[Paper Review] Submodularity on a tree: Unifying $L^ atural$-convex and bisubmodular functions

Vladimir Kolmogorov|arXiv (Cornell University)|Jul 7, 2010

Complexity and Algorithms in Graphs31 references4 citations

TL;DR

This paper introduces a new class of functions called strongly tree-submodular functions, defined on variables whose domains are nodes of rooted binary trees with specific operations $ abla$ and $ abla$ derived from tree structure. The authors present a polynomial-time algorithm combining Murota’s steepest descent for $L^{ atural}$-convex functions and bisubmodular minimization techniques, proving that minimization is tractable for this class, which unifies and generalizes $L^{ atural}$-convex and bisubmodular functions.

ABSTRACT

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 $L^ atural$-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 $L^ atural$-convex functions with bisubmodular minimization algorithms.

Motivation & Objective

• To unify $L^{ atural}$-convex and bisubmodular functions under a common framework using tree-structured domains.
• To define a new class of functions—strongly tree-submodular—based on tree operations $ abla$ and $ abla$ for variables with labels on rooted binary trees.
• To present a polynomial-time algorithm for minimizing such functions in the value oracle model.
• To generalize existing tractable classes of discrete functions by extending the scope of operations beyond lattices and chains.

Proposed method

• Define a rooted binary tree for each variable’s domain, with labels as nodes and a partial order based on ancestry.
• Introduce component-wise operations $ abla$ and $ abla$ using the path between two nodes: $ abla$ and $ abla$ are the two middle nodes (or their floor/ceiling) on the path between $a$ and $b$, with $ abla$ being the ancestor.
• Formulate the strong tree-submodularity condition: $f(\mathbf{x}) + f(\mathbf{y}) \geq f(\mathbf{x} \nabla \mathbf{y}) + f(\mathbf{x} \nabla \mathbf{y})$ for all $\mathbf{x}, \mathbf{y}$.
• Adapt Murota’s steepest descent algorithm for $L^\natural$-convex functions and integrate it with bisubmodular minimization techniques.
• Prove that the resulting algorithm minimizes strongly tree-submodular functions in time polynomial in $n$ and $\max_i |D_i|$.
• Extend the framework to weakly tree-submodular functions using meet ($\wedge$) and join ($\vee$) operations, and show tractability for specific tree structures like the one in Figure 1(d).

Experimental results

Research questions

• RQ1 Can $L^\natural$-convex and bisubmodular functions be unified under a single framework using tree-structured domains and operations?
• RQ2 Is there a polynomial-time algorithm for minimizing functions satisfying the strong tree-submodularity condition on rooted binary trees?
• RQ3 What is the complexity of minimizing strongly tree-submodular functions when the underlying trees are non-binary (i.e., nodes with three or more children)?
• RQ4 Are there additional tree structures beyond chains and the diamond-shaped tree (Figure 1c) for which weakly tree-submodular functions can be minimized efficiently?
• RQ5 Can the tractability of minimization be characterized for operations derived from lattices, trees, and their combinations?

Key findings

• The class of strongly tree-submodular functions generalizes both $L^\natural$-convex and bisubmodular functions, with the former arising from chain-structured trees and the latter from a specific three-node tree.
• A polynomial-time algorithm for minimizing strongly tree-submodular functions is presented, combining Murota’s steepest descent and bisubmodular minimization techniques.
• The algorithm runs in time polynomial in $n$ and $\max_i |D_i|$, establishing tractability for the new class.
• The paper proves that strong tree-submodularity implies weak tree-submodularity, justifying the terminology.
• For weakly tree-submodular functions, tractability is shown for a new tree structure (Figure 1d), where each node has a root and two leaves at depth $K$, using a lifting to a signed ring family.
• The framework suggests that tractable minimization may extend to other tree and lattice-based operations, though the full characterization remains open.