Abstract: We construct a new polynomial commitment scheme for univariate and multivariate polynomials over finite fields, with logarithmic size evaluation proofs and verification time, measured in the number of coefficients of the polynomial. The underlying technique is a Diophantine Argument of Knowledge (DARK), leveraging integer representations of polynomials and groups of unknown order. Security is shown from the strong RSA and the adaptive root assumptions. Moreover, the scheme does not require a trusted setup if instantiated with class groups. We apply this new cryptographic compiler to a restricted class of algebraic linear IOPs, which we call Polynomial IOPs, to obtain doubly-efficient public-coin interactive arguments of knowledge for any NP relation with succinct communication. With linear preprocessing, the online verifier’s work is logarithmic in the circuit complexity of the relation.

Abstract: Machine learning components commonly appear in larger decision-making pipelines; however, the model training process typically focuses only on a loss that measures average accuracy between predicted values and ground truth values. Decision-focused learning explicitly integrates the downstream decision problem when training the predictive model, in order to optimize the quality of decisions induced by the predictions. It has been successfully applied to several limited combinatorial problem classes, such as those that can be expressed as linear programs (LP), and submodular optimization. However, these previous applications have uniformly focused on problems with simple constraints. Here, we enable decision-focused learning for the broad class of problems that can be encoded as a mixed integer linear program (MIP), hence supporting arbitrary linear constraints over discrete and continuous variables. We show how to differentiate through a MIP by employing a cutting planes solution approach, an algorithm that iteratively tightens the continuous relaxation by adding constraints removing fractional solutions. We evaluate our new end-to-end approach on several real world domains and show that it outperforms the standard two phase approaches that treat prediction and optimization separately, as well as a baseline approach of simply applying decision-focused learning to the LP relaxation of the MIP. Lastly, we demonstrate generalization performance in several transfer learning tasks.

Abstract: Counterfactual Explanation (CE) is one of the posthoc explanation methods that provides a perturbation vector so as to alter the prediction result obtained from a classifier. Users can directly interpret the perturbation as an ”action” for obtaining their desired decision results. However, an action extracted by existing methods often becomes unrealistic for users because they do not adequately care about the characteristics corresponding to the empirical data distribution such as feature-correlations and outlier risk. To suggest an executable action for users, we propose a new framework of CE for extracting an action by evaluating its reality on the empirical data distribution. The key idea of our proposed method is to define a new cost function based on the Mahalanobis’ distance and the local outlier factor. Then, we propose a mixed-integer linear optimization approach to extracting an optimal action by minimizing our cost function. By experiments on real datasets, we confirm the effectiveness of our method in comparison with existing methods for CE.

Abstract: In this paper, a new one-dimensional discrete-space chaotic map based on the multiplication of integer numbers and circular shift is presented. Dynamical properties of the proposed map are analyzed, and it exhibits chaotic behavior. The proposed map has fixed points for certain settings, but it is easy to completely avoid them. This map preserves all desirable properties of previous discrete-space chaotic maps and has improved characteristics related to orbit length, computational complexity and memory requirements. These improvements can be particularly useful when implementation in digital devices, which have limited memory and computational resources, is needed. S-box design method based on this chaotic map is presented as an example of its application in cryptography. The results of performance tests show that S-boxes with good cryptographic properties can be generated on the basis of this discrete-space chaotic map.

Abstract: This paper considers the multi-vehicle automated driving coordination problem. We develop a distributed, hybrid decision-making framework for safe and efficient autonomous driving of selfish vehicles on multi-lane highways, where each dynamics is modeled as a mixed-logical–dynamical system. We formalize the coordination problem as a generalized mixed-integer potential game, seeking an equilibrium solution that generates almost individually optimal mixed-integer decisions, given the safety constraints. Finally, we embed the proposed best-response-based algorithms within the distributed open- and closed-loop control policies.

Abstract: In this paper, we describe a comprehensive algorithmic framework for solving mixed integer bilevel linear optimization problems (MIBLPs) using a generalized branch-and-cut approach. The framework presented merges features from existing algorithms (for both traditional mixed integer linear optimization and MIBLPs) with new techniques to produce a flexible and robust framework capable of solving a wide range of bilevel optimization problems. The framework has been fully implemented in the open-source solver MibS. The paper describes the algorithmic options offered by MibS and presents computational results evaluating the effectiveness of the various options for the solution of a number of classes of bilevel optimization problems from the literature.

