A Scalable Approach to Exact Resource-Constrained Scheduling Based on a Joint SDC and SAT Formulation

Scheduling and variable binding for improved testability in high level synthesis

We describe a design problem arising in the color printing industry and discuss a number of integer linear programming and constraint programming approaches to its solution. Despite the apparent simplicity of the problem it presents a challenge for both approaches. We present results for three typical cases and show that the constraint programming approach provides better results, although in some cases after considerable handcrafting. We also show that the results obtained by constraint programming can be improved by a simple goal programming model.

This paper addresses the problem of counting models in integer linear programming (ILP) using Boolean Satisfiability (SAT) techniques, and proposes two approaches to solve this problem. The first approach consists of encoding ILP instances into pseudo-Boolean (PB) instances. Moreover, the paper introduces a model counter for PB constraints, which can be used for counting models in PB as well as in ILP. A second alternative approach consists of encoding instances of ILP into instances of SAT. A two-step procedure is proposed, consisting of first mapping the ILP instance into PB constraints and then encoding the PB constraints into SAT. One key observation is that not all existing PB to SAT encodings can be used for counting models. The paper provides conditions for PB to SAT encodings that can be safely used for model counting, and proves that some of the existing encodings are safe for model counting while others are not. Finally, the paper provides experimental results, comparing the PB and SAT approaches, as well as existing alternative solutions.

Optimized solvers for the Boolean satisfiability (SAT) problem have many applications in areas such as hardware and software verification, FPGA routing, planning, and so forth. Further uses are complicated by the need to express "counting constraints" in conjunctive normal form (CNF). Expressing such constraints by pure CNF leads to more complex SAT instances. Alternatively, those constraints can be handled by integer linear programming (ILP), but generic ILP solvers may ignore the Boolean nature of 0-1 variables. Therefore, specialized 0-1 ILP solvers extend SAT solvers to handle these so-called "pseudo-Boolean" (PB) constraints. This work provides an update on the on-going competition between generic ILP techniques and specialized 0-1 ILP techniques. To make a fair comparison, we generalize recent ideas for fast SAT-solving to more general 0-1 ILP problems that may include counting constraints and optimization. This generalization is embodied in our PB constraint solver and optimizer PBS, which is compared with state-of-the-art CNF and generic ILP solvers. Another aspect of our comparison is the evaluation on 0-1 ILP benchmarks that originate in electronic design automation (EDA) but that cannot be directly solved by an SAT solver. Specifically, we solve instances of the max-SAT and max-ONEs optimization problems, which seek to maximize the number of satisfied clauses and the "true" values over all satisfying assignments, respectively. Those problems have straightforward applications to SAT-based routing and are additionally important due to reductions from max-cut, max-clique, and min vertex cover. Our experimental results show that specialized 0-1 techniques implemented in PBS tend to outperform generic ILP techniques on Boolean optimization problems, as well as on general EDA SAT problems.

With the advent of deep sub-micron era, there is a demand to consider the design closure problem in high-level synthesis. It is well known that the slack is an effective means of tolerating the uncertainties in operation delays. Previous work ever attempted to increase the usable slack based on a given initial schedule. Instead of the post-processing approach, this paper is the first attempt to the simultaneous application of operation scheduling and slack optimization. We use a 0-1 integer linear programming (0-1 ILP) approach to formally formulate the problem. Under the design constraints (timing and resource), our approach is applicable to two different objective functions: the maximization of the total usable slack and the maximization of the number of non-zero slack operations. Compared with previous work, our approach has the following two advantages: first, our approach guarantees the optimality; second, our approach is more suitable for the design space exploration.

The periodic event scheduling problem (PESP) is a well studied problem known as intrinsically hard. Its main application is for designing periodic timetables in public transportation. To this end, the passengers' paths are required as input data. This is a drawback since the final paths which are used by the passengers depend on the timetable to be designed. Including the passengers' routing in the PESP hence improves the quality of the resulting timetables. However, this makes PESP even harder. Formulating the PESP as satisfiability problem and using SAT solvers for its solution has been shown to be a highly promising approach. The goal of this paper is to exploit if SAT solvers can also be used for the problem of integrated timetabling and passenger routing. In our model of the integrated problem we distribute origin-destination (OD) pairs temporally through the network by using time-slices in order to make the resulting model more realistic. We present a formulation of this integrated problem as integer program which we are able to transform to a satisfiability problem. We tested the latter formulation within numerical experiments, which are performed on Germany's long-distance passenger railway network. The computation's analysis in which we compare the integrated approach with the traditional one with fixed passengers' weights, show promising results for future scientific investigations.

