Monochromatic and Zero-Sum Sets of Nondecreasing Modified Diameter

Let m be a positive integer whose smallest prime divisor is denoted by p, and let Zm denote the cyclic group of residues modulo m. For a set B = {x1, x2, ..., xm} of m integers satisfying x1 {0, 1} (every coloring Delta : {1, ..., N} -> Zm), there exist two m-sets [see Abstract in the PDF]

On monochromatic sets with nondecreasing diameter

Our problem comes from the field of combinatorics known as Ramsey theory. Ramsey theory, in a general sense, is about identifying the threshold for which a family of objects, associated with a particular parameter, goes from never or sometimes satisfying a certain property to always satisfying that property. Research in Ramsey theory has applications in design theory and coding theory. For integers m, r, and t, we say that a set of n integers colored with r colors is (m, r, t)-permissible if there exist t monochromatic subsets B1, B2, . . . , Bt such that (a) |B1| = |B2| = · · · = |Bt | = m, (b) the largest element in Bi is less than the smallest element in Bi+1 for 1 ≤ i ≤ t − 1, and (c) the diameters of the subsets are nondecreasing. We define f(m, r, t) to be the smallest integer n such that every string of length n is (m, r, t)-permissible. In this thesis, we first look at some preliminary results for values of f(m, r, t), specifically when each individual parameter is 1 as the others vary. We then show that f(m, r, t) exists for all possible positive parameters. We proceed by determining f(2, 2, t) for all positive integers t. We conclude by considering colorings with more than two colors and monochromatic sets that have more than 2 elements, as well as investigating an enumeration of the number of ways a string could be realized as (m, r, t)-permissible

On Monochromatic Pairs with Nondecreasing Diameters

Let n m r t be positive integers and Δ : [n] → [r]. We say Δ is (m, r, t) - permissible if there exist t disjoint m-sets B1,…,Bt contained in [n] for which |Δ(Bi)| = 1 for each i = 1,2,…, t. max(Bi) \u3c min(Bi+1) for each i = 1,2,…, t − 1, and max(Bi) − min(Bi) ≤ max(Bi+1) − max(Bi+1) for each i = 1, 2,…, t − 1. Let f(m ,r, t) be the smallest such n so that all colorings Δ are (m, r, t)-permissible. In this paper, we show that f(2, 2, t) = 5t − 4

Tournament Directed Graphs

Paired comparison is the process of comparing objects two at a time. A tournament in Graph Theory is a representation of such paired comparison data. Formally, an n-tournament is an oriented complete graph on n vertices; that is, it is the representation of a paired comparison, where the winner of the comparison between objects x and y (x and y are called vertices) is depicted with an arrow or arc from the winner to the other. In this thesis, we shall prove several results on tournaments. In Chapter 2, we will prove that the maximum number of vertices that can beat exactly m other vertices in an n-tournament is min{2m + 1,2n - 2m - 1}. The remainder of this thesis will deal with tournaments whose arcs have been colored. In Chapter 3 we will define what it means for a k-coloring of a tournament to be k-primitive. We will prove that the maximum k such that some strong n-tournament can be k-colored to be k-primitive lies in the interval [(n-12), (n2) - [n/4]). In Chapter 4, we shall prove special cases of the following 1982 conjecture of Sands, Sauer, and Woodrow from [14]: Let T be a 3-arc-colored tournament containing no 3-cycle C such that each arc in C is a different color. Then T contains a vertex v such that for any other vertex x, x has a monochromatic path to v

Fine-Grained Completeness for Optimization in P

We initiate the study of fine-grained completeness theorems for exact and approximate optimization in the polynomial-time regime. Inspired by the first completeness results for decision problems in P (Gao, Impagliazzo, Kolokolova, Williams, TALG 2019) as well as the classic class MaxSNP and MaxSNP-completeness for NP optimization problems (Papadimitriou, Yannakakis, JCSS 1991), we define polynomial-time analogues MaxSP and MinSP, which contain a number of natural optimization problems in P, including Maximum Inner Product, general forms of nearest neighbor search and optimization variants of the $k$-XOR problem. Specifically, we define MaxSP as the class of problems definable as $\max_{x_1,\dots,x_k} \#\{ (y_1,\dots,y_\ell) : \phi(x_1,\dots,x_k, y_1,\dots,y_\ell) \}$, where $\phi$ is a quantifier-free first-order property over a given relational structure (with MinSP defined analogously). On $m$-sized structures, we can solve each such problem in time $O(m^{k+\ell-1})$. Our results are: - We determine (a sparse variant of) the Maximum/Minimum Inner Product problem as complete under *deterministic* fine-grained reductions: A strongly subquadratic algorithm for Maximum/Minimum Inner Product would beat the baseline running time of $O(m^{k+\ell-1})$ for *all* problems in MaxSP/MinSP by a polynomial factor. - This completeness transfers to approximation: Maximum/Minimum Inner Product is also complete in the sense that a strongly subquadratic $c$-approximation would give a $(c+\varepsilon)$-approximation for all MaxSP/MinSP problems in time $O(m^{k+\ell-1-\delta})$, where $\varepsilon > 0$ can be chosen arbitrarily small. Combining our completeness with~(Chen, Williams, SODA 2019), we obtain the perhaps surprising consequence that refuting the OV Hypothesis is *equivalent* to giving a $O(1)$-approximation for all MinSP problems in faster-than-$O(m^{k+\ell-1})$ time.Comment: Full version of APPROX'21 paper, abstract shortened to fit ArXiv requirement