Principal Sequence of Partitions (PSP)

• Principal Sequence of Partitions (PSP) is a hierarchical structure for submodular functions that defines optimal partitions using a penalized cost function.
• The framework extends classical partitioning by incorporating {s,t}-separation constraints, ensuring controlled laminar refinements and unique breakpoints.
• Efficient parametric algorithms, including the Newton–Dinkelbach method, enable polynomial-time construction and approximation in combinatorial optimization applications.

A Principal Sequence of Partitions (PSP) is a fundamental structure associated with a submodular function on a finite set, capturing hierarchical decompositions of the set based on a parametric penalized cost. The PSP and its recently developed $\{s,t\}$-Separating variant extend classical notions of set-partition optimization to settings with additional separation constraints between distinguished elements. This structure enables efficient algorithms for a range of combinatorial optimization problems, including submodular $k$-partitioning and constrained hypergraph orientation, providing both theoretical insights and practical algorithms for problems involving submodularity and separation requirements (Bérczi et al., 29 Oct 2025).

1. Formal Definitions

Let $V$ be a finite ground set and $f: 2^V \rightarrow \mathbb{R}$ a submodular function. Submodularity is defined by

$$f(A) + f(B) \geq f(A \cap B) + f(A \cup B), \quad \forall A, B \subseteq V,$$

and equivalently reflects the diminishing returns property:

$$f(A \cup \{x\}) - f(A) \geq f(B \cup \{x\}) - f(B), \quad x \notin B \supseteq A.$$

A function is normalized if $f(\emptyset)=0$, monotone if $f(A) \leq f(B)$ for $A \subseteq B$, symmetric if $f(S) = f(V \setminus S)$, and posimodular if $$f(A) + f(B) \geq f(A \setminus B) + f(B \setminus A)$$ for all $A,B$.

The classical PSP studies the family of partitions of $V$ with cost

$g_P(\lambda) := f(P) - \lambda \cdot |P|,$

where $f(P) = \sum_{X \in P} f(X)$ and $|P|$ is the number of parts. As the penalty $\lambda$ sweeps over $\mathbb{R}$, the optimal partition minimizes $g_P(\lambda)$. The Principal Sequence of Partitions is the indexed chain of partitions corresponding to the breakpoints of the piecewise-linear, concave lower envelope

$$g_f(\lambda) := \min_{P\,\,\text{partition of}\, V} g_P(\lambda).$$

This sequence begins with the trivial partition $\{V\}$ and ends with all singletons.

The $\{s,t\}$-Separating Principal Partition Sequence generalizes the PSP by restricting to partitions that separate two distinguished elements $s,t \in V$. A partition $P$ is $\{s,t\}$-separating if $s$ and $t$ are in different parts. Defining

$$g^{s,t}(\lambda) := \min_{P~\{s,t\}\textrm{–separating}} g_P(\lambda),$$

the resulting sequence of $\{s,t\}$-separating partitions (the $\{s,t\}$-PSP) is indexed by breakpoints $\mu_1 < \cdots < \mu_{\ell-1}$, and exhibits structural refinements adapted to the separation constraint.

2. Existence and Structural Properties

The existence of the classical PSP relies on submodular-minimization and yields a unique chain of partitions $\Pi_1 \prec \Pi_2 \prec \ldots \prec \Pi_{r+1}$, with each $\Pi_{j+1}$ refining $\Pi_j$ by splitting exactly one part. For the $\{s,t\}$-PSP, analogous results hold but with the following additional properties:

• Each $Q_{j+1}$ is obtained from $Q_j$ either by a one-part split (classical refinement), or by an $\{s, t\}$-refinement up to two sets: exactly two parts $X$ and $Y$ intersect such that $X \cup Y$ is the region being refined, corresponding to a laminar family with at most one crossing at each refinement.
• Between breakpoints, the optimal partition does not change.
• All parts in all $Q_j$ form an "almost laminar" family: any crossing parts must separate $s$ and $t$, and only one such crossing is allowed at each move.

A tightness/uncrossing lemma guarantees that at each breakpoint, the transition between minimizing partitions can be realized via these controlled refinements. This enables a stepwise construction of the entire PSP by progressing $\lambda$ through its breakpoints and applying appropriate uncrossing arguments associated with submodular functions (Bérczi et al., 29 Oct 2025).

