arXiv subjects

Matthew Drescher

Publications and source records attributed to Matthew Drescher.

7 recordsLinked to original sources

arXiv2024

Constructions, bounds, and algorithms for peaceable queens

The peaceable queens problem asks to determine the maximum number $a(n)$ such that there is a placement of $a(n)$ white queens and $a(n)$ black queens on an $n \times n$ chessboard so that no queen can capture any queen of the opposite color. In this paper, we consider the peaceable queens problem and its variant on the toroidal board. For the regular board, we show that $a(n) \leq 0.1716n^2$, for all sufficiently large $n$. This improves on the bound $a(n) \leq 0.25n^2$ of van Bommel and MacEachern. For the toroidal board, we provide new upper and lower bounds. Somewhat surprisingly, our bounds show that there is a sharp contrast in behaviour between the odd torus and the even torus. Our lower bounds are given by explicit constructions. For the upper bounds, we formulate the problem as a non-linear optimization problem with at most $100$ variables, regardless of the size of the board. We solve our non-linear program exactly using modern optimization software. We also provide a local search algorithm and a software implementation which converges very rapidly to solutions which appear optimal. Our algorithm is sufficiently robust that it works on both the regular and toroidal boards. For example, for the regular board, the algorithm quickly finds the so-called Ainley construction. Thus, our work provides some further evidence that the Ainley construction is indeed optimal.

math.CO↗

arXiv2023

BOBA: A Parallel Lightweight Graph Reordering Algorithm with Heavyweight Implications

We describe a simple parallel-friendly lightweight graph reordering algorithm for COO graphs (edge lists). Our ``Batched Order By Attachment'' (BOBA) algorithm is linear in the number of edges in terms of reads and linear in the number of vertices for writes through to main memory. It is highly parallelizable on GPUs\@. We show that, compared to a randomized baseline, the ordering produced gives improved locality of reference in sparse matrix-vector multiplication (SpMV) as well as other graph algorithms. Moreover, it can substantially speed up the conversion from a COO representation to the compressed format CSR, a very common workflow. Thus, it can give \emph{end-to-end} speedups even in SpMV\@. Unlike other lightweight approaches, this reordering does not rely on explicitly knowing the degrees of the vertices, and indeed its runtime is comparable to that of computing degrees. Instead, it uses the structure and edge distribution inherent in the input edge list, making it a candidate for default use in a pragmatic graph creation pipeline. This algorithm is suitable for road-type networks as well as scale-free. It improves cache locality on both CPUs and GPUs, achieving hit rates similar to the heavyweight techniques (e.g., for SpMV, 7--52\% and 11--67\% in the L1 and L2 caches, respectively). Compared to randomly labeled graphs, BOBA-reordered graphs achieve end-to-end speedups of up to 3.45. The reordering time is approximately one order of magnitude faster than existing lightweight techniques and up to 2.5 orders of magnitude faster than heavyweight techniques.

cs.DC↗

arXiv2020

A simple $(2+ε)$-approximation algorithm for Split Vertex Deletion

A split graph is a graph whose vertex set can be partitioned into a clique and a stable set. Given a graph $G$ and weight function $w: V(G) \to \mathbb{Q}_{\geq 0}$, the Split Vertex Deletion (SVD) problem asks to find a minimum weight set of vertices $X$ such that $G-X$ is a split graph. It is easy to show that a graph is a split graph if and only it it does not contain a $4$-cycle, $5$-cycle, or a two edge matching as an induced subgraph. Therefore, SVD admits an easy $5$-approximation algorithm. On the other hand, for every $δ>0$, SVD does not admit a $(2-δ)$-approximation algorithm, unless P=NP or the Unique Games Conjecture fails. For every $ε>0$, Lokshtanov, Misra, Panolan, Philip, and Saurabh recently gave a randomized $(2+ε)$-approximation algorithm for SVD. In this work we give an extremely simple deterministic $(2+ε)$-approximation algorithm for SVD.

math.CO↗

arXiv2020

A simple 7/3-approximation algorithm for feedback vertex set in tournaments

We show that performing just one round of the Sherali-Adams hierarchy gives an easy 7/3-approximation algorithm for the Feedback Vertex Set (FVST) problem in tournaments. This matches the best deterministic approximation algorithm for FVST due to Mnich, Williams, and Végh, and is a significant simplification and runtime improvement of their approach.

math.CO↗

arXiv2020

A Tight Approximation Algorithm for the Cluster Vertex Deletion Problem

We give the first $2$-approximation algorithm for the cluster vertex deletion problem. This is tight, since approximating the problem within any constant factor smaller than $2$ is UGC-hard. Our algorithm combines the previous approaches, based on the local ratio technique and the management of true twins, with a novel construction of a 'good' cost function on the vertices at distance at most $2$ from any vertex of the input graph. As an additional contribution, we also study cluster vertex deletion from the polyhedral perspective, where we prove almost matching upper and lower bounds on how well linear programming relaxations can approximate the problem.

math.CO↗

arXiv2019

The Adaptive Sampling Revisited

The problem of estimating the number $n$ of distinct keys of a large collection of $N$ data is well known in computer science. A classical algorithm is the adaptive sampling (AS). $n$ can be estimated by $R.2^D$, where $R$ is the final bucket (cache) size and $D$ is the final depth at the end of the process. Several new interesting questions can be asked about AS (some of them were suggested by P.Flajolet and popularized by J.Lumbroso). The distribution of $W=\log (R2^D/n)$ is known, we rederive this distribution in a simpler way. We provide new results on the moments of $D$ and $W$. We also analyze the final cache size $R$ distribution. We consider colored keys: assume that among the $n$ distinct keys, $n_C$ do have color $C$. We show how to estimate $p=\frac{n_C}{n}$. We also study colored keys with some multiplicity given by some distribution function. We want to estimate mean an variance of this distribution. Finally, we consider the case where neither colors nor multiplicities are known. There we want to estimate the related parameters. An appendix is devoted to the case where the hashing function provides bits with probability different from $1/2$.

cs.DS↗

arXiv2018

Short rainbow cycles in graphs and matroids

Let $G$ be a simple $n$-vertex graph and $c$ be a colouring of $E(G)$ with $n$ colours, where each colour class has size at least $2$. We prove that $(G,c)$ contains a rainbow cycle of length at most $\lceil \frac{n}{2} \rceil$, which is best possible. Our result settles a special case of a strengthening of the Caccetta-H\"aggkvist conjecture, due to Aharoni. We also show that the matroid generalization of our main result also holds for cographic matroids, but fails for binary matroids.

math.CO↗