arXiv subjects

Robert Hildebrand

Publications and source records attributed to Robert Hildebrand.

At least 19 recordsLinked to original sources

arXiv2026

Bounded Cubic Integer Programming in Fixed Dimension

We consider the exact minimization of a rational cubic polynomial over the integer points of a rational polytope in fixed dimension. Del Pia, Hildebrand, Weismantel, and Zemmer proved polynomial-time solvability in dimension two, while quartic polynomial minimization is already NP-hard in dimension two. We show that the bounded cubic result extends to every fixed dimension. The proof combines two exact identities for cubic polynomials with three ingredients from the recent fixed-dimensional algorithms for integer quadratic programming of Ari and Hildebrand: symmetric displacement covers, negative-displacement search for quadratic forms, and integer-query convex feasibility. At an integer query point, a negative Hessian direction yields a linear curvature cut valid for every global minimizer in the current cell, while absence of such a direction yields a linear objective cut through an exact endpoint-Hessian identity. These cuts define an integer-query separation oracle for the convex hull of the curvature-admissible sublevel points. Integer convex feasibility via ellipsoids and lattice algorithms then gives exact optimization without computing lattice centerpoints. A real-algebraic localization bound extends the result to unbounded polyhedra with a bounded real improving sublevel, including all coercive objectives. The argument also explains why degree three is a natural boundary for this approach: the symmetric second difference of a cubic is exactly its directional Hessian, which is affine in the basepoint; for quartics an additional fourth-order term appears.

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arXiv2026

Curvature batching for integer and mixed-integer quadratic programming

We give an explicit algorithm for integer quadratic programming with running time $(nL)^{O(n^2)} \mathrm{poly}(ϕ)$, where $n$ is the number of variables, $L$ bounds the absolute entries of the constraint and quadratic matrices, and $ϕ$ is the total input encoding length. The linear objective and constraint right-hand sides enter only through $ϕ$. The algorithm refines Lokshtanov's branching framework. Small integer kernel directions identify optima near constraint boundaries. When all chosen basis directions have nonnegative curvature, their gradient values can be imposed in one batch. The quadratic form then vanishes on the remaining kernel, and the residual problem is an integer linear program. Thus determinant growth after the batch does not feed into further branching. Over a polytope, the method extends to mixed-integer quadratic programming with $q$ continuous variables in time $(nL)^{O(n^2 (q+1)^2)} \mathrm{poly}(ϕ)$, without a convexity assumption on the continuous block. We also decide and certify integer and mixed-integer unboundedness in $(m+n)^{O(n)} \mathrm{poly}(ϕ)$ time for arbitrary rational polyhedra with $m$ inequalities. Noncopositivity on polyhedral cones and integer unboundedness are W[1]-hard in the dimension and admit no $(m+ϕ)^{o(n)}$ algorithm under ETH, matching the linear order of dimension in the recession-test exponent. Companion results give a right-hand-side-independent bound for concave integer minimization and explicit runtimes for applications of IQP.

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arXiv2026

Bounded Integer Quadratic Programming through Parallelepiped Covers and Discrete Convic Optimization

We give an exact algorithm for minimizing an arbitrary rational quadratic polynomial $x^T Q x + c^T x + γ$ over the integer points of a bounded rational polyhedron $\{x : Ax \le b\}$. For $n$ variables and $m$ inequalities, the running time is $$2^{O(n \log(n+1))} (m+1)^{O(n)} β^{O(n)} (1+L)^{O(1)},$$ where $β$ is one plus the maximum binary encoding length of an entry of $A$ or $Q$, and $L$ is the full input encoding length. The polynomial degree in $L$ is absolute; in particular, the encoding of $b$, $c$, and $γ$ enters only this fixed-degree factor. Our algorithm refines the integral parallelepiped cover of Goemans and Rothvoss into cells with controlled displacement sets. On each cell, either an integer displacement of negative quadratic value rules out every point as a global minimizer, or the quadratic identity supplies a separation oracle for the convex hull of each integer objective sublevel set. The integer-query feasibility theorem of Hildebrand and Goess then finds the cell minimum. A refinement through scaled lattices solves the displacement search, and a localization argument separates the encoding dependence of the constraint matrix and quadratic part from that of the remaining data. Combining the bounded algorithm with the unboundedness test of Ari and Hildebrand extends the method to arbitrary rational polyhedra.

