scholarly journals On the Approximability of Weighted Model Integration on DNF Structures

2020 ◽  
Author(s):  
Ralph Abboud ◽  
İsmail İlkan Ceylan ◽  
Radoslav Dimitrov
Keyword(s):  
Approximation Scheme ◽  
General Purpose ◽  
Model Integration ◽  
Weight Functions ◽  
Propositional Formula ◽  
Weighted Sum ◽  
Large Problem ◽  
Model Counting ◽  
Weighted Model ◽  
Problem Instances

Weighted model counting (WMC) consists of computing the weighted sum of all satisfying assignments of a propositional formula. WMC is well-known to be #P-hard for exact solving, but admits a fully polynomial randomised approximation scheme (FPRAS) when restricted to DNF structures. In this work, we study weighted model integration, a generalization of weighted model counting which involves real variables in addition to propositional variables, and pose the following question: Does weighted model integration on DNF structures admit an FPRAS? Building on classical results from approximate volume computation and approximate weighted model counting, we show that weighted model integration on DNF structures can indeed be approximated for a class of weight functions. Our approximation algorithm is based on three subroutines, each of which can be a weak (i.e., approximate), or a strong (i.e., exact) oracle, and in all cases, comes along with accuracy guarantees. We experimentally verify our approach over randomly generated DNF instances of varying sizes, and show that our algorithm scales to large problem instances, involving up to 1K variables, which are currently out of reach for existing, general-purpose weighted model integration solvers.

scholarly journals Exact and Approximate Weighted Model Integration with Probability Density Functions Using Knowledge Compilation

2019 ◽  
Vol 33 ◽  
pp. 7825-7833 ◽  
Author(s):  
Pedro Zuidberg Dos Martires ◽  
Anton Dries ◽  
Luc De Raedt
Keyword(s):  
Probability Density ◽  
Probabilistic Reasoning ◽  
Probability Distributions ◽  
Probability Density Functions ◽  
Knowledge Compilation ◽  
Model Integration ◽  
Weight Functions ◽  
Density Functions ◽  
Model Counting ◽  
Weighted Model

Weighted model counting has recently been extended to weighted model integration, which can be used to solve hybrid probabilistic reasoning problems. Such problems involve both discrete and continuous probability distributions. We show how standard knowledge compilation techniques (to SDDs and d-DNNFs) apply to weighted model integration, and use it in two novel solvers, one exact and one approximate solver. Furthermore, we extend the class of employable weight functions to actual probability density functions instead of mere polynomial weight functions.

scholarly journals The pywmi Framework and Toolbox for Probabilistic Inference using Weighted Model Integration

2019 ◽  
Author(s):  
Samuel Kolb ◽  
Paolo Morettin ◽  
Pedro Zuidberg Dos Martires ◽  
Francesco Sommavilla ◽  
Andrea Passerini ◽  
...  
Keyword(s):  
State Of The Art ◽  
Probabilistic Inference ◽  
Model Integration ◽  
Modeling Languages ◽  
Continuous Variables ◽  
Open Source Framework ◽  
Model Counting ◽  
Weighted Model ◽  
Common Interface ◽  
Discrete Domains

Weighted Model Integration (WMI) is a popular technique for probabilistic inference that extends Weighted Model Counting (WMC) -- the standard inference technique for inference in discrete domains -- to domains with both discrete and continuous variables.  However, existing WMI solvers each have different interfaces and use different formats for representing WMI problems.  Therefore, we introduce pywmi (http://pywmi.org), an open source framework and toolbox for probabilistic inference using WMI, to address these shortcomings.  Crucially, pywmi fixes a common internal format for WMI problems and introduces a common interface for WMI solvers.  To assist users in modeling WMI problems, pywmi introduces modeling languages based on SMT-LIB.v2 or MiniZinc and parsers for both.  To assist users in comparing WMI solvers, pywmi includes implementations of several state-of-the-art solvers, a fast approximate WMI solver, and a command-line interface to solve WMI problems.  Finally, to assist developers in implementing new solvers, pywmi provides Python implementations of commonly used subroutines.

scholarly journals Efficient Weighted Model Integration via SMT-Based Predicate Abstraction

