相关论文: Structure and randomness in combinatorics

In combinatorics, the probabilistic method is a very powerful tool to prove the existence of combinatorial objects with interesting and useful properties. Explicit constructions of objects with such properties are often very difficult, or…

We are often interested in decomposing complex, structured data into simple components that explain the data. The linear version of this problem is well-studied as dictionary learning and factor analysis. In this work, we propose a…

Many random combinatorial objects have a component structure whose joint distribution is equal to that of a process of mutually independent random variables, conditioned on the value of a weighted sum of the variables. It is interesting to…

Assembling parts into an object is a combinatorial problem that arises in a variety of contexts in the real world and involves numerous applications in science and engineering. Previous related work tackles limited cases with identical unit…

Combinatorial optimization can be described as the problem of finding a feasible subset that maximizes a objective function. The paper discusses combinatorial optimization problems, where for each dimension the set of feasible subsets is…

Composition is an important feature of a specification language, as it enables the design of a complex system in terms of a product of its parts. Decomposition is equally important in order to reason about structural properties of a system.…

Framed combinatorial topology is a novel theory describing combinatorial phenomena arising at the intersection of stratified topology, singularity theory, and higher algebra. The theory synthesizes elements of classical combinatorial…

A quantum algorithm for general combinatorial search that uses the underlying structure of the search space to increase the probability of finding a solution is presented. This algorithm shows how coherent quantum systems can be matched to…

Algorithms to generate various combinatorial structures find tremendous importance in computer science. In this paper, we begin by reviewing an algorithm proposed by Rohl that generates all unique permutations of a list of elements which…

In many applications of the probabilistic method, one looks to study phenomena that occur ``with high probability''. More recently however, in an attempt to understand some of the most fundamental problems in combinatorics, researchers have…

Random graphs have proven to be one of the most important and fruitful concepts in modern Combinatorics and Theoretical Computer Science. Besides being a fascinating study subject for their own sake, they serve as essential instruments in…

This chapter studies the problem of decomposing a tensor into a sum of constituent rank one tensors. While tensor decompositions are very useful in designing learning algorithms and data analysis, they are NP-hard in the worst-case. We will…

The major challenge in designing a discriminative learning algorithm for predicting structured data is to address the computational issues arising from the exponential size of the output space. Existing algorithms make different assumptions…

Two important pieces of nuclear structure are many-body collective deformations and single-particle spin-orbit splitting. The former can be well-described microscopically by simple SU(3) irreps, but the latter mixes SU(3) irreps, which…

The problem of ranking can be described as follows. We have a set of combinatorial objects $S$, such as, say, the k-subsets of n things, and we can imagine that they have been arranged in some list, say lexicographically, and we want to…

We explore various combinatorial problems mostly borrowed from physics, that share the property of being continuously or discretely integrable, a feature that guarantees the existence of conservation laws that often make the problems…

Chaotic dynamics are ubiquitous in nature and useful in engineering, but their geometric design can be challenging. Here, we propose a method using reservoir computing to generate chaos with a desired shape by providing a periodic orbit as…

Hypergraphs are structures that can be decomposed or described; in other words they are recursively countable. Here, we get exact and asymptotic enumeration results on hypergraphs by means of exponential generating functions. The number of…

We deal with the random combinatorial structures called assemblies. By weakening the logarithmic condition which assures regularity of the number of components of a given order, we extend the notion of logarithmic assemblies. Using the…

With this work we aim to show how Mathematica can be a useful tool to investigate properties of combinatorial structures. Specifically, we will face enumeration problems on independent subsets of powers of paths and cycles, trying to…