Matching Items (22)

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Post-Optimization of Permutation Coverings

Description

Covering subsequences with sets of permutations arises in many applications, including event-sequence testing. Given a set of subsequences to cover, one is often interested in knowing the fewest number of

Covering subsequences with sets of permutations arises in many applications, including event-sequence testing. Given a set of subsequences to cover, one is often interested in knowing the fewest number of permutations required to cover each subsequence, and in finding an explicit construction of such a set of permutations that has size close to or equal to the minimum possible. The construction of such permutation coverings has proven to be computationally difficult. While many examples for permutations of small length have been found, and strong asymptotic behavior is known, there are few explicit constructions for permutations of intermediate lengths. Most of these are generated from scratch using greedy algorithms. We explore a different approach here. Starting with a set of permutations with the desired coverage properties, we compute local changes to individual permutations that retain the total coverage of the set. By choosing these local changes so as to make one permutation less "essential" in maintaining the coverage of the set, our method attempts to make a permutation completely non-essential, so it can be removed without sacrificing total coverage. We develop a post-optimization method to do this and present results on sequence covering arrays and other types of permutation covering problems demonstrating that it is surprisingly effective.

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Created

Date Created
  • 2014-12

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A Study of Some Edge-Deletion Algorithms for Reducing Disease Spread on Networks

Description

This thesis discusses three recent optimization problems that seek to reduce disease spread on arbitrary graphs by deleting edges, and it discusses three approximation algorithms developed for these problems. Important

This thesis discusses three recent optimization problems that seek to reduce disease spread on arbitrary graphs by deleting edges, and it discusses three approximation algorithms developed for these problems. Important definitions are presented including the Linear Threshold and Triggering Set models and the set function properties of submodularity and monotonicity. Also, important results regarding the Linear Threshold model and computation of the influence function are presented along with proof sketches. The three main problems are formally presented, and NP-hardness results along with proof sketches are presented where applicable. The first problem seeks to reduce spread of infection over the Linear Threshold process by making use of an efficient tree data structure. The second problem seeks to reduce the spread of infection over the Linear Threshold process while preserving the PageRank distribution of the input graph. The third problem seeks to minimize the spectral radius of the input graph. The algorithms designed for these problems are described in writing and with pseudocode, and their approximation bounds are stated along with time complexities. Discussion of these algorithms considers how these algorithms could see real-world use. Challenges and the ways in which these algorithms do or do not overcome them are noted. Two related works, one which presents an edge-deletion disease spread reduction problem over a deterministic threshold process and the other which considers a graph modification problem aimed at minimizing worst-case disease spread, are compared with the three main works to provide interesting perspectives. Furthermore, a new problem is proposed that could avoid some issues faced by the three main problems described, and directions for future work are suggested.

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Created

Date Created
  • 2018-05

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An exploration of proofs of the Szemerédi regularity lemma

Description

This paper focuses on the Szemerédi regularity lemma, a result in the field of extremal graph theory. The lemma says that every graph can be partitioned into bounded equal parts

This paper focuses on the Szemerédi regularity lemma, a result in the field of extremal graph theory. The lemma says that every graph can be partitioned into bounded equal parts such that most edges of the graph span these partitions, and these edges are distributed in a fairly uniform way. Definitions and notation will be established, leading to explorations of three proofs of the regularity lemma. These are a version of the original proof, a Pythagoras proof utilizing elemental geometry, and a proof utilizing concepts of spectral graph theory. This paper is intended to supplement the proofs with background information about the concepts utilized. Furthermore, it is the hope that this paper will serve as another resource for students and others to begin study of the regularity lemma.

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Created

Date Created
  • 2015-05

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On the Bounds of Van der Waerden Numbers

Description

Van der Waerden’s Theorem asserts that for any two positive integers k and r, one may find an integer w=w(k,r) known as the Van der Waerden Number such that for

Van der Waerden’s Theorem asserts that for any two positive integers k and r, one may find an integer w=w(k,r) known as the Van der Waerden Number such that for every r-coloring of the integers from 1 to w there exists a monochromatic arithmetic progression of length k. This groundbreaking theorem in combinatorics has greatly impacted the field of discrete math for decades. However, it is quite difficult to find the exact values of w. As such, it would be worth more of our time to try and bound such a value, both from below and above, in order to restrict the possible values of the Van der Waerden Numbers. In this thesis we will endeavor to bound such a number; in addition to proving Van der Waerden’s Theorem, we will discuss the unique functions that bound the Van der Waerden Numbers.

