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.

This paper provides a novel framework for solving multiobjective discrete optimization problems with an arbitrary number of objectives. Our framework represents 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 network models 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. Summary of Contribution: Multiobjective optimization has a long history of research with applications in several domains. Our paper provides an alternative modeling and solution approach for multiobjective discrete optimization problems by leveraging graphical structures. Specifically, we encode the decision space of a problem as a layered network and propose graph reduction operators to preserve only solutions whose image are part of the Pareto frontier. The nondominated solutions can then be extracted through shortest-path algorithms on such a network. Numerical results comparing our method with state-of-the-art approaches on several problem classes, including the knapsack, set covering, and the traveling salesperson problem (TSP), suggest orders-of-magnitude runtime speed-ups for exactly enumerating the Pareto frontier, especially when the number of objective functions grows.

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.

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 an important shortcoming: 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 three challenging problems: the traveling salesman problem with time windows, the 4-moments portfolio optimization problem, and the 0-1 knapsack problem. Results obtained show that the framework introduced outperforms the stand-alone RL and CP solutions, while being competitive with industrial solvers.

Networks undergo continuous changes to introduce new services and improve existing ones. Network change deployment involves carefully deciding when each change activity will be executed and who will be executing the change. This is a complex process because each service group has to plan its activities following a set of operational and technological constraints. Besides, multiple groups may be working on the same or dependent nodes at the same time, and they must coordinate their deployment plans. If they do not co-ordinate, conflicting change execution could result in unexpected impacts. Traditionally, change deployment has been a tedious and time-consuming task. To address this, we propose an innovative solution Zapper that aims for minimal human effort to coordinate the changes, minimal risk to service quality, and efficient plans to rapidly deploy the changes. Zapper maps change scheduling constraints into mathematical equations and then uses optimization algorithms to generate conflict-free change plans that satisfy all constraints across service groups. We have deployed Zapper at a large service provider and it is being used regularly by the network operations teams for more than two years to schedule over 4.5 million change activities.

We address the one-to-one multicommodity pickup-and-delivery traveling salesman problem, a challenging variant of the traveling salesman problem that includes the transportation of commodities between locations. The goal is to find a minimum cost tour such that each commodity is delivered to its destination and the maximum capacity of the vehicle is never exceeded. We propose an exact approach that uses a discrete relaxation based on multivalued decision diagrams (MDDs) to better represent the combinatorial structure of the problem. We enhance our relaxation by using the MDDs as a subproblem to a Lagrangian relaxation technique, leading to significant improvements in both bound quality and run-time performance. Our work extends the use of MDDs for solving routing problems by presenting new construction methods and filtering rules based on capacity restrictions. Experimental results show that our approach outperforms state-of-the-art methodologies, closing 33 open instances from the literature, with 27 of those closed by our best variant.

We investigate the use of relaxed decision diagrams (DDs) for computing admissible heuristics for the cost-optimal delete-free planning (DFP) problem. Our main contributions are the introduction of two novel DD encodings for a DFP task: a multivalued decision diagram that includes the sequencing aspect of the problem and a binary decision diagram representation of its sequential relaxation. We present construction algorithms for each DD that leverage these different perspectives of the DFP task and provide theoretical and empirical analyses of the associated heuristics. We further show that relaxed DDs can be used beyond heuristic computation to extract delete-free plans, find action landmarks, and identify redundant actions. Our empirical analysis shows that while DD-based heuristics trail the state of the art, even small relaxed DDs are competitive with the linear programming heuristic for the DFP task, thus, revealing novel ways of designing admissible heuristics.

Constraint-based 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 (MDD) 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.

Linear programming has been successfully used to compute admissible heuristics for cost-optimal classical planning. Although one of the strengths of linear programming is the ability to express and reason about numeric variables and constraints, their use in numeric planning is limited. In this work, we extend linear programming-based heuristics for classical planning to support numeric state variables. In particular, we propose a model for the interval relaxation, coupled with landmarks and state equation constraints. We consider both linear programming models and their harder-to-solve, yet more informative, integer programming versions. Our experimental analysis shows that considering an NP-Hard heuristic often pays off and that A* search using our integer programming heuristics establishes a new state of the art in cost-optimal numeric planning.