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arXiv2026

Integer Maximization over $\ell_p$ Balls: Hardness and Exact Algorithms

We study the problem of maximizing a linear function over the integer points of an origin-centered $\ell_p$ ball, which we call \BallIPp{p}. For every fixed integer $p\ge2$, we prove that the decision problem over an $\ell_p$-ball is NP-complete. We then focus on the Euclidean case and study how the difficulty of the problem depends on the numerical parameters of the instance. We give two complementary pseudo-polynomial exact algorithms. The first specializes a known radius-budget dynamic program; the same nonlinear-knapsack framework also applies to every fixed finite integer $p$. We then develop a complementary dynamic program over candidate objective values for the Euclidean case. The latter is polynomial in the encoding size of the radius and pseudo-polynomial in the magnitude of the cost coefficients. For a fixed objective value, feasibility can be formulated as a closest vector problem (CVP) instance. This connection gives an exact algorithm whose running time depends on the covering radius of that lattice. Conversely, we show that a rank-$k$ Euclidean closest-vector instance in ambient dimension $d$ reduces to the decision version of a Euclidean \BallIP{} instance in dimension $d+k\le2d$, transferring known bounds under the Exponential Time Hypothesis (ETH). Finally, we study dimension-dependent approaches based on proximity to the continuous optimizer and describe the fixed-level constructions that extend to ellipsoids.

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arXiv2026

Complexity of Integer Programming in Reverse Convex Sets via Boundary Hyperplane Cover

We study the complexity of identifying the integer feasibility of reverse convex sets. We present various settings where the complexity can be either NP-Hard or efficiently solvable when the dimension is fixed. Of particular interest is the case of bounded reverse convex constraints with a polyhedral domain. We introduce a structure, \emph{Boundary Hyperplane Cover}, that permits this problem to be solved in polynomial time in fixed dimension provided the number of nonlinear reverse convex sets is fixed.

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arXiv2026

Cooperative Integer Programming Games: Core Stability and Optimal Coalition Structures

We introduce cooperative integer programming games (CIPGs), in which agents pool budget constraints to accomplish indivisible tasks jointly and the characteristic function maps every coalition to the optimal value of a pooled integer program. Our goal is to identify an optimal coalition structure (OCS) and a stable one (OSCS). We derive a stability inequality that keeps each formed coalition in the Core with respect to itself, and present two mixed-integer OCS formulations, aggregated and disaggregated, proving that the disaggregated formulation is integer-equivalent yet yields a tighter LP relaxation. Building on the stability inequality we develop lifted stability cuts, several separation strategies inside a cutting-plane algorithm, an SCS-feasible primal heuristic that constructs warm starts with guaranteed stability, and a payoff-refinement step computing the Shapley value and the nucleolus of every formed coalition. On benchmark cooperative knapsack games, the method certifies optimality with up to 16 players and reaches MIP gaps below 1% at 30 players while evaluating 766 of the roughly $10^9$ coalition values.

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arXiv2026

Automating Idealness Proofs for Binary Programs with Application to Rectangle Packing

An integer program is called ideal if its continuous relaxation coincides with its convex hull allowing the problem to be solved as a continuous program and offering substantial computational advantages. Proving idealness analytically can be extraordinarily tedious -- even for small formulations -- such proofs often span many pages of intricate case analysis which motivates the development of automated verification methods. We develop a general-purpose framework for certifying idealness in Mixed Binary Linear Programs (MBLPs), formulating the verification problem as a linear program when the data is fixed and as a nonconvex quadratic program when the data is parametric. We apply this framework to study several formulations of the rectangle packing problem that are conjectured to be pairwise-ideal, obtaining computational proofs where analytic proofs were previously unknown or impractical. As our second contribution, we introduce and model a novel generalization of the rectangle packing problem that enforces edge clearances between selected rectangles. We present both existing and novel MBLP formulations which arise from different encodings of the underlying disjunctive constraints. We perform some computational experiments on these formulations under a strip-packing objective to determine the importance of pairwise-idealness in practice.

