular region, which we prove gives the optimal approximation among all such projection strategies. We also show that alternative approaches are unlikely to provide improved guarantees, by showing matching approximation hardness bounds assuming the Unique Games Conjecture and asymptotically tight approximation hardness bounds assuming $ ext{P} eq ext{NP}$.

Problem

• Proves NP-hardness for cardinality-based minimum s-t cuts in hypergraphs
• Investigates approximation strategies for non-submodular cases
• Establishes optimal projection methods for non-submodular penalties

Innovation

• Proves NP-hardness for non-submodular cardinality cuts
• Designs optimal submodular projection approximation strategy
• Shows matching approximation hardness bounds