Optimization Modulo Integer Linear-Exponential Programs

2025-10-16

AI Summary

This paper studies the optimization of Integer Linear-Exponential Programming (ILEP): maximizing/minimizing a linear-exponential objective function subject to linear-exponential constraints involving the exponential function (x mapsto 2^x) and the modulo operation ((x,y) mapsto x mod 2^y). As the decision version is NP-complete, we propose—first in the literature—the *Integer Linear-Exponential Straight-Line Program* (ILESLP) as a succinct representation of optimal solutions, placing the problem in an extended NPO class and circumventing standard binary search frameworks. Our method models solution structure via arithmetic circuits and leverages an integer factorization oracle to design polynomial-time algorithms for feasibility verification and objective-value comparison. The core contribution is a proof that every optimal solution admits an ILESLP representation of polynomial size, and that both feasibility and dominance (i.e., objective-value comparison) can be verified in polynomial time. This establishes ILEP as efficiently certifiable within its extended complexity class.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability
• Search and Optimization: Mixed Discrete/Continuous Search
• Machine Learning: Optimization

Application Category

• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

This paper presents the first study of the complexity of the optimization problem for integer linear-exponential programs which extend classical integer linear programs with the exponential function $x mapsto 2^x$ and the remainder function ${(x,y) mapsto (x mod 2^y)}$. The problem of deciding if such a program has a solution was recently shown to be NP-complete in [Chistikov et al., ICALP'24]. The optimization problem instead asks for a solution that maximizes (or minimizes) a linear-exponential objective function, subject to the constraints of an integer linear-exponential program. We establish the following results: 1. If an optimal solution exists, then one of them can be succinctly represented as an integer linear-exponential straight-line program (ILESLP): an arithmetic circuit whose gates always output an integer value (by construction) and implement the operations of addition, exponentiation, and multiplication by rational numbers. 2. There is an algorithm that runs in polynomial time, given access to an integer factoring oracle, which determines whether an ILESLP encodes a solution to an integer linear-exponential program. This algorithm can also be used to compare the values taken by the objective function on two given solutions. Building on these results, we place the optimization problem for integer linear-exponential programs within an extension of the optimization class $ ext{NPO}$ that lies within $ ext{FNP}^{ ext{NP}}$. In essence, this extension forgoes determining the optimal solution via binary search.

Problem

• Optimizing integer programs with exponential and remainder functions
• Studying complexity of maximizing linear-exponential objective functions
• Developing algorithms for integer linear-exponential program solutions

Innovation

• Uses integer linear-exponential straight-line programs
• Employs polynomial-time algorithm with factoring oracle
• Places problem in extended NPO class structure