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U.S. presidential elections are decided by the Electoral College, established in 1789, and designed to mitigate potential risks arising from the collusion of large groups of citizens. A statewide winner-take-all popular voting system for electors is implemented in all but two states, which has led to instances where narrow victories in key states were decisive in several recent elections. Small groups of voters can significantly impact the election, for example, through voter turnout. However, another dynamic can also influence this: a surprisingly small number of dedicated voters moving short distances across state lines. The extent to which the election's outcome is sensitive to small and well-coordinated movements of people has not been investigated in detail. Using a combination of forecasting, simulation, and optimization, we show that a candidate's probability of winning can be increased by 1% through the strategic relocation of approximately 10,000 people no farther than 100 miles from their current county of residence, less than 0.006% of the eligible voting population. Moreover, an 8% probability increase can be realized by a mere 50,000 voters relocating across state lines, or 0.03% of the voting population. Given the remarkably small number of people involved and the fact that establishing electoral residence in many states takes about a month, this coordinated relocation of voters is not nearly as challenging as previously thought. As it stands, U.S. presidential elections may be vulnerable to the exploitation of the aforementioned loophole. Therefore, we anticipate our findings will have direct consequences on policymaking and campaign strategy, as well as motivate new operations research methods within the political sciences., Comment: 16 pages, 2 figures, under review

We investigate relaxations for a class of discrete bilevel programs where the interaction constraints linking the leader and the follower are linear. Our approach reformulates the upper-level optimality constraints by projecting the leader's decisions onto vectors that map to distinct follower solution values, each referred to as a state. Based on such a state representation, we develop a network-flow linear program via a decision diagram that captures the convex hull of the follower's value function graph, leading to a new single-level reformulation of the bilevel problem. We also present a reduction procedure that exploits symmetry to identify the reformulation of minimal size. For large networks, we introduce parameterized relaxations that aggregate states by considering tractable hyperrectangles based on lower and upper bounds associated with the interaction constraints, and can be integrated into existing mixed-integer bilevel linear programming (MIBLP) solvers. Numerical experiments suggest that the new relaxations, whether used within a simple cutting-plane procedure or integrated into state-of-the-art MIBLP solvers, significantly reduce runtimes or solve additional benchmark instances. Our findings also highlight the correlation between the quality of relaxations and the properties of the interaction matrix, underscoring the potential of our approach in enhancing solution methods for structured bilevel optimization instances.

Many discrete optimization problems are amenable to constrained shortest-path reformulations in an extended network space, a technique that has been key in convexification, bound strengthening, and search. In this paper, we propose a constrained variant of these models for two challenging classes of discrete two-stage optimization problems, where traditional methods (e.g., dualize-and-combine) are not applicable compared to their continuous counterparts. Specifically, we propose a framework that models problems as decision diagrams and introduces side constraints either as linear inequalities in the underlying polyhedral representation, or as state variables in shortest-path dynamic programming models. For our first structured class, we investigate two-stage problems with interdiction constraints. We show that such constraints can be formulated as indicator functions in the arcs of the diagram, providing an alternative single-level reformulation of the problem via a network-flow representation. Our second structured class is classical robust optimization, where we leverage the decision diagram network to iteratively identify label variables, akin to an L-shaped method. We evaluate these strategies on a competitive project selection problem and the robust traveling salesperson with time windows, observing considerable improvements in computational efficiency as compared to general methods in the respective areas.

This paper develops a memory-efficient approach for Sequential Pattern Mining (SPM), a fundamental topic in knowledge discovery that faces a well-known memory bottleneck for large data sets. Our methodology involves a novel hybrid trie data structure that exploits recurring patterns to compactly store the data set in memory; and a corresponding mining algorithm designed to effectively extract patterns from this compact representation. Numerical results on small to medium-sized real-life test instances show an average improvement of 85% in memory consumption and 49% in computation time compared to the state of the art. For large data sets, our algorithm stands out as the only capable SPM approach within 256GB of system memory, potentially saving 1.7TB in memory consumption.

