arXiv subjects

Aleksandr M. Kazachkov

Publications and source records attributed to Aleksandr M. Kazachkov.

10 recordsLinked to original sources

arXiv2025

Parametric Disjunctive Cuts for Sequences of Mixed Integer Linear Optimization Problems

Many applications require solving sequences of related mixed-integer linear programs. We introduce a class of parametric disjunctive inequalities (PDIs), obtained by reusing the disjunctive proofs of optimality from prior solves to construct cuts valid for perturbed instances. We describe several methods of generating such cuts that navigate the tradeoff between computational expense and strength. We provide sufficient conditions under which PDIs support the disjunctive hull and a tightening step that guarantees support when needed. On perturbed instances from MIPLIB 2017, augmenting branch-and-cut with PDIs substantially improves performance, reducing total solve times on the majority of challenging cases.

math.OC↗

arXiv2025

Approximating value functions via corner Benders' cuts

We introduce a novel technique to generate Benders' cuts from a conic relaxation ("corner") derived from a basis of a higher-dimensional polyhedron that we aim to outer approximate in a lower-dimensional space. To generate facet-defining inequalities for the epigraph associated to this corner, we develop a computationally-efficient algorithm based on a compact reverse polar formulation and a row generation scheme that handles the redundant inequalities. Via a known connection between arc-flow and path-flow formulations, we show that our method can recover the linear programming bound of a Dantzig-Wolfe formulation using multiple cuts in the projected space. In computational experiments, our generic technique enhances the performance of a problem-specific state-of-the-art algorithm for the vehicle routing problem with stochastic demands, a well-studied variant of the classic capacitated vehicle routing problem that accounts for customer demand uncertainty.

math.OC↗

arXiv2024

Machine Learning for Optimization-Based Separation of Mixed-Integer Rounding Cuts

Mixed-integer rounding (MIR) cutting planes (cuts) are effective at improving the strength of a linear relaxation for mixed-integer linear programming (MIP) problems. The cuts in this family are derived by aggregating constraints then rounding coefficients, but finding the strongest MIR cuts requires optimizing a costly MIP for the aggregation step, so in practice, heuristic strategies for separating fractional points are employed. We propose to improve MIR cut generation in the context of a common scenario in applications, where constraints remain fixed but costs are varied. We present a hybrid cut generation framework in which we train a machine learning (ML) model to classify which constraints are involved in useful MIR cuts based on fractional points from relaxations of the problem. At test time, the predictions of the ML model create a reduced MIP-based generator of MIR cuts. In our experiments, we create an instance family from each of three benchmark MIP instances by performing a careful and costly perturbation of objective coefficients to build a dataset of 1,000 fractional points to be separated over the same constraint set. The results indicate that the reduced separator better strengthens the bound in each round of cut generation, particularly for instances in which the full separator failed to find strong cuts.

math.OC↗

arXiv2024

Symmetrically Fair Allocations of Indivisible Goods

We consider allocating indivisible goods with provable fairness guarantees that are satisfied regardless of which bundle of items each agent receives. Symmetrical allocations of this type are known to exist for divisible resources, such as consensus splitting of a cake into parts, each having equal value for all agents, ensuring that in any allocation of the cake slices, no agent would envy another. For indivisible goods, one analogous concept relaxes envy freeness to guarantee the existence of an allocation in which any bundle is worth as much as any other, up to the value of a bounded number of items from the other bundle. Previous work has studied the number of items that need to be removed. In this paper, we improve upon these bounds for the specific setting in which the number of bundles equals the number of agents. Concretely, we develop the theory of symmetrically envy free up to one good, or symEF1, allocations. We prove that a symEF1 allocation exists if the vertices of a related graph can be partitioned (colored) into as many independent sets as there are agents. This sufficient condition always holds for two agents, and for agents that have identical, disjoint, or binary valuations. We further prove conditions under which exponentially-many distinct symEF1 allocations exist. Finally, we perform computational experiments to study the incidence of symEF1 allocations as a function of the number of agents and items when valuations are drawn uniformly at random.

cs.GT↗

arXiv2024

$\mathcal{V}$-Polyhedral Disjunctive Cuts

We introduce $\mathcal{V}$-polyhedral disjunctive cuts (VPCs) for generating valid inequalities from general disjunctions. Cuts are critical to integer programming solvers, but the benefit from many families is only realized when the cuts are applied recursively, causing numerical instability and "tailing off" of cut strength after several rounds. To mitigate these difficulties, the VPC framework offers a practical method for generating strong cuts without resorting to recursion. The framework starts with a disjunction whose terms partition the feasible region into smaller subproblems, then obtains a collection of points and rays from the disjunctive terms, from which we build a linear program whose feasible solutions correspond to valid disjunctive cuts. Though a naïve implementation would result in an exponentially-sized optimization problem, we show how to efficiently construct this linear program, such that it is much smaller than the one from the alternative higher-dimensional cut-generating linear program. This enables us to test strong multiterm disjunctions that arise from the leaf nodes of a partial branch-and-bound tree. In addition to proving useful theoretical properties of the cuts, we evaluate their performance computationally through an implementation in the open-source COIN-OR framework. In the results, VPCs from a strong disjunction significantly improve the gap closed compared to existing cuts in solvers, and they also decrease some instances' solving time when used with branch and bound.

