Given a basic block of instructions, finding a schedule that requires the minimum number of registers for evaluation is a well-known problem. The problem is NP-complete when the dependences among instructions form a directed-acyclic graph instead of a tree. We are striving to find efficient approximation algorithms for this problem not simply because it is an interesting graph optimization problem in theory. A good solution to this problem is also an essential component in solving the more complex instruction scheduling problem on GPU. In this paper, we start with explanations on why this problem is important in GPU instruction scheduling. We then explore two different approaches to tackling this problem. First we model this problem as a constraint-programming problem. Using a state-of-the-art CP-SAT solver, we can find optimal answers for much larger cases than previous works on a modest desktop PC. Second, guided by the optimal answers, we design and evaluate heuristics that can be applied to the polynomial-time list scheduling algorithms. A combination of those heuristics can achieve the register-pressure results that are about 17\% higher than the optimal minimum on average. However, there are still near 6\% cases in which the register pressure by the heuristic approach is 50\% higher than the optimal minimum.
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P-time event graphs are discrete event systems suitable for modeling processes in which tasks must be executed in predefined time windows. Their dynamics can be represented by max-plus linear-dual inequalities (LDIs), i.e., systems of linear dynamical inequalities in the primal and dual operations of the max-plus algebra. We define a new class of models called switched LDIs (SLDIs), which allow to switch between different modes of operation, each corresponding to a set of LDIs, according to a sequence of modes called schedule. In this paper, we focus on the analysis of SLDIs when the considered schedule is fixed and either periodic or intermittently periodic. We show that SLDIs can model a wide range of applications including single-robot multi-product processing networks, in which every product has different processing requirements and corresponds to a specific mode of operation. Based on the analysis of SLDIs, we propose algorithms to compute: i. minimum and maximum cycle times for these processes, improving the time complexity of other existing approaches; ii. a complete trajectory of the robot including start-up and shut-down transients.
In this work we propose an extension of physics informed supervised learning strategies to parametric partial differential equations. Indeed, even if the latter are indisputably useful in many applications, they can be computationally expensive most of all in a real-time and many-query setting. Thus, our main goal is to provide a physics informed learning paradigm to simulate parametrized phenomena in a small amount of time. The physics information will be exploited in many ways, in the loss function (standard physics informed neural networks), as an augmented input (extra feature employment) and as a guideline to build an effective structure for the neural network (physics informed architecture). These three aspects, combined together, will lead to a faster training phase and to a more accurate parametric prediction. The methodology has been tested for several equations and also in an optimal control framework.
Hybrid dynamical systems, i.e. systems that have both continuous and discrete states, are ubiquitous in engineering, but are difficult to work with due to their discontinuous transitions. For example, a robot leg is able to exert very little control effort while it is in the air compared to when it is on the ground. When the leg hits the ground, the penetrating velocity instantaneously collapses to zero. These instantaneous changes in dynamics and discontinuities (or jumps) in state make standard smooth tools for planning, estimation, control, and learning difficult for hybrid systems. One of the key tools for accounting for these jumps is called the saltation matrix. The saltation matrix is the sensitivity update when a hybrid jump occurs and has been used in a variety of fields including robotics, power circuits, and computational neuroscience. This paper presents an intuitive derivation of the saltation matrix and discusses what it captures, where it has been used in the past, how it is used for linear and quadratic forms, how it is computed for rigid body systems with unilateral constraints, and some of the structural properties of the saltation matrix in these cases.
Recently, hybrid metaheuristics have become a trend in operations research. A successful example combines the Greedy Randomized Adaptive Search Procedures (GRASP) and data mining techniques, where frequent patterns found in high-quality solutions can lead to an efficient exploration of the search space, along with a significant reduction of computational time. In this work, a GRASP-based state-of-the-art heuristic for the Minimum Latency Problem (MLP) is improved by means of data mining techniques for two MLP variants. Computational experiments showed that the approaches with data mining were able to match or improve the solution quality for a large number of instances, together with a substantial reduction of running time. In addition, 88 new cost values of solutions are introduced into the literature. To support our results, tests of statistical significance, impact of using mined patterns, equal time comparisons and time-to-target plots are provided.
Line coverage is the task of servicing a given set of one-dimensional features in an environment. It is important for the inspection of linear infrastructure such as road networks, power lines, and oil and gas pipelines. This paper addresses the single robot line coverage problem for aerial and ground robots by modeling it as an optimization problem on a graph. The problem belongs to the broad class of arc routing problems and is closely related to the rural postman problem (RPP) on asymmetric graphs. The paper presents an integer linear programming formulation with proofs of correctness. Using the minimum cost flow problem, we develop approximation algorithms with guarantees on the solution quality. These guarantees also improve the existing results for the asymmetric RPP. The main algorithm partitions the problem into three cases based on the structure of the required graph, i.e., the graph induced by the features that require servicing. We evaluate our algorithms on road networks from the 50 most populous cities in the world, consisting of up to 730 road segments. The algorithms, augmented with improvement heuristics, run within 3s and generate solutions that are within 10% of the optimum. We experimentally demonstrate our algorithms with commercial UAVs.