In the last decade, decision diagrams (DDs) have been the basis for a large array of novel approaches for modeling and solving optimization problems. Many techniques now use DDs as a key tool to achieve state-of-the-art performance within other optimization paradigms, such as integer programming and constraint programming. This paper provides a survey of the use of DDs in discrete optimization, particularly focusing on recent developments. We classify these works into two groups based on the type of diagram (i.e., exact or approximate) and present a thorough description of their use. We discuss the main advantages of DDs, point out major challenges, and provide directions for future work.

This paper investigates sequencing policies for file reading requests in linear storage devices, such as magnetic tapes. Tapes are the technology of choice for long-term storage in data centers due to their low cost and reliability. However, their physical structure imposes challenges to data retrieval operations reflected in classic optimization and operations research problems. In this work, we provide a theoretical and numerical performance analysis of low-complexity algorithms under deterministic, stochastic, and online settings, which are key in practice due to their interpretability and the large scale of existing data services. In the deterministic setting, we show that traditional policies, such as first-in first-out (FIFO), have arbitrarily poor performance, and we develop and investigate new constant-factor approximations. For the stochastic setting, we present a fully polynomial-time approximation scheme that weighs files based on their access frequencies. Finally, we investigate an online extension and propose a new algorithm with constant competitive-factor guarantees. Our numerical analysis on synthetic and real-world data suggest that the proposed algorithms may significantly outperform policies currently adopted in practice with respect to average reading times., Comment: 32 pages, 8 tables, 9 figures

This paper presents and evaluates two pruning techniques to reinforce the efficiency of constraint optimization solvers based on multi-valued decision-diagrams (MDD). It adopts the branch-and-bound framework proposed by Bergman et al. in 2016 to solve dynamic programs to optimality. In particular, our paper presents and evaluates the effectiveness of the local-bound (LocB) and rough upper-bound pruning (RUB). LocB is a new and effective rule that leverages the approximate MDD structure to avoid the exploration of non-interesting nodes. RUB is a rule to reduce the search space during the development of bounded-width-MDDs. The experimental study we conducted on the Maximum Independent Set Problem (MISP), Maximum Cut Problem (MCP), Maximum 2 Satisfiability (MAX2SAT) and the Traveling Salesman Problem with Time Windows (TSPTW) shows evidence indicating that rough-upper-bound and local-bound pruning have a high impact on optimization solvers based on branch-and-bound with MDDs. In particular, it shows that RUB delivers excellent results but requires some effort when defining the model. Also, it shows that LocB provides a significant improvement automatically; without necessitating any user-supplied information. Finally, it also shows that rough-upper-bound and local-bound pruning are not mutually exclusive, and their combined benefit supersedes the individual benefit of using each technique.

We study the problem when a firm sets prices for products based on the transaction data, i.e., which product past customers chose from an assortment and what were the historical prices that they observed. Our approach does not impose a model on the distribution of the customers' valuations and only assumes, instead, that purchase choices satisfy incentive-compatible constraints. The individual valuation of each past customer can then be encoded as a polyhedral set, and our approach maximizes the worst-case revenue assuming that new customers' valuations are drawn from the empirical distribution implied by the collection of such polyhedra. We show that the optimal prices in this setting can be approximated at any arbitrary precision by solving a compact mixed-integer linear program. Moreover, we study the single-product case and relate it to the traditional model-based approach. We also design three approximation strategies that are of low computational complexity and interpretable. Comprehensive numerical studies based on synthetic and real data suggest that our pricing approach is uniquely beneficial when the historical data has a limited size or is susceptible to model misspecification.