Abstract: Multiobjective mixed integer convex optimization refers to mathematical programming problems where more than one convex objective function needs to be optimized simultaneously and some of the varia...

Abstract: We extend recent work on solving mixed-integer nonlinear optimal control problems (MIOCPs) to the case of integer control functions subject to constraints that involve a pointwise coupling of the s...

Abstract: Secure integer comparison has been one of the first problems introduced in cryptography, both for its simplicity to describe and for its applications. The first formulation of the problem was to enable two parties to compare their inputs without revealing the exact value of those inputs, also called the Millionaires’ problem [45]. The recent rise of fully homomorphic encryption has given a new formulation to this problem. In this new setting, one party blindly computes an encryption of the boolean \((a

Abstract: Selecting the most pertinent countermeasures to secure a system is one of the ultimate goals of risk assessment. In this context, it is important to rely on modeling methods that the security experts are already familiar with, so that the solution can be smoothly adopted within industry.We propose a full-fledged framework, relying on attack–defense trees and integer linear programming, to find an optimal set of countermeasures. We use attack–defense trees formalized with directed acyclic graphs. This enables us to conveniently reason about attacker’s actions that can contribute to several distinct attacks, and countermeasures that can block different ways of attacking. We provide a constructive way of extracting all reasonable behaviors of the two actors from such models. We then exploit this extracted information to formulate a generic solution, based on integer linear programing, to address a wide class of optimization problems. We show how to instantiate it for specific security-relevant optimization criteria. We cover deterministic and probabilistic cases. The framework has been implemented in a prototype tool, and validated in a real-life case study.

Abstract: Mixed Integer Programming (MIP) solvers rely on an array of sophisticated heuristics developed with decades of research to solve large-scale MIP instances encountered in practice. Machine learning offers to automatically construct better heuristics from data by exploiting shared structure among instances in the data. This paper applies learning to the two key sub-tasks of a MIP solver, generating a high-quality joint variable assignment, and bounding the gap in objective value between that assignment and an optimal one. Our approach constructs two corresponding neural network-based components, Neural Diving and Neural Branching, to use in a base MIP solver such as SCIP. Neural Diving learns a deep neural network to generate multiple partial assignments for its integer variables, and the resulting smaller MIPs for un-assigned variables are solved with SCIP to construct high quality joint assignments. Neural Branching learns a deep neural network to make variable selection decisions in branch-and-bound to bound the objective value gap with a small tree. This is done by imitating a new variant of Full Strong Branching we propose that scales to large instances using GPUs. We evaluate our approach on six diverse real-world datasets, including two Google production datasets and MIPLIB, by training separate neural networks on each. Most instances in all the datasets combined have $10^3-10^6$ variables and constraints after presolve, which is significantly larger than previous learning approaches. Comparing solvers with respect to primal-dual gap averaged over a held-out set of instances, the learning-augmented SCIP is 2x to 10x better on all datasets except one on which it is $10^5$x better, at large time limits. To the best of our knowledge, ours is the first learning approach to demonstrate such large improvements over SCIP on both large-scale real-world application datasets and MIPLIB.

Abstract: Mixed integer dynamic approximation scheme (MIDAS) is a new sampling-based algorithm for solving finite-horizon stochastic dynamic programs with monotonic Bellman functions. MIDAS approximates these value functions using step functions, leading to stage problems that are mixed integer programs. We provide a general description of MIDAS, and prove its almost-sure convergence to a $$2T\varepsilon $$-optimal policy for problems with T stages when the Bellman functions are known to be monotonic, and the sampling process satisfies standard assumptions.

Abstract: In this study, we extend newly introduced numerical method to partial differential and integral equations with integer and non-integer order. This numerical approximation suggested by Atangana and Seda was constructed with Newton polynomial. Moreover it is accurate and efficient for solving partial differential and integral equations. Also, we present numerical simulation for solution of the considered equation. The numerical results show that this numerical approach is useful and accurate for obtaining numerical solution of such equations.

Abstract: We study the problem of decomposing the Hessian matrix of a mixed integer convex quadratic program (MICQP) into the sum of positive semidefinite 2 × 2 matrices. Solving this problem enables the use...

