Separable convex optimization over indegree polytopes

2025-09-07

AI Summary

This paper investigates the fair acyclic orientation problem on undirected graphs—optimizing convex objective functions over the vertices of the indegree polytope (i.e., acyclic orientations). Specifically, it considers minimizing the sum of a strictly convex function φ applied to vertex indegrees, decreasing minimization (dec-min), and increasing maximization (inc-max). The work reveals that acyclicity breaks the equivalence among these three fairness criteria—a key structural insight—and provides the first systematic proof of NP-hardness across multiple settings: arbitrary discrete strictly convex φ, bounded indegree k ≥ 2, unbounded indegree, and complementary inc-min/dec-max on 3-regular graphs. Methodologically, it introduces weighted minimum-priority ordering, an exact exponential-time algorithm, and approximation strategies. It achieves polynomial-time solvability for maximum weighted indegree and devises a 3-approximation for the product-sum optimization. The study precisely characterizes the tractability boundary, with implications for scheduling and deadlock-free routing.

Technology Category

• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Planning, Routing, and Scheduling: Scheduling under Uncertainty
• Search and Optimization: Combinatorial Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Economics, Online Markets and Human Computation: Fairness, privacy, and diversity in economic environments
• Social Networks and Social Media: Fairness and bias in social network and social media analysis

Abstract

We study egalitarian (acyclic) orientations of undirected graphs under indegree-based objectives, such as minimizing the $varphi$-sum of indegrees for a strictly convex function $varphi$, decreasing minimization (dec-min), and increasing maximization (inc-max). In the non-acyclic setting of Frank and Murota (2022), a single orientation simultaneously optimizes these three objectives, however, restricting to acyclic orientations confines us to the corners of the indegree polytope, where these fairness objectives do diverge. We establish strong hardness results across a broad range of settings: minimizing the $varphi$-sum of indegrees is NP-hard for every discrete strictly convex function $varphi$; dec-min and inc-max are NP-hard for every indegree bound $k geq 2$, as well as without a bound; and the complementary inc-min and dec-max problems are NP-hard even on $3$-regular graphs. On the algorithmic side, we give a polynomial-time algorithm for minimizing the maximum weighted indegree via a weighted smallest-last ordering. We also provide an exact exponential-time algorithm for minimizing general separable discrete convex objectives over indegrees, and a polynomial-time algorithm for the non-acyclic case. Finally, for maximizing the sum of the products of indegrees and outdegrees, we prove NP-hardness on graphs of maximum degree $4$, give an algorithm for maximum degree $3$, and provide a $3$-approximation algorithm. Our results delineate the algorithmic frontier of convex integral optimization over indegree (base-)polytopes, and highlight both theoretical consequences and practical implications, notably for scheduling and deadlock-free routing.

Problem

• Optimizing separable convex objectives over acyclic indegree polytopes
• Resolving NP-hardness for fairness objectives like dec-min and inc-max
• Developing algorithms for weighted indegree minimization and approximation methods

Innovation

• Weighted smallest-last ordering minimizes maximum indegree
• Exponential algorithm for separable convex indegree optimization
• Polynomial algorithm for non-acyclic convex optimization cases