3. Construction Algorithms

The construction of both the classical PSP and $\{s,t\}$-PSP proceeds via parametric search:

• Parametric Search: For each candidate $\lambda$, compute a partition minimizing $f(P) - \lambda|P|$ among all (or all $\{s,t\}$-separating) partitions. This is achieved using modular penalties, Dilworth truncation, or submodular-minimization over extended ground sets. Each instance can be solved in polynomial time using a submodular function minimization oracle.
• Newton–Dinkelbach Method: The piecewise-linear concave function $g^{s,t}(\lambda)$ (resp. $g_f(\lambda)$) allows efficient detection of breakpoints using iterative techniques. The complexity is $O(r T_f \log N)$, where $r$ is the number of breakpoints, $T_f$ is the time for one submodular-minimization, and $N$ is the bit-precision.
• Overall Complexity: The full sequence (partitions and thresholds) is obtained in polynomial time in $|V|$ and $\log B$ (with $B$ a bound on numerical values).

4. Algorithmic Applications

Principal Sequences of Partitions, particularly the $\{s,t\}$-PSP, underpin efficient algorithms for several combinatorial optimization problems.

Application 1: $\{s,t\}$-Separating Submodular $k$-Partition

Given monotone or posimodular submodular $f\geq 0$ and $s,t \in V$, the goal is to compute a $\{s,t\}$-separating partition into exactly $k$ parts minimizing $\sum_{X \in P} f(X)$. This is NP-hard in general, but the $\{s,t\}$-PSP enables constant-factor approximation:

• If some partition in the sequence has $k$ parts, it is optimal for all $k$-partitions.
• Otherwise, for classical refinements, select the $k-|Q_{i-1}|$ cheapest subparts by $f$; for $\{s,t\}$-refinements, generate candidates by including/excluding crossing pieces.
• Achieved approximations: $2$-approximation for posimodular $f$; $4/3$-approximation for monotone $f$. These match the best-known unconstrained results.

Application 2: Hypergraph Orientation with $(k,(s,t),\ell)$ Demands

Given a hypergraph $G=(V,E)$, the problem is to orient each hyperedge so that the resulting directed hypergraph is $k$-hyperarc-connected and contains at least $\ell$ edge-disjoint $s \to t$ paths. This orientation exists if and only if

$$|\delta_G(P)| \geq \sum_{X \in P} p_{k,\ell}^{s,t}(X) \text{ for every partition } P,$$

where $p_{k,\ell}^{s,t}(X) = k$ except when $t \in X \subseteq V \setminus \{s\}$, in which case $p = \ell$ (Frank–Király–Király characterization).

Algorithmic steps include:

• Verifying the cut-condition via (submodular) partition minimization problems over all partitions and all $\{s,t\}$-separating partitions.
• For each vertex $v \in V$, solve minimum-cut problems on $G-v$ to identify necessary indegree assignments.
• Apply the Hakimi/Frank covering-min theorem for submodular functions to realize feasible indegree assignments matching the cut-conditions.
• The overall orientation is computable in polynomial time and satisfies both $k$-hyperarc-connectivity and $\ell$ $s \to t$ path connectivity.

5. Illustrative Example

Consider $V = \{s,a,t\}$, and $f$ the cut-function of a graph with edge weights $w(s,a)=2$, $w(a,t)=3$, $w(s,t)=4$. Then $f(S)$ equals the sum of weights crossing $(S, V\setminus S)$. For terminals $s,t$:

• For small $\lambda$, the best bipartition separating $s,t$ is $\{\{s\},\{a,t\}\}$ with cost $6-2\lambda$; $\mu_1 = 1$.
• For $\lambda > 1$, the best tripartition is $\{\{s\},\{a\},\{t\}\}$ with cost $9-3\lambda$.

In this instance, the $\{s,t\}$-PSP coincides with the classical PSP, as only these two partitions separate $s$ and $t$.

6. Theoretical Significance and Context

The development of the $\{s,t\}$-Separating Principal Sequence of Partitions generalizes existing PSP theory by incorporating explicit separation constraints. This structure retains laminarity and refinement properties of classical PSP while introducing novel uncrossing phenomena dictated by separation. The algorithmic results substantiate the broader utility of parametric submodular partition frameworks for constrained partitioning and orientation problems in combinatorial optimization, with demonstrated exactness and approximation guarantees.

For details of proofs, construction, and further extensions, see (Bérczi et al., 29 Oct 2025).