Integer Linear Programming (ILP) is commonly used in high level and system level synthesis. It is an NP-Complete problem (in general cases). There exists some tools that give an optimal solution for small ILP formulations. Nevertheless, these tools may not give solutions for complex formulations. In this paper, we present a solution to overcome the problem of complexity in ILP formulations. We propose a partitioning methodology based on a constraint graph representing all the constraints included in any ILP formulation. To direct the partitioning, the constraint graph nodes are grouped to represent Data Flow Graph (DFG) nodes. This reduced constraint graph can be used to partition any ILP formulation based on DFG. We illustrate this method on ILP formulation for scheduling. Results on ILP scheduling formulations are promising.

This paper presents a new heuristic, concurrent, iterative loop-based scheduling and allocation algorithm for high-level synthesis of digital signal processing (DSP) architectures using heterogeneous functional units. In a heterogeneous architecture, functional units could be either bit-serial or digit-serial or bit-parallel. We assume that a library of functional units based on heterogeneous implementation style is available. Experiments show that this new heuristic synthesis approach generates optimal and near-optimal area solutions. Although optimum synthesis of such architectures were proposed recently using an integer linear programming (ILP) model, our method can produce similar solutions in one to two orders of magnitude less time, at the expense of sacrificing the cost optimality. We compare the solutions generated by the proposed algorithm with the optimal solutions generated by the ILP approach and other recent techniques. We have incorporated this new algorithm into the Minnesota ARchitecture Synthesis (MARS-II) system.

This study is concerned with the Boolean satisfiability (SAT) problem and its solution in setting a hybrid computational intelligence environment of genetic and fuzzy computing. In this framework, fuzzy sets realize an embedding principle meaning that original two-valued (Boolean) functions under investigation are extended to their continuous counterparts resulting in the form of fuzzy (multivalued) functions. In the sequel, the SAT problem is reformulated for the fuzzy functions and solved using a genetic algorithm (GA). It is shown that a GA, especially its recursive version, is an efficient tool for handling multivariable SAT problems. Thorough experiments revealed that the recursive version of the GA can solve SAT problems with more than 1000 variables.

This dissertation is to illustrate the current state of the art of DNA computing achievements, especially of new approaches to solve theoretical 3-SAT problems and the hitting-set problem. Beginning with Adleman’s breakthrough which is an molecular algorithm for the solution of a NP-complete, combinatorial problem, the directed Hamiltonian path problem (HPP). Today, many researchers all over the world concentrate on proposing new methods to solve engineering or application problems with a DNA computing approach. Satisfiability problem is given a Boolean formula, and decide if a satisfying truth assignment exists. ( ) ( ) … ( ) ( ) is an example of Boolean formula. k-SAT problem means that each clause has exactly k literals. Not-All-Equal (NAE) 3-SAT problem and One-In-Three (1IN3) 3-SAT problem are both NP-complete problems. In this dissertation, we present molecular solutions to find all true assignments (3-SAT problem) and furthermore find Not-All-Equal (NAE) solutions and One-In-Three (1IN3) solutions in DNA-based Supercomputing. Hitting-set problem assume that there exists a collection C of subsets of a finite set S, and a positive integer K  |S|, and we need to know if there is a subset  S with | |  K such that contains at least one element of each subset in C. In other words, is the subset that intersects every subset in C and is called the hitting-set. In this dissertation, a DNA-based algorithm is proposed to solve the small hitting-set problem. A small hitting-set is a hitting-set with the smallest K value, i.e., the hitting-set with the smallest number of elements. Furthermore, an algorithm is introduced to find the number of ones from 2n combinations and minimum numbers of ones represents the small hitting-set since K is expected to be as small as possible. The complexity of all the presented DNA-based algorithms is also discussed. We describe time complexity and volume complexity of three Algorithms, numbers of test tube used and the longest library strand in solution space of all three Algorithms. Finally, the simulated experiment is applied to verify correctness of the proposed DNA-based algorithm for solving the One-In-Three (1IN3) 3-SAT problem, and simulation of Not-All-Equal (NAE) 3-SAT problem is similar. Also, another simulated experiment is applied to our proposed DNA-based algorithm 6-2, in order to solve the well-known hitting-set problem. This research has been motivated by the benefit and the application of DNA computing and gives new methods to solve two 3-SAT problems and the hitting-set problem which are NP-complete.

