Universal Sorting: Finding a DAG using Priced Comparisons

We resolve two open problems in sorting with priced information, introduced by [Charikar, Fagin, Guruswami, Kleinberg, Raghavan, Sahai (CFGKRS), STOC 2000]. In this setting, different comparisons have different (potentially infinite) costs. The goal is to sort with small competitive ratio (algorithmic cost divided by cheapest proof). 1) When all costs are in $\{0,1,n,\infty\}$, we give an algorithm that has $\widetilde{O}(n^{3/4})$ competitive ratio. Our algorithm generalizes the algorithms for generalized sorting (all costs are either $1$ or $\infty$), a version initiated by [Huang, Kannan, Khanna, FOCS 2011] and addressed recently by [Kuszmaul, Narayanan, FOCS 2021]. 2) We answer the problem of bichromatic sorting posed by [CFGKRS]: The input is split into $A$ and $B$, and $A-A$ and $B-B$ comparisons are more expensive than an $A-B$ comparisons. We give a randomized algorithm with a O(polylog n) competitive ratio. These results are obtained by introducing the universal sorting problem, which generalizes the existing framework in two important ways. We remove the promise of a directed Hamiltonian path in the DAG of all comparisons. Instead, we require that an algorithm outputs the transitive reduction of the DAG. For bichromatic sorting, when $A-A$ and $B-B$ comparisons cost $\infty$, this generalizes the well-known nuts and bolts problem. We initiate an instance-based study of the universal sorting problem. Our definition of instance-optimality is inherently more algorithmic than that of the competitive ratio in that we compare the cost of a candidate algorithm to the cost of the optimal instance-aware algorithm. This unifies existing lower bounds, and opens up the possibility of an $O(1)$-instance optimal algorithm for the bichromatic version.Comment: 40 pages, 5 figure

Coloring problems in combinatorics and descriptive set theory

In this dissertation we study problems related to colorings of combinatorial structures both in the “classical” finite context and in the framework of descriptive set theory, with applications to topological dynamics and ergodic theory. This work consists of two parts, each of which is in turn split into a number of chapters. Although the individual chapters are largely independent from each other (with the exception of Chapters 4 and 6, which partially rely on some of the results obtained in Chapter 3), certain common themes feature throughout—most prominently, the use of probabilistic techniques. In Chapter 1, we establish a generalization of the Lovász Local Lemma (a powerful tool in probabilistic combinatorics), which we call the Local Cut Lemma, and apply it to a variety of problems in graph coloring. In Chapter 2, we study DP-coloring (also known as correspondence coloring)—an extension of list coloring that was recently introduced by Dvorák and Postle. The goal of that chapter is to gain some understanding of the similarities and the differences between DP-coloring and list coloring, and we find many instances of both. In Chapter 3, we adapt the Lovász Local Lemma for the needs of descriptive set theory and use it to establish new bounds on measurable chromatic numbers of graphs induced by group actions. In Chapter 4, we study shift actions of countable groups on spaces of the form A, where A is a finite set, and apply the Lovász Local Lemma to find “large” closed shift-invariant subsets X A on which the induced action of is free. In Chapter 5, we establish precise connections between certain problems in graph theory and in descriptive set theory. As a corollary of our general result, we obtain new upper bounds on Baire measurable chromatic numbers from known results in finite combinatorics. Finally, in Chapter 6, we consider the notions of weak containment and weak equivalence of probability measure-preserving actions of a countable group—relations introduced by Kechris that are combinatorial in spirit and involve the way the action interacts with finite colorings of the underlying probability space. This work is based on the following papers and preprints: [Ber16a; Ber16b; Ber16c; Ber17a; Ber17b; Ber17c; Ber18a; Ber18b], [BK16; BK17a] (with Alexandr Kostochka), [BKP17] (with Alexandr Kostochka and Sergei Pron), and [BKZ17; BKZ18] (with Alexandr Kostochka and Xuding Zhu)

LIPIcs, Volume 258, SoCG 2023, Complete Volume

LIPIcs, Volume 258, SoCG 2023, Complete Volum

Symmetry reduction in convex optimization with applications in combinatorics

This dissertation explores different approaches to and applications of symmetry reduction in convex optimization. Using tools from semidefinite programming, representation theory and algebraic combinatorics, hard combinatorial problems are solved or bounded. The first chapters consider the Jordan reduction method, extend the method to optimization over the doubly nonnegative cone, and apply it to quadratic assignment problems and energy minimization on a discrete torus. The following chapter uses symmetry reduction as a proving tool, to approach a problem from queuing theory with redundancy scheduling. The final chapters propose generalizations and reductions of flag algebras, a powerful tool for problems coming from extremal combinatorics