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Created

Date Created
  • 2019-12

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Skipping Turns on the Ordering Game

Description

In the ordering game on a graph G, Alice and Bob take turns placing the vertices of G into a linear ordering. The score of the game is the maximum

In the ordering game on a graph G, Alice and Bob take turns placing the vertices of G into a linear ordering. The score of the game is the maximum number of neighbors that any vertex has before it in the ordering. Alice's goal in the ordering game is to minimize the score, while Bob's goal is to maximize it. The game coloring number is the least score that Alice can always guarantee in the ordering game, no matter how Bob plays. This paper examines what happens to the game coloring number if Alice or Bob skip turns on the ordering game. Preliminary definitions and examples are provided to give context to the ordering game. These include a polynomial time algorithm to compute the coloring number, a non-competitive version of the game coloring number. The notion of the preordered game is introduced as the primary tool of the paper, along with its naturally defined preordered game coloring number. To emphasize the complex relationship between the coloring number and the preordered game coloring number, a non-polynomial time strategy is given to Alice and Bob that yields the preordered game coloring number on any graph. Additionally, the preordered game coloring number is shown to be monotonic, one of the most useful properties for turn-skipping. Techniques developed throughout the paper are then used to determine that Alice cannot reduce the score and Bob cannot improve the score by skipping any number of their respective turns. This paper can hopefully be used as a stepping stone towards bounding the score on graphs when vertices are removed, as well as in deciphering further properties of the asymmetric marking game.

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Created

Date Created
  • 2019-05

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Coloring graphs from almost maximum degree sized palettes

Description

Every graph can be colored with one more color than its maximum degree. A well-known theorem of Brooks gives the precise conditions under which a graph can be colored with

Every graph can be colored with one more color than its maximum degree. A well-known theorem of Brooks gives the precise conditions under which a graph can be colored with maximum degree colors. It is natural to ask for the required conditions on a graph to color with one less color than the maximum degree; in 1977 Borodin and Kostochka conjectured a solution for graphs with maximum degree at least 9: as long as the graph doesn't contain a maximum-degree-sized clique, it can be colored with one fewer than the maximum degree colors. This study attacks the conjecture on multiple fronts. The first technique is an extension of a vertex shuffling procedure of Catlin and is used to prove the conjecture for graphs with edgeless high vertex subgraphs. This general approach also bears more theoretical fruit. The second technique is an extension of a method Kostochka used to reduce the Borodin-Kostochka conjecture to the maximum degree 9 case. Results on the existence of independent transversals are used to find an independent set intersecting every maximum clique in a graph. The third technique uses list coloring results to exclude induced subgraphs in a counterexample to the conjecture. The classification of such excludable graphs that decompose as the join of two graphs is the backbone of many of the results presented here. The fourth technique uses the structure theorem for quasi-line graphs of Chudnovsky and Seymour in concert with the third technique to prove the Borodin-Kostochka conjecture for claw-free graphs. The fifth technique adds edges to proper induced subgraphs of a minimum counterexample to gain control over the colorings produced by minimality. The sixth technique adapts a recoloring technique originally developed for strong coloring by Haxell and by Aharoni, Berger and Ziv to general coloring. Using this recoloring technique, the Borodin-Kostochka conjectured is proved for graphs where every vertex is in a large clique. The final technique is naive probabilistic coloring as employed by Reed in the proof of the Borodin-Kostochka conjecture for large maximum degree. The technique is adapted to prove the Borodin-Kostochka conjecture for list coloring for large maximum degree.

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Created

Date Created
  • 2013

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Hash Families and Applications to t-Restrictions

Description

The construction of many families of combinatorial objects remains a challenging problem. A t-restriction is an array where a predicate is satisfied for every t columns; an example is a

The construction of many families of combinatorial objects remains a challenging problem. A t-restriction is an array where a predicate is satisfied for every t columns; an example is a perfect hash family (PHF). The composition of a PHF and any t-restriction satisfying predicate P yields another t-restriction also satisfying P with more columns than the original t-restriction had. This thesis concerns three approaches in determining the smallest size of PHFs.