A drawback of the classic approach for complexity analysis of distributed graph problems is that it mostly informs about the complexity of notorious classes of ``worst case'' graphs. Algorithms that are used to prove a tight (existential) bound are essentially optimized to perform well on such worst case graphs. However, such graphs are often either unlikely or actively avoided in practice, where benign graph instances usually admit much faster solutions. To circumnavigate these drawbacks, the concept of universal complexity analysis in the distributed setting was suggested by [Kutten and Peleg, PODC'95] and actively pursued by [Haeupler et al., STOC'21]. Here, the aim is to gauge the complexity of a distributed graph problem depending on the given graph instance. The challenge is to identify and understand the graph property that allows to accurately quantify the complexity of a distributed problem on a given graph. In the present work, we consider distributed shortest paths problems in the HYBRID model of distributed computing, where nodes have simultaneous access to two different modes of communication: one is restricted by locality and the other is restricted by congestion. We identify the graph parameter of neighborhood quality and show that it accurately describes a universal bound for the complexity of certain class of shortest paths problems in the HYBRID model.
The Tucker tensor decomposition is a natural extension of the singular value decomposition (SVD) to multiway data. We propose to accelerate Tucker tensor decomposition algorithms by using randomization and parallelization. We present two algorithms that scale to large data and many processors, significantly reduce both computation and communication cost compared to previous deterministic and randomized approaches, and obtain nearly the same approximation errors. The key idea in our algorithms is to perform randomized sketches with Kronecker-structured random matrices, which reduces computation compared to unstructured matrices and can be implemented using a fundamental tensor computational kernel. We provide probabilistic error analysis of our algorithms and implement a new parallel algorithm for the structured randomized sketch. Our experimental results demonstrate that our combination of randomization and parallelization achieves accurate Tucker decompositions much faster than alternative approaches. We observe up to a 16X speedup over the fastest deterministic parallel implementation on 3D simulation data.
Minimizing the weight of an edge set satisfying parity constraints is a challenging branch of combinatorial optimization as witnessed by the binary hypergraph chapter of Alexander Schrijver's book ``Combinatorial Optimization'' (Chapter 80). This area contains relevant graph theory problems including open cases of the Max Cut problem, or some multiflow problems. We clarify the interconnections of some problems and establish three levels of difficulties. On the one hand, we prove that the Shortest Odd Path problem in an undirected graph without cycles of negative total weight and several related problems are NP-hard, settling a long-standing open question asked by Lov\'asz (Open Problem 27 in Schrijver's book ``Combinatorial Optimization''. On the other hand, we provide a polynomial-time algorithm to the closely related and well-studied Minimum-weight Odd $\{s,t\}$-Join problem for non-negative weights, whose complexity, however, was not known; more generally, we solve the Minimum-weight Odd $T$-Join problem in FPT time when parameterized by $|T|$. If negative weights are also allowed, then finding a minimum-weight odd $\{s,t\}$-join is equivalent to the Minimum-weight Odd $T$-Join problem for arbitrary weights, whose complexity is only conjectured to be polynomially solvable. The analogous problems for digraphs are also considered.
In this paper we investigate formal verification problems for Neural Network computations. Various reachability problems will be in the focus, such as: Given symbolic specifications of allowed inputs and outputs in form of Linear Programming instances, one question is whether valid inputs exist such that the given network computes a valid output? Does this property hold for all valid inputs? The former question's complexity has been investigated recently by S\"alzer and Lange for nets using the Rectified Linear Unit and the identity function as their activation functions. We complement their achievements by showing that the problem is NP-complete for piecewise linear functions with rational coefficients that are not linear, NP-hard for almost all suitable activation functions including non-linear ones that are continuous on an interval, complete for the Existential Theory of the Reals $\exists \mathbb R$ for every non-linear polynomial and $\exists \mathbb R$-hard for the exponential function and various sigmoidal functions. For the completeness results, linking the verification tasks with the theory of Constraint Satisfaction Problems turns out helpful.
Constraint Programming (CP) is a declarative programming paradigm that allows for modeling and solving combinatorial optimization problems, such as the Job-Shop Scheduling Problem (JSSP). While CP solvers manage to find optimal or near-optimal solutions for small instances, they do not scale well to large ones, i.e., they require long computation times or yield low-quality solutions. Therefore, real-world scheduling applications often resort to fast, handcrafted, priority-based dispatching heuristics to find a good initial solution and then refine it using optimization methods. This paper proposes a novel end-to-end approach to solving scheduling problems by means of CP and Reinforcement Learning (RL). In contrast to previous RL methods, tailored for a given problem by including procedural simulation algorithms, complex feature engineering, or handcrafted reward functions, our neural-network architecture and training algorithm merely require a generic CP encoding of some scheduling problem along with a set of small instances. Our approach leverages existing CP solvers to train an agent learning a Priority Dispatching Rule (PDR) that generalizes well to large instances, even from separate datasets. We evaluate our method on seven JSSP datasets from the literature, showing its ability to find higher-quality solutions for very large instances than obtained by static PDRs and by a CP solver within the same time limit.
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