Abstract: We study DatalogMTL—an extension of Datalog with metric temporal operators—under integer semantics, where the temporal domain of both interpretations and temporal operators consists of integer time points only. This is in contrast to the standard semantics, which is defined over the rational timeline. DatalogMTL under integer semantics is an interesting KR language: on the one hand, one can often assume the integer timeline in applications; on the other hand, it captures prominent temporal extensions of Datalog such as Datalog1S. We show that the choice of integer semantics leads to more favourable computational properties. We first show that reasoning over integers is at most as hard as reasoning over rationals for DatalogMTL and its natural fragments. Then, we investigate fragments of DatalogMTL where adopting the integer semantics makes reasoning easier. In particular, we show that complexity drops from P-hard to NC1-complete for the propositional fragment (where all object variables are grounded), and from TC0-hard to ACC0 for the linear fragment where the past diamond operator is the only metric operator allowed in rule bodies. Thus, reasoning in such fragments is both tractable and highly parallelisable, which suggests their appropriateness for data-intensive applications.

Abstract: We consider integer and linear programming problems for which the linear constraints exhibit a (recursive) block-structure: The problem decomposes into independent and efficiently solvable sub-problems if a small number of constraints is deleted. A prominent example are $n$-fold integer programming problems and their generalizations which have received considerable attention in the recent literature. The previously known algorithms for these problems are based on the augmentation framework, a tailored integer programming variant of local search. In this paper we propose a different approach. Our algorithm relies on parametric search and a new proximity bound. We show that block-structured linear programming can be solved efficiently via an adaptation of a parametric search framework by Norton, Plotkin, and Tardos in combination with Megiddo's multidimensional search technique. This also forms a subroutine of our algorithm for the integer programming case by solving a strong relaxation of it. Then we show that, for any given optimal vertex solution of this relaxation, there is an optimal integer solution within $\ell_1$-distance independent of the dimension of the problem. This in turn allows us to find an optimal integer solution efficiently. We apply our techniques to integer and linear programming with $n$-fold structure or bounded dual treedepth, two benchmark problems in this field. We obtain the first algorithms for these cases that are both near-linear in the dimension of the problem and strongly polynomial. Moreover, unlike the augmentation algorithms, our approach is highly parallelizable.

Abstract: Brain-inspired Hyperdimensional (HD) computing is an emerging technique that computes with either binary or integer HD vectors. However, both vector representations confront an extreme trade-off between accuracy and energy efficiency. This issue limits the generalizability of HD computing for many applications. In this paper, we propose a threshold-based dynamic HD computing framework (TD-HDC) to improve the accuracy-energy efficiency trade-offs. Standard HD computing always executes the same processing flow regardless of input data. On the contrary, TD-HDC dynamically selects the execution path between the binary and integer HD models based on the classification difficulty of input data. In other words, TD-HDC utilizes both HD models and manages to efficiently allocate their computational resources. On the MNIST dataset, we demonstrate that our proposed framework is flexible and can reduce energy consumption and execution time by 51.3% and 15%, respectively, under the same accuracy level.

Abstract: We introduce the monitoring of trace properties under assumptions. An assumption limits the space of possible traces that the monitor may encounter. An assumption may result from knowledge about the system that is being monitored, about the environment, or about another, connected monitor. We define monitorability under assumptions and study its theoretical properties. In particular, we show that for every assumption A, the boolean combinations of properties that are safe or co-safe relative to A are monitorable under A. We give several examples and constructions on how an assumption can make a non-monitorable property monitorable, and how an assumption can make a monitorable property monitorable with fewer resources, such as integer registers.

Abstract: Linnik considered about 70 years ago the following approximation to the binary Goldbach problem. Is it possible to give a fixed integer K such that every sufficiently large even integer could be written as the sum of two primes and K powers of two? He solved the problem affirmatively with an unspecified large K. The first explicit result $$(K=54\,000)$$ appeared at the end of the 1990's. In the present work we show that $$K=8$$ powers of two suffices without supposing any unproved hypothesis.