Combinatorial optimization has found applications in numerous fields, from aerospace to transportation planning and economics. The goal is to find an optimal solution among a finite set of possibilities. The well-known challenge one faces with combinatorial optimization is the state-space explosion problem: the number of possibilities grows exponentially with the problem size, which makes solving intractable for large problems. In the last years, deep reinforcement learning (DRL) has shown its promise for designing good heuristics dedicated to solve NP-hard combinatorial optimization problems. However, current approaches have two shortcomings: (1) they mainly focus on the standard travelling salesman problem and they cannot be easily extended to other problems, and (2) they only provide an approximate solution with no systematic ways to improve it or to prove optimality. In another context, constraint programming (CP) is a generic tool to solve combinatorial optimization problems. Based on a complete search procedure, it will always find the optimal solution if we allow an execution time large enough. A critical design choice, that makes CP non-trivial to use in practice, is the branching decision, directing how the search space is explored. In this work, we propose a general and hybrid approach, based on DRL and CP, for solving combinatorial optimization problems. The core of our approach is based on a dynamic programming formulation, that acts as a bridge between both techniques. We experimentally show that our solver is efficient to solve two challenging problems: the traveling salesman problem with time windows, and the 4-moments portfolio optimization problem. Results obtained show that the framework introduced outperforms the stand-alone RL and CP solutions, while being competitive with industrial solvers.

Cut generation and lifting are key components for the performance of state-of-the-art mathematical programming solvers. This work proposes a new general cut-and-lift procedure that exploits the combinatorial structure of 0-1 problems via a binary decision diagram (BDD) encoding of their constraints. We present a general framework that can be applied to a wide range of binary optimization problems and show its applicability for second-order conic inequalities. We identify conditions for which our lifted inequalities are facet-defining and derive a new BDD-based cut generation linear program. Such a model serves as a basis for a max-flow combinatorial algorithm over the BDD that can be applied to derive valid cuts more efficiently. Our numerical results show encouraging performance when incorporated into a state-of-the-art mathematical programming solver, significantly reducing the root node gap, increasing the number of problems solved, and reducing the run-time by a factor of three on average.

Constrained sequential pattern mining aims at identifying frequent patterns on a sequential database of items while observing constraints defined over the item attributes. We introduce novel techniques for constraint-based sequential pattern mining that rely on a multi-valued decision diagram representation of the database. Specifically, our representation can accommodate multiple item attributes and various constraint types, including a number of non-monotone constraints. To evaluate the applicability of our approach, we develop an MDD-based prefix-projection algorithm and compare its performance against a typical generate-and-check variant, as well as a state-of-the-art constraint-based sequential pattern mining algorithm. Results show that our approach is competitive with or superior to these other methods in terms of scalability and efficiency., Comment: AAAI2019

This paper provides a novel framework for solving multiobjective discrete optimization problems with an arbitrary number of objectives. Our framework formulates these problems as network models, in that enumerating the Pareto frontier amounts to solving a multicriteria shortest path problem in an auxiliary network. We design techniques for exploiting the network model in order to accelerate the identification of the Pareto frontier, most notably a number of operations to simplify the network by removing nodes and arcs while preserving the set of nondominated solutions. We show that the proposed framework yields orders-of-magnitude performance improvements over existing state-of-the-art algorithms on five problem classes containing both linear and nonlinear objective functions., Comment: 32 pages, 15 figures, 5 tables

A graph is chordal if every cycle of length at least four contains a chord, that is, an edge connecting two nonconsecutive vertices of the cycle. Several classical applications in sparse linear systems, database management, computer vision, and semidefinite programming can be reduced to finding the minimum number of edges to add to a graph so that it becomes chordal, known as the minimum chordal completion problem (MCCP). In this article we propose a new formulation for the MCCP which does not rely on finding perfect elimination orderings of the graph, as has been considered in previous work. We introduce several families of facet-defining inequalities for cycle subgraphs and investigate the underlying separation problems, showing that some key inequalities are NP-Hard to separate. We also show general properties of the proposed polyhedra, indicating certain conditions and methods through which facets and inequalities associated with the polytope of a certain graph can be adapted in order to become valid and eventually facet-defining for some of its subgraphs or supergraphs. Numerical studies combining heuristic separation methods based on a threshold rounding and lazy-constraint generation indicate that our approach substantially outperforms existing methods for the MCCP, solving many benchmark graphs to optimality for the first time.