Constrainedness in stable matching

In constraint satisfaction problems, constrainedness provides a way to predict the number of solutions: for instances of a same size, the number of constraints is inversely correlated with the number of solutions. However, there is no obvious equivalent metric for stable matching problems. We introduce the contrarian score, a simple metric that is to matching problems what constrainedness is to constraint satisfaction problems. In addition to comparing the contrarian score against other potential tightness metrics, we test it for different instance sizes as well as extremely distinct versions of the stable matching problem. In all cases, we find that the correlation between contrarian score and number of solutions is very strong

Phase Transitions and Backbones of the Asymmetric Traveling Salesman Problem

In recent years, there has been much interest in phase transitions of combinatorial problems. Phase transitions have been successfully used to analyze combinatorial optimization problems, characterize their typical-case features and locate the hardest problem instances. In this paper, we study phase transitions of the asymmetric Traveling Salesman Problem (ATSP), an NP-hard combinatorial optimization problem that has many real-world applications. Using random instances of up to 1,500 cities in which intercity distances are uniformly distributed, we empirically show that many properties of the problem, including the optimal tour cost and backbone size, experience sharp transitions as the precision of intercity distances increases across a critical value. Our experimental results on the costs of the ATSP tours and assignment problem agree with the theoretical result that the asymptotic cost of assignment problem is pi ^2 /6 the number of cities goes to infinity. In addition, we show that the average computational cost of the well-known branch-and-bound subtour elimination algorithm for the problem also exhibits a thrashing behavior, transitioning from easy to difficult as the distance precision increases. These results answer positively an open question regarding the existence of phase transitions in the ATSP, and provide guidance on how difficult ATSP problem instances should be generated

Cable Tree Wiring -- Benchmarking Solvers on a Real-World Scheduling Problem with a Variety of Precedence Constraints

Cable trees are used in industrial products to transmit energy and information between different product parts. To this date, they are mostly assembled by humans and only few automated manufacturing solutions exist using complex robotic machines. For these machines, the wiring plan has to be translated into a wiring sequence of cable plugging operations to be followed by the machine. In this paper, we study and formalize the problem of deriving the optimal wiring sequence for a given layout of a cable tree. We summarize our investigations to model this cable tree wiring Problem (CTW) as a traveling salesman problem with atomic, soft atomic, and disjunctive precedence constraints as well as tour-dependent edge costs such that it can be solved by state-of-the-art constraint programming (CP), Optimization Modulo Theories (OMT), and mixed-integer programming (MIP) solvers. It is further shown, how the CTW problem can be viewed as a soft version of the coupled tasks scheduling problem. We discuss various modeling variants for the problem, prove its NP-hardness, and empirically compare CP, OMT, and MIP solvers on a benchmark set of 278 instances. The complete benchmark set with all models and instance data is available on github and is accepted for inclusion in the MiniZinc challenge 2020

Automatic Adjacency Grammar Generation from User Drawn Sketches

http://www.ieee.orgIn this paper we present an innovative approach to automatically generate adjacency grammars describing graphical symbols. A grammar production is formulated in terms of rulesets of geometrical constraints among symbol primitives. Given a set of symbol instances sketched by a user using a digital pen, our approach infers the grammar productions consisting of the ruleset most likely to occur. The performance of our work is evaluated using a comprehensive benchmarking database of on-line symbols

A model-aware inexact Newton scheme for electrical impedance tomography

This work gives new insights into the EIT model. Firstly, a novel relation between the conductivity and the data is derived, giving quantitative insights about the instability of the inverse problem. Secondly, a reconstruction framework is introduced which estimates unknown model parameters and then solves the problem with a tailored Newton method. Additional problem-specific optimizations are incorporated into the framework. Simulations verify its efficiency for simulated and measured data