math.OC↗

arXiv2023

An Abstract Model for Branch and Cut

Branch and cut is the dominant paradigm for solving a wide range of mathematical programming problems -- linear or nonlinear -- combining efficient search (via branch and bound) and relaxation-tightening procedures (via cutting planes, or cuts). While there is a wealth of computational experience behind existing cutting strategies, there is simultaneously a relative lack of theoretical explanations for these choices, and for the tradeoffs involved therein. Recent papers have explored abstract models for branching and for comparing cuts with branch and bound. However, to model practice, it is crucial to understand the impact of jointly considering branching and cutting decisions. In this paper, we provide a framework for analyzing how cuts affect the size of branch-and-cut trees, as well as their impact on solution time. Our abstract model captures some of the key characteristics of real-world phenomena in branch-and-cut experiments, regarding whether to generate cuts only at the root or throughout the tree, how many rounds of cuts to add before starting to branch, and why cuts seem to exhibit nonmonotonic effects on the solution process.

math.OC↗

arXiv2023

Monoidal strengthening of simple $\mathcal{V}$-polyhedral disjunctive cuts

Disjunctive cutting planes can tighten a relaxation of a mixed-integer linear program. Traditionally, such cuts are obtained by solving a higher-dimensional linear program, whose additional variables cause the procedure to be computationally prohibitive. Adopting a $\mathcal{V}$-polyhedral perspective is a practical alternative that enables the separation of disjunctive cuts via a linear program with only as many variables as the original problem. The drawback is that the classical approach of monoidal strengthening cannot be directly employed without the values of the extra variables appearing in the extended formulation. We derive how to compute these values from a solution to the linear program generating $\mathcal{V}$-polyhedral disjunctive cuts. We then present computational experiments with monoidal strengthening of cuts from disjunctions with as many as 64 terms. Some instances are dramatically impacted, with strengthening increasing the gap closed by the cuts from 0 to 100%. However, for larger disjunctions, monoidal strengthening appears to be less effective, for which we identify a potential cause.

math.OC↗

arXiv2022

The Machine Learning for Combinatorial Optimization Competition (ML4CO): Results and Insights

Combinatorial optimization is a well-established area in operations research and computer science. Until recently, its methods have focused on solving problem instances in isolation, ignoring that they often stem from related data distributions in practice. However, recent years have seen a surge of interest in using machine learning as a new approach for solving combinatorial problems, either directly as solvers or by enhancing exact solvers. Based on this context, the ML4CO aims at improving state-of-the-art combinatorial optimization solvers by replacing key heuristic components. The competition featured three challenging tasks: finding the best feasible solution, producing the tightest optimality certificate, and giving an appropriate solver configuration. Three realistic datasets were considered: balanced item placement, workload apportionment, and maritime inventory routing. This last dataset was kept anonymous for the contestants.

cs.LG↗

arXiv2018

Partial hyperplane activation for generalized intersection cuts

The generalized intersection cut (GIC) paradigm is a recent framework for generating cutting planes in mixed integer programming with attractive theoretical properties. We investigate this computationally unexplored paradigm and observe that a key hyperplane activation procedure embedded in it is not computationally viable. To overcome this issue, we develop a novel replacement to this procedure called partial hyperplane activation (PHA), introduce a variant of PHA based on a notion of hyperplane tilting, and prove the validity of both algorithms. We propose several implementation strategies and parameter choices for our PHA algorithms and provide supporting theoretical results. We computationally evaluate these ideas in the COIN-OR framework on MIPLIB instances. Our findings shed light on the the strengths of the PHA approach as well as suggest properties related to strong cuts that can be targeted in the future.

math.OC↗

arXiv2016

Small Representations of Big Kidney Exchange Graphs

Kidney exchanges are organized markets where patients swap willing but incompatible donors. In the last decade, kidney exchanges grew from small and regional to large and national---and soon, international. This growth results in more lives saved, but exacerbates the empirical hardness of the $\mathcal{NP}$-complete problem of optimally matching patients to donors. State-of-the-art matching engines use integer programming techniques to clear fielded kidney exchanges, but these methods must be tailored to specific models and objective functions, and may fail to scale to larger exchanges. In this paper, we observe that if the kidney exchange compatibility graph can be encoded by a constant number of patient and donor attributes, the clearing problem is solvable in polynomial time. We give necessary and sufficient conditions for losslessly shrinking the representation of an arbitrary compatibility graph. Then, using real compatibility graphs from the UNOS nationwide kidney exchange, we show how many attributes are needed to encode real compatibility graphs. The experiments show that, indeed, small numbers of attributes suffice.

cs.AI↗