A partial-order plan (POP) compactly encodes a set of sequential plans that can be dynamically chosen by an agent at execution time. One natural measure of the quality of a POP is its flexibility, which is defined to be the total number of sequential plans it embodies (i.e., its linearizations). As this criteria is hard to optimize, existing work has instead optimized proxy functions that are correlated with the number of linearizations. In this paper, we develop and strengthen mixed-integer linear programming (MILP) models for three proxy functions: two from the POP literature and a third novel function based on the temporal flexibility criteria from the scheduling literature. We show theoretically and empirically that none of the three proxy measures dominate the others in terms of number of sequential plans. Compared to the state-of-the-art MaxSAT model for the problem, we empirically demonstrate that two of our MILP models result in equivalent or slightly better solution quality with savings of approximately one order of magnitude in computation time.

We propose a general branch-and-bound algorithm for discrete optimization in which binary decision diagrams (BDDs) play the role of the traditional linear programming relaxation. In particular, relaxed BDD representations of the problem provide bounds and guidance for branching, and restricted BDDs supply a primal heuristic. Each problem is given a dynamic programming model that allows one to exploit recursive structure, even though the problem is not solved by dynamic programming. A novel search scheme branches within relaxed BDDs rather than on values of variables. Preliminary testing shows that a rudimentary BDD-based solver is competitive with or superior to a leading commercial integer programming solver for the maximum stable set problem, the maximum cut problem on a graph, and the maximum 2-satisfiability problem. Specific to the maximum cut problem, we tested the BDD-based solver on a classical benchmark set and identified tighter relaxation bounds than have ever been found by any technique, nearly closing the entire optimality gap on four large-scale instances.

Recent research in hybrid optimization shows that a combination of technologies that exploits their complementary strengths can significantly speed up computation. The use of high-level metaconstraints in the problem formulation can achieve a substantial share of these computational gains by better communicating problem structure to the solver. During the solution process, however, metaconstraints give rise to reformulations or relaxations that introduce auxiliary variables, and some of the variables in one metaconstraint’s reformulation may be functionally the same as or related to variables in another metaconstraint’s reformulation. These relationships must be recognized to obtain a tight overall relaxation. We propose a modeling scheme based on semantic typing that systematically addresses this problem while providing simpler, self-documenting models. It organizes the model around predicates and declares variables by associating each with a predicate through a keyword that is analogous to a database query. We present a series of examples to illustrate this idea over a wide variety of applications.

We consider the problem of parallelizing restarted backtrack search. With few notable exceptions, most commercial and academic constraint programming solvers do not learn no-goods during search. Depending on the branching heuristics used, this means that there are little to no side-effects between restarts, making them an excellent target for parallelization. We develop a simple technique for parallelizing restarted search deterministically and demonstrate experimentally that we can achieve near-linear speed-ups in practice.

We explore the idea of obtaining bounds on the value of an optimization problem from a discrete relaxation based on binary decision diagrams (BDDs). We show how to construct a BDD that represents a relaxation of a 0-1 optimization problem, and how to obtain a bound for a separable objective function by solving a shortest (or longest) path problem in the BDD. As a test case we apply the method to the maximum independent set problem on a graph. We find that for most problem instances, it delivers tighter bounds in less computation time, than state-of-the-art integer programming software obtains by solving a continuous relaxation augmented with cutting planes.

We study propagation for the Sequence constraint in the context of constraint programming based on limited-width MDDs. Our first contribution is proving that establishing MDD-consistency for Sequence is NP-hard. Yet, we also show that this task is fixed parameter tractable with respect to the length of the sub-sequences. In addition, we propose a partial filtering algorithm that relies on a specific decomposition of the constraint and a novel extension of MDD filtering to node domains. We experimentally evaluate the performance of our proposed filtering algorithm, and demonstrate that the strength of the MDD propagation increases as the maximum width is increased. In particular, MDD propagation can outperform conventional domain propagation for Sequence by reducing the search tree size and solving time by several orders of magnitude. Similar improvements are observed with respect to the current best MDD approach that applies the decomposition of Sequence into Among constraints.

Recent contributions to advancing planning from the classical model to more realistic problems include using temporal logic such as LTL to express desired properties of a solution plan. This paper introduces a planning model that combines temporally extended goals and uncontrollable events. The planning task is to reach a state such that all event sequences generated from that state satisfy the problem's temporally extended goal. A real-life application that motivates this work is to use planning to configure a system in such a way that its subsequent, non-deterministic internal evolution (nominal behavior) is guaranteed to satisfy a condition expressed in temporal logic.