This chapter presents a linear-programming-based optimal solution for an Optimal Placement Problem (OPP) of Synchrophasors or Phasor Measurement Units (PMU). With the increasing use of Synchrophasors for smart-grid monitoring and control, the problem with PMU placement has become a major concern due to its large installation costs. This chapter utilizes an Integer Linear Programming (ILP) approach to find optimal solutions. We show that the performance of our LP constraints and the formulation makes the ILP approach very attractive for smaller utilities that have limited budgets. This is shown using with and without zero-injection cases in programmed AMPL language using CPLEX solver. We also propose an Optimal Redundancy Criterion (ORC) which assists utilities with providing redundant observability for critical buses in the system. Our approach yields full and redundant observabilities within a few milliseconds for different cases, such as IEEE 300 test systems and an Indian power grid.

Scheduling educational institution courses for multiple departments presents a significant challenge due to the complexity of constraints and requirements. This paper addresses the University Course Scheduling Problem (UCSP) using a binary Integer Linear Programming (ILP) approach to generate semester schedules for winter or summer semesters. The approach incorporates both hard constraints - such as avoiding simultaneous sessions for the same semester, adhering to room capacities, and ensuring lecturers are not doublebooked - and soft constraints, including preferences for lecture times. While the whole methodology integrates tools for data management, optimization modeling, and a structured workflow to create detailed schedules for lecturers, students, and room allocations, this paper focuses on the ILP-formulation of these constraints. The construction of these matrices is systematically explained through practical examples, illustrating how the relationships between courses, lecturers, rooms, and time slots are encoded. To validate the proposed approach, a case study was conducted at the Niederrhein University of Applied Sciences. This study addresses the scheduling requirements of three engineering departments with more than 50 degree programs and 260 modules, demonstrating the applicability of the matrixdriven ILP method.

Propositional Satisfiability (SAT) is a keystone in the history of computer science. SAT was the first problem shown to be NP-complete in 1971 by Stephen Cook. Having passed more than 40 years from then, SAT is now a lively research field where theory and practice have a natural intermixing. In this talk, we overview the use of SAT in practical domains, where SAT is thought in a broad sense, i.e. including SAT extensions such as Maximum Satisfiability (MaxSAT), Pseudo-Boolean Optimization (PBO) and Quantified Boolean Formulas (QBF).

Advances in methods for solving Boolean satisfiability (SAT) for large problems have motivated recent attempts to recast physical design problems as Boolean SAT problems. One persistent criticism of these approaches is their inability to supply partial solutions, i.e., to satisfy most but not all of the constraints cast in the SAT style. In this paper, we present a formulation for "subset satisfiable" Boolean SAT: we transform a "strict" SAT problem with N constraints into a new, "relaxed" SAT problem which is satisfiable just if not more than k/spl Lt/N of these constraints cannot be satisfied in the original problem. We describe a transformation based on explicit thresholding and counting for the necessary SAT relaxation. Examples from field-programmable gate-array routing show how we can determine efficiently when we can satisfy "almost all" of our geometric constraints.

Quantum annealing device has shown a great potential in solving discrete problems that are theoretically and empirically hard. Boolean Satisfiability (SAT) problem, determining if there is an assignment of variables that satisfies a given Boolean function, is the first proven NP-complete problem widely used in various domains. Here, we present a novel mapping of the SAT problem to the quadratic unconstrained binary optimization problem (QUBO), and further develop a tool flow embedding the proposed QUBO to the architecture of the commercialized quantum computer D-Wave. By leveraging electronic design automation techniques including synthesis, placement and routing, this is not only the first work providing the detail flow that embeds the QUBO, but also a technique scalable for real world applications and some hard SAT problems with over 6000 variables in QUBO. Based on our results, we discuss the challenges in solving SAT using the current generation of annealing device, and explore the problem solving capability of future quantum annealing computers.