Related papers: Towards Solving NP-Complete and Other Hard Problem…
We study the problem of learning efficient algorithms that strongly generalize in the framework of neural program induction. By carefully designing the input / output interfaces of the neural model and through imitation, we are able to…
People solve different problems and know that some of them are simple, some are complex and some insoluble. The main goal of this work is to develop a mathematical theory of algorithmic complexity for problems. This theory is aimed at…
This article presents a general solution to the problem of computational complexity. First, it gives a historical introduction to the problem since the revival of the foundational problems of mathematics at the end of the 19th century.…
Bridging logical and algorithmic reasoning with modern machine learning techniques is a fundamental challenge with potentially transformative impact. On the algorithmic side, many NP-hard problems can be expressed as integer programs, in…
Computational complexity is a core theory of computer science, which dictates the degree of difficulty of computation. There are many problems with high complexity that we have to deal, which is especially true for AI. This raises a big…
Max-cut, clustering, and many other partitioning problems that are of significant importance to machine learning and other scientific fields are NP-hard, a reality that has motivated researchers to develop a wealth of approximation…
Optimization is fundamental in many areas of science, from computer science and information theory to engineering and statistical physics, as well as to biology or social sciences. It typically involves a large number of variables and a…
In the first part of this paper, we present a unified framework for analyzing the algorithmic complexity of any optimization problem, whether it be continuous or discrete in nature. This helps to formalize notions like "input", "size" and…
While graphs and abstract data structures can be large and complex, practical instances are often regular or highly structured. If the instance has sufficient structure, we might hope to compress the object into a more succinct…
Decision-theoretic troubleshooting is one of the areas to which Bayesian networks can be applied. Given a probabilistic model of a malfunctioning man-made device, the task is to construct a repair strategy with minimal expected cost. The…
This paper considers the problem of compressive sensing over a finite alphabet, where the finite alphabet may be inherent to the nature of the data or a result of quantization. There are multiple examples of finite alphabet based static as…
Algorithms for continuous optimization problems have a rich history of design and innovation over the past several decades, in which mathematical analysis of their convergence and complexity properties plays a central role. Besides their…
We consider the hardness of approximation of optimization problems from the point of view of definability. For many NP-hard optimization problems it is known that, unless P = NP, no polynomial-time algorithm can give an approximate solution…
Mathematical optimization, although often leading to NP-hard models, is now capable of solving even large-scale instances within reasonable time. However, the primary focus is often placed solely on optimality. This implies that while…
Optimization problems are ubiquitous in our societies and are present in almost every segment of the economy. Most of these optimization problems are NP-hard and computationally demanding, often requiring approximate solutions for…
Research efforts of the past fifty years have led to a development of linear integer programming as a mature discipline of mathematical optimization. Such a level of maturity has not been reached when one considers nonlinear systems subject…
State minimization of combinatorial filters is a fundamental problem that arises, for example, in building cheap, resource-efficient robots. But exact minimization is known to be NP-hard. This paper conducts a more nuanced analysis of this…
Much algorithmic research in NLP aims to efficiently manipulate rich formal structures. An algorithm designer typically seeks to provide guarantees about their proposed algorithm -- for example, that its running time or space complexity is…
Most continuous mathematical formulations arising in science and engineering can only be solved numerically and therefore approximately. We shall always assume that we're dealing with a numerical approximation to the solution. There are two…
A major problem in evaluating stochastic local search algorithms for NP-complete problems is the need for a systematic generation of hard test instances having previously known properties of the optimal solutions. On the basis of…