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arXiv2026

Optimal triangulations for piecewise linear approximations of non-convex variable products

We show optimal triangulations for piecewise linear (PWL) approximations of indefinite quadratic functions over the plane. Optimal triangulations have minimum triangle density while allowing a PWL approximation that fulfills a prescribed error bound measured in the L-infinity norm. In 2000, Pottmann et al. proved optimal triangulations for PWL interpolations and conjectured that these are also optimal for general PWL approximations. This conjecture was refuted in 2018 by Atariah et al., who allowed a constant deviation at the vertices of the triangles and decreased the triangle density by roughly 3%, though they left open whether their construction was optimal. In this paper, we resolve this open question: allowing varying deviations at vertices reduces the optimal triangle density by 25% compared to Atariah et al., and we prove this is globally optimal. We thus show that the potential of general PWL approximations is significantly lower for indefinite than for definite quadratic functions, where the triangle density can be halved when allowing general approximations compared to interpolations. Furthermore, we prove that among parallelogram tilings -- triangulations built from translated copies of a triangle and its point-reflection -- the constant-deviation construction of Atariah et al. is optimal when continuity of the PWL approximation is required. We conjecture that this optimality extends to all continuous triangulations, not just those based on parallelograms.

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arXiv2026

Optimizing Representation in Redistricting: Dual Bounds for Partitioning Problems with Non-Convex Objectives

We investigate optimization models for the purpose of computational redistricting. Our focus is on nonconvex objectives for estimating expected Black Representatives and Political Representation. The objectives are a composition of a ratio of variables and a normal distribution's cumulative distribution function (or ``probit curve"). We extend the work of Validi et al.~\cite{validi2022imposing}, which presented a robust implementation of contiguity constraints. By developing mixed integer linear programming models that closely approximate the parent nonlinear model, our approaches yield tight bounds on these optimization problems. We exhibit the effectiveness of these approaches on county-level data.

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arXiv2026

Random-Restart Best-Response Dynamics for Large-Scale Integer Programming Games and Their Applications

This paper presents scalable algorithms for computing pure Nash equilibria (PNEs) in large-scale integer programming games (IPGs), where existing exact methods typically handle only small numbers of players. Motivated by a county-level aquatic invasive species (AIS) prevention problem with 84 decision makers, we develop and analyze random-restart best-response dynamics (RR-BRD), a randomized search framework for PNEs. For IPGs with finite action sets, we model RR-BRD as a Markov chain on the best-response state graph and show that, whenever a PNE exists and the restart law has positive probability of reaching a PNE within the round cap, RR-BRD finds a PNE almost surely. We also propose a Monte Carlo sampling-and-simulation procedure to estimate success behavior under a fixed round cap, which informs our instance-dependent performance characterization. We then embed RR-BRD as a randomized local-search subroutine within the zero-regret (ZR) framework, yielding BRD-incorporated zero-regret (BZR). Using solver callbacks, RR-BRD searches for and supplies PNEs, while ZR separates and adds equilibrium inequalities to tighten the formulation. We introduce edge-weighted budgeted maximum coverage (EBMC) games to model AIS prevention and establish PNE existence results for both selfish and locally altruistic utilities. Computational experiments on synthetic EBMC and knapsack problem game instances show that RR-BRD and BZR scale equilibrium computation up to $n \le 30$ players. We further solve a real-world EBMC game derived from the Minnesota AIS dataset with $n = 84$ county players.

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arXiv2025

Equivariant Perturbation in Gomory and Johnson's Infinite Group Problem. IV. The General Unimodular Two-Dimensional Case

We study an abstract setting for cutting planes for integer programming called the infinite group problem. In this abstraction, cutting planes are computed via cut generating function that act on the simplex tableau. In this function space, cut generating functions are classified as minimal, extreme, and facets as a proxy for understanding the strength or potential importance of these functions. Prior work developed algorithms for testing minimality, extremality, and facetness for cut generating functions applied to 1-row tableau and to some 2-row tableau in a restricted setting. We complement and generalize this work by giving an algorithm for testing the extremality of a large class of minimal valid functions for the two-dimensional infinite group problem. Along the way, we develop results of independent interest on functional equations and infinite systems of linear equations.

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arXiv2024

Job Shop Scheduling with Integer Programming, Shifting Bottleneck, and Decision Diagrams: A Computational Study

We study heuristic algorithms for job shop scheduling problems. We compare classical approaches, such as the shifting bottleneck heuristic with novel strategies using decision diagrams. Balas' local refinement is used to improve feasible solutions. Heuristic approaches are combined with Mixed Integer Programming and Constraint Programming approaches. We discuss our results via computational experiments.

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arXiv2024

Continuous Equality Knapsack with Probit-Style Objectives

We study continuous, equality knapsack problems with uniform separable, non-convex objective functions that are continuous, antisymmetric about a point, and have concave and convex regions. For example, this model captures a simple allocation problem with the goal of optimizing an expected value where the objective is a sum of cumulative distribution functions of identically distributed normal distributions (i.e., a sum of inverse probit functions). We prove structural results of this model under general assumptions and provide two algorithms for efficient optimization: (1) running in linear time and (2) running in a constant number of operations given preprocessing of the objective function.

