223 research outputs found
LIPIcs, Volume 251, ITCS 2023, Complete Volume
LIPIcs, Volume 251, ITCS 2023, Complete Volum
On Vizing's problem for triangle-free graphs
We prove that for any triangle-free
graph of maximum degree provided . This gives
tangible progress towards an old problem of Vizing, in a form cast by Reed. We
use a method of Hurley and Pirot, which in turn relies on a new counting
argument of the second author.Comment: 10 page
A machine learning approach to constructing Ramsey graphs leads to the Trahtenbrot-Zykov problem.
Attempts at approaching the well-known and difficult problem of constructing Ramsey graphs via machine learning lead to another difficult problem posed by Zykov in 1963 (now commonly referred to as the Trahtenbrot-Zykov problem): For which graphs F does there exist some graph G such that the neighborhood of every vertex in G induces a subgraph isomorphic to F? Chapter 1 provides a brief introduction to graph theory. Chapter 2 introduces Ramsey theory for graphs. Chapter 3 details a reinforcement learning implementation for Ramsey graph construction. The implementation is based on board game software, specifically the AlphaZero program and its success learning to play games from scratch. The chapter ends with a description of how computing challenges naturally shifted the project towards the Trahtenbrot-Zykov problem. Chapter 3 also includes recommendations for continuing the project and attempting to overcome these challenges. Chapter 4 defines the Trahtenbrot-Zykov problem and outlines its history, including proofs of results omitted from their original papers. This chapter also contains a program for constructing graphs with all neighborhood-induced subgraphs isomorphic to a given graph F. The end of Chapter 4 presents constructions from the program when F is a Ramsey graph. Constructing such graphs is a non-trivial task, as Bulitko proved in 1973 that the Trahtenbrot-Zykov problem is undecidable. Chapter 5 is a translation from Russian to English of this famous result, a proof not previously available in English. Chapter 6 introduces Cayley graphs and their relationship to the Trahtenbrot-Zykov problem. The chapter ends with constructions of Cayley graphs Γ in which the neighborhood of every vertex of Γ induces a subgraph isomorphic to a given Ramsey graph, which leads to a conjecture regarding the unique extremal Ramsey(4, 4) graph
Sum-of-squares representations for copositive matrices and independent sets in graphs
A polynomial optimization problem asks for minimizing a polynomial function (cost) given a set of constraints (rules) represented by polynomial inequalities and equations. Many hard problems in combinatorial optimization and applications in operations research can be naturally encoded as polynomial optimization problems. A common approach for addressing such computationally hard problems is by considering variations of the original problem that give an approximate solution, and that can be solved efficiently. One such approach for attacking hard combinatorial problems and, more generally, polynomial optimization problems, is given by the so-called sum-of-squares approximations. This thesis focuses on studying whether these approximations find the optimal solution of the original problem.We investigate this question in two main settings: 1) Copositive programs and 2) parameters dealing with independent sets in graphs. Among our main new results, we characterize the matrix sizes for which sum-of-squares approximations are able to capture all copositive matrices. In addition, we show finite convergence of the sums-of-squares approximations for maximum independent sets in graphs based on their continuous copositive reformulations. We also study sum-of-squares approximations for parameters asking for maximum balanced independent sets in bipartite graphs. In particular, we find connections with the Lovász theta number and we design eigenvalue bounds for several related parameters when the graphs satisfy some symmetry properties.<br/
LIPIcs, Volume 261, ICALP 2023, Complete Volume
LIPIcs, Volume 261, ICALP 2023, Complete Volum
Asymptotics for Palette Sparsification
It is shown that the following holds for each . For an
-vertex graph of maximum degree and "lists" () chosen
independently and uniformly from the ()-subsets of , with probability tending to 1 as .
This is an asymptotically optimal version of a recent "palette
sparsification" theorem of Assadi, Chen, and Khanna.Comment: 29 page
Cluster expansion methods in rigorous statistical mechanics
This draft is intended to be used as class notes for a grad course on
rigorous statistical mechanics at math department of UFMG. It should be
considered as a very prelimivary version and a work in progress. Several
chapters lack references, exercises, and revision
A tight Monte-Carlo algorithm for Steiner Tree parameterized by clique-width
Recently, Hegerfeld and Kratsch [ESA 2023] obtained the first tight
algorithmic results for hard connectivity problems parameterized by
clique-width. Concretely, they gave one-sided error Monte-Carlo algorithms that
given a -clique-expression solve Connected Vertex Cover in time
and Connected Dominating Set in time . Moreover,
under the Strong Exponential-Time Hypothesis (SETH) these results were showed
to be tight. However, they leave open several important benchmark problems,
whose complexity relative to treewidth had been settled by Cygan et al. [SODA
2011 & TALG 2018]. Among which is the Steiner Tree problem. As a key
obstruction they point out the exponential gap between the rank of certain
compatibility matrices, which is often used for algorithms, and the largest
triangular submatrix therein, which is essential for current lower bound
methods. Concretely, for Steiner Tree the -rank is , while no
triangular submatrix larger than was known. This yields time
, while the obtainable impossibility of time
under SETH was already known relative to pathwidth.
We close this gap by showing that Steiner Tree can be solved in time
given a -clique-expression. Hence, for all parameters between
cutwidth and clique-width it has the same tight complexity. We first show that
there is a ``representative submatrix'' of GF(2)-rank (ruling out larger
triangular submatrices). At first glance, this only allows to count (modulo 2)
the number of representations of valid solutions, but not the number of
solutions (even if a unique solution exists). We show how to overcome this
problem by isolating a unique representative of a unique solution, if one
exists. We believe that our approach will be instrumental for settling further
open problems in this research program
- …