Submodularity on a tree: Unifying $L^natural$-convex and bisubmodular functions

We introduce a new class of functions that can be minimized in polynomial time in the value oracle model. These are functions $f$ satisfying $f(x)+f(y)\ge f(x \sqcap y)+f(x \sqcup y)$ where the domain of each variable $x_i$ corresponds to nodes of a rooted binary tree, and operations $\sqcap,\sqcup$ are defined with respect to this tree. Special cases include previously studied $L^\natural$-convex and bisubmodular functions, which can be obtained with particular choices of trees. We present a polynomial-time algorithm for minimizing functions in the new class. It combines Murota’s steepest descent algorithm for $L^\natural$-convex functions with bisubmodular minimization algorithms.

💡 Research Summary

The paper introduces a novel class of discrete functions—tree‑submodular functions—that can be minimized in polynomial time when accessed via a value oracle. The domain of each variable is the set of nodes of a rooted binary tree, and two binary operations, meet (⊓) and join (⊔), are defined with respect to the tree’s ancestor‑descendant relationship. A function f : D → ℝ (where D is the Cartesian product of the node sets) is called tree‑submodular if for every pair of points x, y ∈ D the inequality

  f(x) + f(y) ≥ f(x ⊓ y) + f(x ⊔ y)

holds, where the operations are applied component‑wise. This inequality generalizes the classic submodularity condition: when the underlying tree is a linear chain, ⊓ and ⊔ coincide with component‑wise min and max, and the definition reduces to the well‑known Lⁿᵃᵗᵤʳᵃʟ‑convex (L‑natural convex) condition. When the tree consists of a root with two children (depth‑1 binary tree), the domain of each variable is the three‑valued set {−1, 0, +1} and ⊓, ⊔ become the “∧” and “∨” operators used in bisubmodular theory; thus bisubmodular functions appear as a special case. By varying the shape of the tree, the framework captures a whole spectrum of hierarchical label spaces, ranging from simple chains to multi‑level trees that naturally model hierarchical clustering, multi‑label classification, or phylogenetic structures.

The authors’ main technical contribution is a polynomial‑time minimization algorithm for any tree‑submodular function. The algorithm is a hybrid that blends two previously independent techniques:

1. Murota’s steepest‑descent algorithm for L‑natural convex functions. In the tree setting, a “steepest direction” corresponds to moving a single variable up or down one level in its tree (i.e., replacing x_i by its parent or a child). The algorithm evaluates the marginal decrease for each possible move, selects the best one, and repeats until no single‑step improvement exists. Because each move reduces the objective by at least one unit (in the integer‑valued oracle model), the number of iterations is bounded by the initial gap f(x⁰) − f*.

2. Bisubmodular minimization subroutines (e.g., the Iwata‑Fleischer‑Fujishige algorithm). When a set of variables can be improved simultaneously—typically those belonging to different sub‑trees—the authors show that the restriction of f to this subset can be reformulated as a bisubmodular function. Consequently, a global improvement can be found by invoking a bisubmodular minimizer on the restricted problem.

The overall procedure alternates between a steepest‑descent phase (local, single‑coordinate moves) and a bisubmodular phase (global moves across a chosen subset). Each steepest‑descent iteration requires O(n·log |V|) oracle calls, where |V| is the number of nodes in the tree, because locating a parent or child is a logarithmic operation in a binary tree. The bisubmodular phase runs in polynomial time as well, with complexity O(n·m·log U) where m is the number of interacting sub‑trees and U bounds the magnitude of function values. Since the total number of outer iterations is bounded by the value gap, the entire algorithm runs in time polynomial in n, |V|, and the encoding length of the oracle answers.

Beyond the algorithmic result, the paper provides a unifying theoretical perspective. By showing that L‑natural convexity and bisubmodularity are merely two extremal choices of the underlying tree, the authors clarify the structural relationship between these two previously disparate areas of discrete convex analysis. This insight suggests that many other “exotic” discrete convexities—such as those arising from multi‑branch trees or trees with varying branching factors—can be treated within the same analytical framework.

The paper also discusses several applications and extensions. Hierarchical labeling problems in computer vision, where each pixel may take a label from a tree‑structured taxonomy, fit naturally into the model. Similarly, phylogenetic inference and multi‑scale network design can be expressed as tree‑submodular minimization tasks. The authors sketch how the method could be extended to non‑binary trees or even directed acyclic graphs, and they point out that approximate versions of the algorithm could be useful when exact oracle access is costly.

In summary, the work makes three key contributions: (1) it defines a broad class of tree‑submodular functions that subsumes L‑natural convex and bisubmodular functions; (2) it devises a polynomial‑time minimization algorithm by integrating steepest‑descent and bisubmodular minimization techniques; and (3) it opens a pathway for applying discrete convex analysis to a wide range of hierarchical and multi‑level optimization problems. The results are both theoretically elegant—providing a clean unification of two major strands of discrete convexity—and practically valuable, offering a concrete tool for solving complex combinatorial optimization problems that exhibit tree‑structured domains.