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Despite the recent progress of automated program verification techniques, fully automated verification of programs manipulating recursive data structures remains a challenge. We introduce solvable tuple patterns (STPs) and conjunctive STPs…
A classic result of Lenstra [Math.~Oper.~Res.~1983] says that an integer linear program can be solved in fixed-parameter tractable (FPT) time for the parameter being the number of variables. We extend this result by incorporating…
Loop invariants are fundamental to reasoning about programs with loops. They establish properties about a given loop's behavior. When they additionally are inductive, they become useful for the task of formal verification that seeks to…
This applied research article explores the application of Mixed-Integer Linear Programming (MILP) to address line-balancing challenges in the garment industry, focusing on optimizing production processes under multiple constraints. By…
We consider the problem of finding all enclosing rectangles of minimum area that can contain a given set of rectangles without overlap. Our rectangle packer chooses the x-coordinates of all the rectangles before any of the y-coordinates. We…
Discrete barycenters are the optimal solutions to mass transport problems for a set of discrete measures. Such transport problems arise in many applications of operations research and statistics. The best known algorithms for exact…
Computational models of human language often involve combinatorial problems. For instance, a probabilistic parser may marginalize over exponentially many trees to make predictions. Algorithms for such problems often employ dynamic…
Motivated by recent advances in solution methods for mixed-integer convex optimization (MICP), we study the fundamental and open question of which sets can be represented exactly as feasible regions of MICP problems. We establish several…
Resilience is one of the key algorithmic problems underlying various forms of reverse data management (such as view maintenance, deletion propagation, and various interventions for fairness): What is the minimal number of tuples to delete…
Integer Linear Programming (ILP) has a broad range of applications in various areas of artificial intelligence. Yet in spite of recent advances, we still lack a thorough understanding of which structural restrictions make ILP tractable.…
Provably correct software is one of the key challenges in our softwaredriven society. While formal verification establishes the correctness of a given program, the result of program synthesis is a program which is correct by construction.…
An infinite set is orbit-finite if, up to permutations of the underlying structure of atoms, it has only finitely many elements. We study a generalisation of linear programming where constraints are expressed by an orbit-finite system of…
We present the PML 2 language, which provides a uniform environment for programming, and for proving properties of programs in an ML-like setting. The language is Curry-style and call-by-value, it provides a control operator (interpreted in…
Indefinite quadratic programs (QPs) are known to be very difficult to be solved to global optimality, so are linear programs with linear complementarity constraints. Treating the former as a subclass of the latter, this paper presents a…
A covering integer program (CIP) is a mathematical program of the form: min {c^T x : Ax >= 1, 0 <= x <= u, x integer}, where A is an m x n matrix, and c and u are n-dimensional vectors, all having non-negative entries. In the online…
As machine learning becomes prevalent, mitigating any unfairness present in the training data becomes critical. Among the various notions of fairness, this paper focuses on the well-known individual fairness, which states that similar…
Understanding how the optimal value of an optimisation problem changes when its input data is modified is an old question in mathematical optimisation. This paper investigates the computation of the optimal values of a family of (possibly…
The efficiency of modern optimization methods, coupled with increasing computational resources, has led to the possibility of real-time optimization algorithms acting in safety critical roles. There is a considerable body of mathematical…
In the digital age, ensuring the correctness, safety, and reliability of software through formal verification is paramount, particularly as software increasingly underpins critical infrastructure. Formal verification, split into theorem…
Numerical ``direct'' approaches to time-optimal control often fail to find solutions that are singular in the sense of the Pontryagin Maximum Principle, performing better when searching for saturated (bang-bang) solutions. In previous work…