2017 ◽  
Author(s):  
Paolo Morettin ◽  
Andrea Passerini ◽  
Roberto Sebastiani
Keyword(s):  
Predicate Abstraction ◽  
Model Integration ◽  
General Notion ◽  
Continuous Variables ◽  
Original Formulation ◽  
Strong Reduction ◽  
Real World Data ◽  
Model Counting ◽  
Weighted Model ◽  
Novel Algorithm

Weighted model integration (WMI) is a recent formalism generalizing weighted model counting (WMC) to run probabilistic inference over hybrid domains, characterized by both discrete and continuous variables and relationships between them. Albeit powerful, the original formulation of WMI suffers from some theoretical limitations, and it is computationally very demanding as it requires to explicitly enumerate all possible models to be integrated over. In this paper we present a novel general notion of WMI, which fixes the theoretical limitations and allows for exploiting the power of SMT-based predicate abstraction techniques. A novel algorithm combines a strong reduction in the number of models to be integrated over with their efficient enumeration. Experimental results on synthetic and real-world data show drastic computational improvements over the original WMI formulation as well as existing alternatives for hybrid inference.

Chapter 25. Model Counting

2021 ◽  
Author(s):  
Carla P. Gomes ◽  
Ashish Sabharwal ◽  
Bart Selman
Keyword(s):  
Local Search ◽  
Upper Bound ◽  
Probabilistic Inference ◽  
Efficient Algorithms ◽  
Propositional Formula ◽  
Exact Methods ◽  
Complete Problem ◽  
Model Counting ◽  
New Ideas

Model counting, or counting the number of solutions of a propositional formula, generalizes SAT and is the canonical #P-complete problem. Surprisingly, model counting is hard even for some polynomial-time solvable cases like 2-SAT and Horn-SAT. Efficient algorithms for this problem will have a significant impact on many application areas that are inherently beyond SAT, such as bounded-length adversarial and contingency planning, and, perhaps most importantly, general probabilistic inference. Model counting can be solved, in principle and to an extent in practice, by extending the two most successful frameworks for SAT algorithms, namely, DPLL and local search. However, scalability and accuracy pose a substantial challenge. As a result, several new ideas have been introduced in the last few years that go beyond the techniques usually employed in most SAT solvers. These include division into components, caching, compilation into normal forms, exploitation of solution sampling methods, and certain randomized streamlining techniques using special constraints. This chapter discusses these techniques, exploring both exact methods as well as fast estimation approaches, including those that provide probabilistic or statistical guarantees on the quality of the reported lower or upper bound on the model count.

scholarly journals Two metaheuristic approaches for solving the multi-compartment vehicle routing problem

2018 ◽  
Vol 20 (4) ◽  
pp. 2085-2108 ◽  
Author(s):  
Hiba Yahyaoui ◽  
Islem Kaabachi ◽  
Saoussen Krichen ◽  
Abdulkader Dekdouk
Keyword(s):  
Vehicle Routing ◽  
Vehicle Routing Problem ◽  
Variable Neighborhood Search ◽  
Neighborhood Search ◽  
Large Problem ◽  
Solution Quality ◽  
Routing Problem ◽  
Petrol Station ◽  
Capacity Limitations ◽  
Problem Instances

Abstract We address in this paper a multi-compartment vehicle routing problem (MCVRP) that aims to plan the delivery of different products to a set of geographically dispatched customers. The MCVRP is encountered in many industries, our research has been motivated by petrol station replenishment problem. The main objective of the delivery process is to minimize the total driving distance by the used trucks. The problem configuration is described through a prefixed set of trucks with several compartments and a set of customers with demands and prefixed delivery. Given such inputs, the minimization of the total traveled distance is subject to assignment and routing constraints that express the capacity limitations of each truck’s compartment in terms of the pathways’ restrictions. For the NP-hardness of the problem, we propose in this paper two algorithms mainly for large problem instances: an adaptive variable neighborhood search (AVNS) and a Partially Matched Crossover PMX-based Genetic Algorithm to solve this problem with the goal of ensuring a better solution quality. We compare the ability of the proposed AVNS with the exact solution using CPLEX and a set of benchmark problem instances is used to analyze the performance of the both proposed meta-heuristics.