Firstly, hash families in which the associated property is satisfied at least some number lambda times are considered, called higher-index, which guarantees redundancy when constructing t-restrictions. Some direct and optimal constructions of hash families of higher index are given. A new recursive construction is established that generalizes previous results and generates higher-index PHFs with more columns. Probabilistic methods are employed to obtain an upper bound on the optimal size of higher-index PHFs when the number of columns is large. A new deterministic algorithm is developed that generates such PHFs meeting this bound, and computational results are reported.

Secondly, a restriction on the structure of PHFs is introduced, called fractal, a method from Blackburn. His method is extended in several ways; from homogeneous hash families (every row has the same number of symbols) to heterogeneous ones; and to distributing hash families, a relaxation of the predicate for PHFs. Recursive constructions with fractal hash families as ingredients are given, and improve upon on the best-known sizes of many PHFs.

Thirdly, a method of Colbourn and Lanus is extended in which they horizontally copied a given hash family and greedily applied transformations to each copy. Transformations of existential t-restrictions are introduced, which allow for the method to be applicable to any t-restriction having structure like those of hash families. A genetic algorithm is employed for finding the "best" such transformations. Computational results of the GA are reported using PHFs, as the number of transformations permitted is large compared to the number of symbols. Finally, an analysis is given of what trade-offs exist between computation time and the number of t-sets left not satisfying the predicate.

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Created

Date Created
  • 2019

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Listing combinatorial objects

Description

Gray codes are perhaps the best known structures for listing sequences of combinatorial objects, such as binary strings. Simply defined as a minimal change listing, Gray codes vary greatly both

Gray codes are perhaps the best known structures for listing sequences of combinatorial objects, such as binary strings. Simply defined as a minimal change listing, Gray codes vary greatly both in structure and in the types of objects that they list. More specific types of Gray codes are universal cycles and overlap sequences. Universal cycles are Gray codes on a set of strings of length n in which the first n-1 letters of one object are the same as the last n-1 letters of its predecessor in the listing. Overlap sequences allow this overlap to vary between 1 and n-1. Some of our main contributions to the areas of Gray codes and universal cycles include a new Gray code algorithm for fixed weight m-ary words, and results on the existence of universal cycles for weak orders on [n]. Overlap cycles are a relatively new structure with very few published results. We prove the existence of s-overlap cycles for k-permutations of [n], which has been an open research problem for several years, as well as constructing 1- overlap cycles for Steiner triple and quadruple systems of every order. Also included are various other results of a similar nature covering other structures such as binary strings, m-ary strings, subsets, permutations, weak orders, partitions, and designs. These listing structures lend themselves readily to some classes of combinatorial objects, such as binary n-tuples and m-ary n-tuples. Others require more work to find an appropriate structure, such as k-subsets of an n-set, weak orders, and designs. Still more require a modification in the representation of the objects to fit these structures, such as partitions. Determining when and how we can fit these sets of objects into our three listing structures is the focus of this dissertation.

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Agent

Created

Date Created
  • 2012

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Estimating Low Generalized Coloring Numbers of Planar Graphs

Description

The chromatic number $\chi(G)$ of a graph $G=(V,E)$ is the minimum

number of colors needed to color $V(G)$ such that no adjacent vertices

receive the same color. The coloring number $\col(G)$ of

The chromatic number $\chi(G)$ of a graph $G=(V,E)$ is the minimum

number of colors needed to color $V(G)$ such that no adjacent vertices

receive the same color. The coloring number $\col(G)$ of a graph

$G$ is the minimum number $k$ such that there exists a linear ordering

of $V(G)$ for which each vertex has at most $k-1$ backward neighbors.

