Computing primal solutions with exact arithmetics in SCIP.

The research for exact solutions of mixed integer problems is an active topic in the scientific community. State-of-the-art MIP solvers exploit a floating- point numerical representation, therefore introducing small approximations. Although such MIP solvers yield reliable results for the majority of problems, there are cases in which a higher accuracy is required. Indeed, it is known that for some applications floating-point solvers provide falsely feasible solutions, i.e. solutions marked as feasible because of approximations that would not pass a check with exact arithmetic and cannot be practically implemented. The framework of the current dissertation is SCIP, a mixed integer programs solver mainly developed at Zuse Institute Berlin. In the same site we considered a new approach for exactly solving MIPs. Specifically, we developed a constraint handler to plug into SCIP, with the aim to analyze the accuracy of provided floating-point solutions and compute exact primal solutions starting from floating-point ones. We conducted a few computational experiments to test the exact primal constraint handler through the adoption of two main settings. Analysis mode allowed to collect statistics about current SCIP solutions' reliability. Our results confirm that floating-point solutions are accurate enough with respect to many instances. However, our analysis highlighted the presence of numerical errors of variable entity. By using the enforce mode, our constraint handler is able to suggest exact solutions starting from the integer part of a floating-point solution. With the latter setting, results show a general improvement of the quality of provided final solutions, without a significant loss of performances.

Regular and exact categories

Lo scopo del lavoro è quello di presentare alcune proprietà di base delle categorie regolari ed esatte nel contesto della teoria delle categoria algebrica.

Embodied computation and life: building blocks for open-ended evolution

Il presente lavoro si propone di sviluppare una analogia formale tra sistemi dinamici e teoria della computazione in relazione all’emergenza di proprietà biologiche da tali sistemi. Il primo capitolo sarà dedicato all’estensione della teoria delle macchine di Turing ad un più ampio contesto di funzioni computabili e debolmente computabili. Mostreremo quindi come un sistema dinamico continuo possa essere elaborato da una macchina computante, e come proprietà informative quali l’universalità possano essere naturalmente estese alla fisica attraverso questo ponte formale. Nel secondo capitolo applicheremo i risultati teorici derivati nel primo allo sviluppo di un sistema chimico che mostri tali proprietà di universalità, ponendo particolare attenzione alla plausibilità fisica di tale sistema.

Exact Combinatorial Optimization with Graph Convolutional Neural Networks

Combinatorial optimization problems are typically tackled by the branch-and-bound paradigm. We propose to learn a variable selection policy for branch-and-bound in mixed-integer linear programming, by imitation learning on a diversified variant of the strong branching expert rule. We encode states as bipartite graphs and parameterize the policy as a graph convolutional neural network. Experiments on a series of synthetic problems demonstrate that our approach produces policies that can improve upon expert-designed branching rules on large problems, and generalize to instances significantly larger than seen during training.

Bounded Arithmetic and Randomized Computation

In this work, we develop a randomized bounded arithmetic for probabilistic computation, following the approach adopted by Buss for non-randomized computation. This work relies on a notion of representability inspired by of Buss' one, but depending on a non-standard quantitative and measurable semantic. Then, we establish that the representable functions are exactly the ones in PPT. Finally, we extend the language of our arithmetic with a measure quantifier, which is true if and only if the quantified formula's semantic has measure greater than a given threshold. This allows us to define purely logical characterizations of standard probabilistic complexity classes such as BPP, RP, co-RP and ZPP.