Tight Algorithm and Hardness for Submodular Linear Ordering

2026-06-18

AI Summary

This work addresses the minimum linear ordering problem under general submodular functions, which seeks a permutation minimizing the sum of function values over all prefixes. Moving beyond prior results restricted to symmetric submodular functions, the paper establishes the first tight approximability characterization for the general case. It presents a polynomial-time algorithm achieving an $O(\sqrt{n/\ln n})$ approximation ratio and complements this with an information-theoretic lower bound in the value oracle model, showing that no polynomial-query algorithm can attain an $o(\sqrt{n/\ln n})$ approximation. By integrating techniques from combinatorial optimization, submodular analysis, and computational complexity, this study fully delineates the approximability limits of the problem.

Abstract

We consider the Minimum Linear Ordering Problem: given a ground set $N$ of cardinality $n$ and a non-negative set function $f\colon 2^N\rightarrow \mathbb{R}_{\geq 0}$, the goal is to find an ordering $π$ of $N$ that minimizes the sum of the values of $f$ over all prefixes of $π$. This problem has been studied for various classes of set functions, and the case of a submodular $f$ is of special interest, as it captures classic problems including Minimum Linear Arrangement and Minimum Containing Interval Graph. In this work, we resolve the approximability of the Minimum Linear Ordering Problem for a general submodular $f$ by establishing matching upper and lower bounds and present: $(1)$ a polynomial-time algorithm achieving an $O(\sqrt{n/\ln n})$-approximation; and $(2)$ a matching information-theoretic hardness result, showing that no algorithm evaluating $f$ a polynomial number of times can achieve an $o(\sqrt{n/\ln n})$-approximation. Previously, the best known hardness of approximation was $2$, and an $O(\sqrt{n/\ln n})$-approximation was known only for the special case where $f$ is both submodular and symmetric.

Problem

• Submodular
• Linear Ordering
• Approximability
• Hardness of Approximation
• Minimum Linear Arrangement

Innovation

• submodular optimization
• linear ordering
• approximation algorithm
• hardness of approximation
• tight bounds