It is well known that the coloring number is an upper bound for the

chromatic number. The weak $r$-coloring number $\wcol_{r}(G)$ is

a generalization of the coloring number, and it was first introduced

by Kierstead and Yang \cite{77}. The weak $r$-coloring number $\wcol_{r}(G)$

is the minimum integer $k$ such that for some linear ordering $L$

of $V(G)$ each vertex $v$ can reach at most $k-1$ other smaller

vertices $u$ (with respect to $L$) with a path of length at most

$r$ and $u$ is the smallest vertex in the path. This dissertation proves that $\wcol_{2}(G)\le23$ for every planar graph $G$.

The exact distance-$3$ graph $G^{[\natural3]}$ of a graph $G=(V,E)$

is a graph with $V$ as its set of vertices, and $xy\in E(G^{[\natural3]})$

if and only if the distance between $x$ and $y$ in $G$ is $3$.

This dissertation improves the best known upper bound of the

chromatic number of the exact distance-$3$ graphs $G^{[\natural3]}$

of planar graphs $G$, which is $105$, to $95$. It also improves

the best known lower bound, which is $7$, to $9$.

A class of graphs is nowhere dense if for every $r\ge 1$ there exists $t\ge 1$ such that no graph in the class contains a topological minor of the complete graph $K_t$ where every edge is subdivided at most $r$ times. This dissertation gives a new characterization of nowhere dense classes using generalized notions of the domination number.

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Agent

Created

Date Created
  • 2020

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On the uncrossing partial order on matchings

Description

The uncrossing partially ordered set $P_n$ is defined on the set of matchings on $2n$ points on a circle represented with wires. The order relation is $\tau'\leq \tau$ in $P_n$

The uncrossing partially ordered set $P_n$ is defined on the set of matchings on $2n$ points on a circle represented with wires. The order relation is $\tau'\leq \tau$ in $P_n$ if and only if $\tau'$ is obtained by resolving a crossing of $\tau$. %This partial order has been studied by Alman-Lian-Tran, Huang-Wen-Xie, Kenyon, and Lam. %The posets $P_n$ emerged from studies of circular planar electrical networks. Circular planar electrical networks are finite weighted undirected graphs embedded into a disk, with boundary vertices and interior vertices. By Curtis-Ingerman-Morrow and de Verdi\`ere-Gitler-Vertigan, the electrical networks can be encoded with response matrices. By Lam the space of response matrices for electrical networks has a cell structure, and this cell structure can be described by the uncrossing partial orders. %Lam proves that the posets can be identified with dual Bruhat order on affine permutations of type $(n,2n)$. Using this identification, Lam proves the poset $\hat{P}_n$, the uncrossing poset $P_n$ with a unique minimum element $\hat{0}$ adjoined, is Eulerian. This thesis consists of two sets of results: (1) flag enumeration in intervals in the uncrossing poset $P_n$ and (2) cyclic sieving phenomenon on the set $P_n$.

I identify elements in $P_n$ with affine permutations of type $(0,2n)$. %This identification enables us to explicitly describe the elements in $P_n$ with the elements in $\mathcal{MP}_n$.

Using this identification, I adapt a technique in Reading for finding recursions for the cd-indices of intervals in Bruhat order of Coxeter groups to the uncrossing poset $P_n$. As a result, I produce recursions for the cd-indices of intervals in the uncrossing poset $P_n$. I also obtain a recursion for the ab-indices of intervals in the poset $\hat{P}_n$, the poset $P_n$ with a unique minimum $\hat0$ adjoined. %We define an induced subposet $\mathcal{MP}_n$ of the affine permutations under Bruhat order.

Reiner-Stanton-White defined the cyclic sieving phenomenon (CSP) associated to a finite cyclic group action on a finite set and a polynomial. Sagan observed the CSP on the set of non-crossing matchings with the $q$-Catalan polynomial. Bowling-Liang presented similar results on the set of $k$-crossing matchings for $1\leq k \leq 3$. In this dissertation, I focus on the set of all matchings on $[2n]:=\{1,2,\dots,2n\}$. I find the number of matchings fixed by $\frac{2\pi}{d}$ rotations for $d|2n$. I then find the polynomial $X_n(q)$ such that the set of matchings together with $X_n(q)$ and the cyclic group of order $2n$ exhibits the CSP.

Contributors

Agent

Created

Date Created
  • 2018