scholarly journals Modified Harris Hawks Optimizer for Solving Machine Scheduling Problems

Symmetry ◽  
2020 ◽  
Vol 12 (9) ◽  
pp. 1460
Author(s):  
Hamza Jouhari ◽  
Deming Lei ◽  
Mohammed A. A. Al-qaness ◽  
Mohamed Abd Elaziz ◽  
Robertas Damaševičius ◽  
...  
Keyword(s):  
Computation Time ◽  
Machine Scheduling ◽  
Setup Time ◽  
Parallel Machine Scheduling ◽  
Parallel Machine ◽  
Scheduling Problems ◽  
Large Problem ◽  
Unrelated Parallel Machine Scheduling ◽  
Problem Instances ◽  
Machine Scheduling Problems

Scheduling can be described as a decision-making process. It is applied in various applications, such as manufacturing, airports, and information processing systems. More so, the presence of symmetry is common in certain types of scheduling problems. There are three types of parallel machine scheduling problems (PMSP): uniform, identical, and unrelated parallel machine scheduling problems (UPMSPs). Recently, UPMSPs with setup time had attracted more attention due to its applications in different industries and services. In this study, we present an efficient method to address the UPMSPs while using a modified harris hawks optimizer (HHO). The new method, called MHHO, uses the salp swarm algorithm (SSA) as a local search for HHO in order to enhance its performance and to decrease its computation time. To test the performance of MHHO, several experiments are implemented using small and large problem instances. Moreover, the proposed method is compared to several state-of-art approaches used for UPMSPs. The MHHO shows better performance in both small and large problem cases.

scholarly journals Designing a Supply Chain Network under the Risk of Disruptions

2012 ◽  
Vol 2012 ◽  
pp. 1-23 ◽  
Author(s):  
Armin Jabbarzadeh ◽  
Seyed Gholamreza Jalali Naini ◽  
Hamid Davoudpour ◽  
Nader Azad
Keyword(s):  
Supply Chain ◽  
Supply Chain Design ◽  
Mixed Integer ◽  
Nonlinear Program ◽  
Supply Chain Network ◽  
Large Problem ◽  
Solution Methods ◽  
Total Profit ◽  
Order Quantities ◽  
Problem Instances

This paper studies a supply chain design problem with the risk of disruptions at facilities. At any point of time, the facilities are subject to various types of disruptions caused by natural disasters, man-made defections, and equipment breakdowns. We formulate the problem as a mixed-integer nonlinear program which maximizes the total profit for the whole system. The model simultaneously determines the number and location of facilities, the subset of customers to serve, the assignment of customers to facilities, and the cycle-order quantities at facilities. In order to obtain near-optimal solutions with reasonable computational requirements for large problem instances, two solution methods based on Lagrangian relaxation and genetic algorithm are developed. The effectiveness of the proposed solution approaches is shown using numerical experiments. The computational results, in addition, demonstrate that the benefits of considering disruptions in the supply chain design model can be significant.

scholarly journals A SEMI-ADELIC KUZNETSOV FORMULA OVER NUMBER FIELDS

2013 ◽  
Vol 09 (07) ◽  
pp. 1649-1681 ◽  
Author(s):  
PÉTER MAGA
Keyword(s):  
Fourier Coefficients ◽  
Critical Line ◽  
Analytic Number Theory ◽  
Number Fields ◽  
Cusp Forms ◽  
Weight Functions ◽  
Weighted Sum ◽  
Admissible Weight ◽  
Bessel Transform ◽  
Kuznetsov Formula

In this paper, we prove a semi-adelic version of the Kuznetsov formula over number fields. This formula matches a weighted sum made of Fourier coefficients of cusp forms and Eisenstein series with a weighted sum of Kloosterman sums, the latter weight function is a kind of Bessel transform of the former one. We obtain a variant which is valid over all number fields. The admissible weight functions are important in applications, they depend on the archimedean parameters of the representations and show exponential decay. The automorphic vectors are not necessarily spherical in the archimedean aspect. Such formulas are proven to be useful in analytic number theory, e.g., in the estimate of L-functions on the critical line.

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