Simple Local Computation Algorithms for the General Lovasz Local Lemma

We consider the task of designing Local Computation Algorithms (LCA) for applications of the Lov\'{a}sz Local Lemma (LLL). LCA is a class of sublinear algorithms proposed by Rubinfeld et al.~\cite{Ronitt} that have received a lot of attention in recent years. The LLL is an existential, sufficient condition for a collection of sets to have non-empty intersection (in applications, often, each set comprises all objects having a certain property). The ground-breaking algorithm of Moser and Tardos~\cite{MT} made the LLL fully constructive, following earlier results by Beck~\cite{beck_lll} and Alon~\cite{alon_lll} giving algorithms under significantly stronger LLL-like conditions. LCAs under those stronger conditions were given in~\cite{Ronitt}, where it was asked if the Moser-Tardos algorithm can be used to design LCAs under the standard LLL condition. The main contribution of this paper is to answer this question affirmatively. In fact, our techniques yield LCAs for settings beyond the standard LLL condition

Transformation of propositional calculus statements into integer and mixed integer programs: An approach towards automatic reformulation

A systematic procedure for transforming a set of logical statements or logical conditions imposed on a model into an Integer Linear Progamming (ILP) formulation Mixed Integer Programming (MIP) formulation is presented. An ILP stated as a system of linear constraints involving integer variables and an objective function, provides a powerful representation of decision problems through a tightly interrelated closed system of choices. It supports direct representation of logical (Boolean or prepositional calculus) expressions. Binary variables (hereafter called logical variables) are first introduced and methods of logically connecting these to other variables are then presented. Simple constraints can be combined to construct logical relationships and the methods of formulating these are discussed. A reformulation procedure which uses the extended reverse polish representation of a compound logical form is then described. These reformulation procedures are illustrated by two examples. A scheme of implementation.ithin an LP modelling system is outlined

Tools for reformulating logical forms into zero-one mixed integer programs (MIPS)

A systematic procedure for transforming a set of logical statements or logical conditions imposed on a model into an Integer Linear Programming (ILP) formulation or a Mixed Integer Programming (MIP) formulation is presented. A reformulation procedure which uses the extended reverse polish representation of a compound logical form is then described. A prototype user interface by which logical forms can be reformulated and the corresponding MIP constructed and analysed within an existing Mathematical Programming modelling system is illustrated. Finally, the steps to formulate a discrete optimisation model in this way are demonstrated by means of an example

PRIMJENA „EINSTEINOVE ZAGONETKE“ PRI RJEŠAVANJU PROBLEMA ALOKACIJE GRAĐEVINSKIH STROJEVA

“Einstein’s riddle” is a popular example of constraints satisfaction problem. Since its introduction, different forms and variations of the riddle have been presented. Regardless of the variant of the riddle, its solution is considered a tough challenge for humans. Researchers have developed and are still developing mathematical models, as well as computational simulation models for solving it. In this article, the authors have modified a previously published mathematical model and developed a computational spreadsheet model for solving the riddle, which provides a unique solution for the riddle. The model was also tested in a small and medium-scaled form for solving constraint satisfaction problems regarding the allocation of construction machines. The authors have also highlighted the model’s limitations for solving such problems and made suggestions regarding necessary modifications in the model to solve more complex problems in the same domain.„Einsteinova zagonetka“ je prepoznatljiv primjer kombinatornog problema ispunjenja ograničenja. Ova zagonetka je imala više verzija, no bez obzira na formulaciju, uglavnom se smatra vrlo teškim zadatkom. Znanstvenici su razvijali i dalje razvijaju matematičke modele, a potom i računalne simulacijske modele za rješavanje spomenutog problema. Autori su u ovome radu modificirali ranije predstavljeni matematički model, a potom prema njemu izradili računalni model, koristeći se proračunskim tablicama kako bi riješili zagonetku. Model je ponudio jedinstveno rješenje u vrlo kratkom vremenu, a potom je ispitan pri rješavanju sličnog problema u građevinskoj praksi. Istaknuta su ograničenja u primjeni modela u obliku kojim je riješena „Einstenova zagonetka“ te koje su modifikacije nužne za aplikaciju pri rješavanju kompleksnijih problema u istoj domeni