Abstract: When building a road, it is critical to select a vertical alignment which ensures design and safety constraints. Finding such a vertical alignment is not necessarily a feasible problem, and the models describing it generally involve a large number of variables and constraints. This paper is dedicated to rapidly proving the feasibility or the infeasibility of a Mixed Integer Linear Program (MILP) modeling the vertical alignment problem. To do so, we take advantage of the particular structure of the MILP, and we prove that only a few of the MILP’s constraints determine the feasibility of the problem. In addition, we propose a method to build a feasible solution to the MILP that does not involve integer variables. This enables time saving to proving the feasibility of the vertical alignment problem and to find a feasible vertical alignment, as emphasized by numerical results. It is on average 75 times faster to prove the feasibility and 10 times faster to build a feasible solution.

Abstract: In the past years, several QRS complex (Q, R and S wave) detecting algorithms have been implemented in software, but they are not applicable for real-time operation due to their mathematical complexity. Hence, this paper focuses on developing an algorithm which enables detection of QRS complex in real time. Here, Integer Haar Wavelet Transform is employed which is used for the purpose of filtering ECG signal and detecting the R-peak frequency of QRS complex. Various blocks of the proposed architecture are implemented in a Digilent Nexys 4 double data rate field-programmable gate array board with only 501 flip-flops and 557 look-up tables utilized which makes it suitable for directly installing it into medical equipment or further developing a smart internet of things system for bio-medical applications. To make the designed system more generic, several similar blocks or components have been used. At first, the principle of using wavelet to detect QRS complex is explored theoretically and accordingly used for developing our system. An efficient hardware architecture is implemented incorporating many simplifications, a significant one being approximation of floating point arithmetic to integer arithmetic, required for wavelets. The architecture of the designed system is represented with demonstration in behavioral simulation as well as hardware testing. In the end, the system is analyzed and results are obtained with an error percentage in RR interval computation of less than 1.4% and QRS detection accuracy of 98.76%. The proposed architecture is also synthesized using 130 nm technology which produces 0.717% of leakage power. The developed architecture can be used for analysis of other bio-medical signals where the operation of wavelet transform in hardware is required.

Abstract: We consider a discrete optimization formulation for learning sparse classifiers, where the outcome depends upon a linear combination of a small subset of features. Recent work has shown that mixed integer programming (MIP) can be used to solve (to optimality) $\ell_0$-regularized regression problems at scales much larger than what was conventionally considered possible. Despite their usefulness, MIP-based global optimization approaches are significantly slower compared to the relatively mature algorithms for $\ell_1$-regularization and heuristics for nonconvex regularized problems. We aim to bridge this gap in computation times by developing new MIP-based algorithms for $\ell_0$-regularized classification. We propose two classes of scalable algorithms: an exact algorithm that can handle $p\approx 50,000$ features in a few minutes, and approximate algorithms that can address instances with $p\approx 10^6$ in times comparable to the fast $\ell_1$-based algorithms. Our exact algorithm is based on the novel idea of \textsl{integrality generation}, which solves the original problem (with $p$ binary variables) via a sequence of mixed integer programs that involve a small number of binary variables. Our approximate algorithms are based on coordinate descent and local combinatorial search. In addition, we present new estimation error bounds for a class of $\ell_0$-regularized estimators. Experiments on real and synthetic data demonstrate that our approach leads to models with considerably improved statistical performance (especially, variable selection) when compared to competing methods.

Abstract: While integer arithmetic has been widely adopted for improved performance in deep quantized neural network inference, training remains a task primarily executed using floating point arithmetic. This is because both high dynamic range and numerical accuracy are central to the success of most modern training algorithms. However, due to its potential for computational, storage and energy advantages in hardware accelerators, neural network training methods that can be implemented with low precision integer-only arithmetic remains an active research challenge. In this paper, we present NITI, an efficient deep neural network training framework that stores all parameters and intermediate values as integers, and computes exclusively with integer arithmetic. A pseudo stochastic rounding scheme that eliminates the need for external random number generation is proposed to facilitate conversion from wider intermediate results to low precision storage. Furthermore, a cross-entropy loss backpropagation scheme computed with integer-only arithmetic is proposed. A proof-of-concept open-source software implementation of NITI that utilizes native 8-bit integer operations in modern GPUs to achieve end-to-end training is presented. When compared with an equivalent training setup implemented with floating point storage and arithmetic, NITI achieves negligible accuracy degradation on the MNIST and CIFAR10 datasets using 8-bit integer storage and computation. On ImageNet, 16-bit integers are needed for weight accumulation with an 8-bit datapath. This achieves training results comparable to all-floating-point implementations.