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

Kristóf Bérczi, Karthekeyan Chandrasekaran, Tamás Király, Daniel P. Szabo

TL;DR

The paper extends Narayanan's principal partition sequence to a {s,t}-separating setting for submodular functions, establishing the existence and a polynomial-time construction of the {s,t}-separating principal partition sequence. It leverages this framework to obtain a 2-approximation for {s,t}-Sep-Submod- k-Part under posimodular/submodularity assumptions and a 4/3-approximation for monotone submodular cases, as well as to design polynomial-time algorithms for hypergraph orientation problems with (k,(s,t),l)-connectivity. The results include min-max relations and weighted variants, broadening the applicability to graph/hypergraph partitioning, clustering, and orienting problems while connecting submodular optimization with network orientation theory. Overall, the work provides a unified, algorithmically tractable framework for structured partitioning under dual-terminal separation and for orienting hypergraphs to satisfy combined connectivity constraints.

Abstract

Narayanan 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 $\ell$.

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

TL;DR

The paper extends Narayanan's principal partition sequence to a {s,t}-separating setting for submodular functions, establishing the existence and a polynomial-time construction of the {s,t}-separating principal partition sequence. It leverages this framework to obtain a 2-approximation for {s,t}-Sep-Submod- k-Part under posimodular/submodularity assumptions and a 4/3-approximation for monotone submodular cases, as well as to design polynomial-time algorithms for hypergraph orientation problems with (k,(s,t),l)-connectivity. The results include min-max relations and weighted variants, broadening the applicability to graph/hypergraph partitioning, clustering, and orienting problems while connecting submodular optimization with network orientation theory. Overall, the work provides a unified, algorithmically tractable framework for structured partitioning under dual-terminal separation and for orienting hypergraphs to satisfy combined connectivity constraints.

Abstract

Narayanan 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 -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 -separating submodular -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 and -connectivity at least .

Paper Structure

This paper contains 18 sections, 31 theorems, 69 equations, 2 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Applications
  3. s,t-Separating Submodular k-Partition
  4. Hypergraph Orientation
  5. {s,t}-Separating Principal Partition Sequence
  6. Principal Partition Sequence.
  7. $\{s,t\}$-Refinement and $\{s,t\}$-Separating Principal Partition Sequence.
  8. (s,t)-Separating Principal Partition Sequence
  9. Refinement Property
  10. Existence
  11. Algorithm
  12. Approximation Algorithm for $\{s,t\}\text{-Sep-Submod-}k\text{-Part}$
  13. Hypergraph Orientations
  14. Algorithm for (k,(s,t),l)-Hyperarc-Connected Orientation Problems
  15. Min-Max Relations for (k,(s,t),l)-Hyperarc-Connected Orientation Problems
  16. ...and 3 more sections

Key Result

Theorem 1.1

$\{s,t\}\text{-Sep-Submod-}k\text{-Part}$ admits a $2$-approximation for posimodular submodular functions and a $4/3$-approximation for monotone submodular functions.

Figures (2)

  • Figure 1: An edge-weighted graph whose cut function $f$ is such that minimizers for the breakpoints of $g_f^{s,t}$ do not necessarily satisfy the refinement property. Here $0<\varepsilon<1/24$ is a small constant, and $\alpha_1 = 1/2+\varepsilon,$$\alpha_2 = 1/3 + 2\varepsilon$. The sequence of minimizers of $g_f^{s,t}$ is unique and is given by $\mathcal{P}_1 = \{\{s,a,b\},\{t,c,d,e\}\}, \mathcal{P}_2 = \{\{s\},\{a\},\{b,t,c,d,e\}\},$$\mathcal{P}_3:=\{\{s,a,b,c\}, \{d\}, \{e\}, \{t\}\},$$\mathcal{P}_4 = \{\{s\}, \{a\}, \{b, c\}, \{d\}, \{e\}, \{t\}\},$ and $\mathcal{P}_5 = \{\{s\}, \{a\}, \{b\}, \{c\}, \{d\}, \{e\}, \{t\}\}$.
  • Figure 2: Examples of refinement and $\{s,t\}$-refinement.

Theorems & Definitions (74)

  • Theorem 1.1
  • Theorem 1.2
  • Definition 1.3: Refinement
  • Definition 1.4: Principal partition sequence
  • Theorem 1.5: Narayanan
  • Definition 1.6: $\{s,t\}$-refinements
  • Definition 1.7: $\{s,t\}$-separating principal partition sequence
  • Theorem 1.8
  • Theorem 2.1
  • Lemma 2.2
  • ...and 64 more