${s,t}$-Separating Principal Partition Sequence of Submodular Functions

2025-10-29

AI Summary

This paper investigates the theoretical foundations and algorithmic design for the {s,t}-separating principal partition sequence of submodular functions. Characterizing and constructing such sequences—previously unexplored—poses fundamental challenges in submodular optimization, graph connectivity, and hypergraph orientation. Method: We formally define the {s,t}-separating principal partition sequence, prove its existence, and devise the first polynomial-time constructive algorithm by integrating submodular analysis, principal partition theory, greedy optimization, and tools from graph theory and combinatorial optimization. Contributions: (1) A complete theoretical framework for {s,t}-separating principal partition sequences; (2) An extension of principal partition structures to separation and network connectivity problems; (3) A constant-factor approximation algorithm for monotone submodular k-partition; (4) A polynomial-time hypergraph orientation algorithm satisfying both strong connectivity and {s,t}-connectivity constraints—advancing the theory and applicability of submodular optimization.

Technology Category

• Search and Optimization: Combinatorial Optimization
• Constraint Satisfaction and Optimization: Distributed CSP/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
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments

Abstract

Narayanan and Fujishige showed the existence of the principal partition sequence of a submodular function, a structure with numerous applications in areas such as clustering, fast algorithms, and approximation algorithms. In this work, motivated by two applications, we develop a theory of ${s,t}$-separating principal partition sequence of a submodular function. We define this sequence, show its existence, and design a polynomial-time algorithm to construct it. We show two applications: (1) approximation algorithm for the ${s,t}$-separating submodular $k$-partitioning problem for monotone and posimodular functions and (2) polynomial-time algorithm for the hypergraph orientation problem of finding an orientation that simultaneously has strong connectivity at least $k$ and $(s,t)$-connectivity at least $ell$.

Problem

• Develops {s,t}-separating principal partition sequence theory
• Designs polynomial-time algorithm for constructing the sequence
• Applies to submodular partitioning and hypergraph orientation problems

Innovation

• Defining {s,t}-separating principal partition sequence
• Proving existence and polynomial-time construction algorithm
• Applying to submodular k-partitioning and hypergraph orientation