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arXiv2024

Automating Idealness Proofs for Binary Programs with Application to Rectangle Packing

We develop an optimization framework for identifying ideal Mixed Binary Linear Programs (MBLP) which is linear when using known input data and nonconvex quadratic over parametric input data. These techniques are applied to various formulations for rectangle packing, conjectured to be pairwise-ideal. Additionally, we address a variation of the rectangle packing problem which incorporates clearances along selected edges of the packed objects. We present both existing and novel MBLP formulations for the underlying disjunctive program and investigate the poor performance of Gurobi's default branch-and-cut methodology. We operate under a strip-packing objective that aims to minimize the overall height of the packed objects.

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arXiv2023

Enhancements of Discretization Approaches for Non-Convex Mixed-Integer Quadratically Constraint Quadratic Programming: Part I

We study mixed-integer programming (MIP) relaxation techniques for the solution of non convex mixed-integer quadratically constrained quadratic programs (MIQCQPs). We present MIP relaxation methods for non convex continuous variable products. In Part I, we consider MIP relaxations based on separable reformulation. The main focus is the introduction of the enhanced separable MIP relaxation for nonconvex quadratic products of the form z=xy, called hybrid separable (HybS). Additionally, we introduce a logarithmic MIP relaxation for univariate quadratic terms, called sawtooth relaxation. We combine the latter with HybS and existing separable reformulations to derive MIP relaxations of MIQCQPs. We provide a comprehensive theoretical analysis of these techniques, underlining the theoretical advantages of HybS compared to its predecessors. We perform a broad computational study to demonstrate the effectiveness of the enhanced MIP relaxation in terms of producing tight dual bounds for MIQCQPs. In Part II, we study MIP relaxations that extend the well-known MIP relaxation normalized multiparametric disaggregation technique (NMDT) and present further theoretical and computational analyses.

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arXiv2023

Enhancements of Discretization Approaches for Non-Convex Mixed-Integer Quadratically Constraint Quadratic Programming: Part II

This is Part II of a study on mixed-integer programming (MIP) relaxation techniques for the solution of non-convex mixed-integer quadratically constrained quadratic programs (MIQCQPs). We set the focus on MIP relaxation methods for non-convex continuous variable products and extend the well-known MIP relaxation normalized multi-parametric disaggregation technique (NMDT), applying a sophisticated discretization to both variables. We refer to this approach as doubly discretized normalized multiparametric disaggregation technique (D-NMDT). In a comprehensive theoretical analysis, we underline the theoretical advantages of the enhanced method D-NMDT compared to NMDT. Furthermore, we perform a broad computational study to demonstrate its effectiveness in terms of producing tight dual bounds for MIQCQPs. Finally, we compare D-NMDT to the separable MIP relaxations from Part I and a state-of-the-art MIQCQP solver.

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arXiv2022

Extension Complexity Lower Bounds for Mixed-Integer Extended Formulations

We prove that any mixed-integer linear extended formulation for the matching polytope of the complete graph on $n$ vertices, with a polynomial number of constraints, requires $Ω(\sqrt{\sfrac{n}{\log n}})$ many integer variables. By known reductions, this result extends to the traveling salesman polytope. This lower bound has various implications regarding the existence of small mixed-integer mathematical formulations of common problems in operations research. In particular, it shows that for many classic vehicle routing problems and problems involving matchings, any compact mixed-integer linear description of such a problem requires a large number of integer variables. This provides a first non-trivial lower bound on the number of integer variables needed in such settings.

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arXiv2022

Complexity, Exactness, and Rationality in Polynomial Optimization

We focus on rational solutions or nearly-feasible rational solutions that serve as certificates of feasibility for polynomial optimization problems. We show that, under some separability conditions, certain cubic polynomially constrained sets admit rational solutions. However, we show in other cases that it is NP Hard to detect if rational solutions exist or if they exist of any reasonable size. We extend this idea to various settings including near feasible, but super optimal solutions and detecting rational rays on which a cubic function is unbounded. Lastly, we show that in fixed dimension, the feasibility problem over a set defined by polynomial inequalities is in NP by providing a simple certificate to verify feasibility. We conclude with several related examples of irrationality and encoding size issues in QCQPs and SOCPs.

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