Optimization Modulo Integer Linear-Exponential Programs

TL;DR

This work studies the optimization problem for Integer Linear-Exponential Programs (ILEP), which augment classical ILP with the exponential map $2^x$ and the remainder $x \bmod 2^y$. It establishes that every optimal solution, when it exists, has a succinct polynomial-size representation as an Integer Linear-Exponential Straight-Line Program (ILESLP), and it develops an OptILEP framework that preserves optimality while eliminating variables via a Bareiss-based scheme and monotone decompositions. A polynomial-time algorithm with a factoring oracle is provided to decide ILEP feasibility and to compare objective values without fully evaluating them, and the optimization problem is placed in the complexity class NPO-cmp, reflecting the need to compare rather than binary-search for optima. The paper also connects ILEP to broader logical frameworks (Semenov/Büchi-Semenov arithmetic), discusses certificate-based solvability, and outlines open questions, including fixed-dimension behavior and potential FP$\,NP$ improvements. Collectively, these results illuminate a distinct computational regime at the intersection of integer programming and exponential/remainder extensions, with certificates enabling efficient verification of optimal solutions even when bit-size blow-up is unavoidable in explicit encodings.

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 ${(x,y) \mapsto (x \bmod 2^y)}$. 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 $\text{NPO}$ that lies within $\text{FNP}^{\text{NP}}$. In essence, this extension forgoes determining the optimal solution via binary search.

Figures (3)

• Figure 1: Flowchart of the main loop of ChistikovMS24.
• Figure 2: Structure of the matrix $M_0$. Observe that all coefficients of the variables in $\bm q_k$ have $\mu$ as a common factor. This is because $(C,\langle{\gamma} \mathbin{;} {\psi}\rangle\xspace)$ belongs to $\mathcal{I}\xspace_k^{0}$.
• Figure : Instantiation of $\textsc{ElimVars}$ (\ref{['algo:gaussopt']}).

Theorems & Definitions (162)

• Example 1
• Theorem 0
• Lemma 0
• Lemma 0
• Proposition 1
• Proposition 2
• Corollary 1
• Example 2: ILP with divisibility constraints
• Example 3
• Example 4