Zipei Nie - Academia.edu (original) (raw)

Papers by Zipei Nie

Zenodo (CERN European Organization for Nuclear Research), Oct 9, 2023

arXiv (Cornell University), Feb 12, 2014

The polynomial method has been used recently to obtain many striking results in combinatorial geo... more The polynomial method has been used recently to obtain many striking results in combinatorial geometry. In this paper, we use affine Hilbert functions to obtain an estimation theorem in finite field geometry. The most natural way to state the theorem is via a sort of bounded degree Zariski closure operation: given a set, we consider all polynomials of some bounded degree vanishing on that set, and then the common zeros of these polynomials. For example, the degree d closure of d + 1 points on a line will contain the whole line, as any polynomial of degree at most d vanishing on the d + 1 points must vanish on the line. Our result is a bound on the size of a finite degree closure of a given set. Finally, we adapt our use of Hilbert functions to the method of multiplicities.

arXiv (Cornell University), Feb 28, 2018

In this paper, we prove that the fundamental group of the manifold obtained by Dehn surgery along... more In this paper, we prove that the fundamental group of the manifold obtained by Dehn surgery along a (−2, 3, 2s + 1)-pretzel knot (s ≥ 3) with slope p q is not left orderable if p q ≥ 2s + 3, and that it is left orderable if p q is in a neighborhood of zero depending on s.

arXiv (Cornell University), Jun 8, 2021

This paper studies the online correlated selection (OCS) problem. It was introduced by Fahrbach, ... more This paper studies the online correlated selection (OCS) problem. It was introduced by Fahrbach, Huang, Tao, and Zadimoghaddam (2020) to obtain the first edge-weighted online bipartite matching algorithm that breaks the 0.5 barrier. Suppose that we receive a pair of elements in each round and immediately select one of them. Can we select with negative correlation to be more effective than independent random selections? Our contributions are threefold. For semi-OCS, which considers the probability that an element remains unselected after appearing in k rounds, we give an optimal algorithm that minimizes this probability for all k. It leads to 0.536-competitive unweighted and vertex-weighted online bipartite matching algorithms that randomize over only two options in each round, improving the 0.508-competitive ratio by Fahrbach et al. (2020). Further, we develop the first multi-way semi-OCS that allows an arbitrary number of elements with arbitrary masses in each round. As an application, it rounds the Balance algorithm in unweighted and vertex-weighted online bipartite matching and is 0.593-competitive. Finally, we study OCS, which further considers the probability that an element is unselected in an arbitrary subset of rounds. We prove that the optimal "level of negative correlation" is between 0.167 and 0.25, improving the previous bounds of 0.109 and 1 by Fahrbach et al. (2020). Our OCS gives a 0.519-competitive edge-weighted online bipartite matching algorithm, improving the previous 0.508-competitive ratio by Fahrbach et al. (2020). * This is the second version on arXiv. Compared to the first version, this one adds a discussion on two concurrent works on the same topic, gives a more accurate description of previous results, and improves the presentation based on the feedbacks by anonymous reviewers. The conference version appears in FOCS 2021.

arXiv (Cornell University), Nov 10, 2021

We provide a polynomial lower bound on the minimum singular value of an m × m random matrix M wit... more We provide a polynomial lower bound on the minimum singular value of an m × m random matrix M with jointly Gaussian entries, under a polynomial bound on the matrix norm and a global small-ball probability bound inf x,y∈S m−1 P |x * M y| > m −O(1) ≥ 1 2. With the additional assumption that M is self-adjoint, the global small-ball probability bound can be replaced by a weaker version. We establish two matrix anti-concentration inequalities, which lower bound the minimum singular values of the sum of independent positive semidefinite selfadjoint matrices and the linear combination of independent random matrices with independent Gaussian coefficients. Both are under a global small-ball probability assumption. As a major application, we prove a better singular value bound for the Krylov space matrix, which leads to a faster and simpler algorithm for solving sparse linear systems. Our algorithm runs inÕ n 3ω−4 ω−1 = O(n 2.2716) time where ω < 2.37286 is the matrix multiplication exponent, improving on the previous fastest one inÕ n 5ω−4 ω+1 = O(n 2.33165) time by Peng and Vempala.

arXiv (Cornell University), Apr 21, 2023

We study the unit-demand capacitated vehicle routing problem in the random setting of the Euclide... more We study the unit-demand capacitated vehicle routing problem in the random setting of the Euclidean plane. The objective is to visit n random terminals in a square using a set of tours of minimum total length, such that each tour visits the depot and at most k terminals. We design an elegant algorithm combining the classical sweep heuristic and Arora's framework for the Euclidean traveling salesman problem [Journal of the ACM 1998]. We show that our algorithm is a polynomial-time approximation of ratio at most 1.55 asymptotically almost surely. This improves on previous approximation ratios of 1.995 due to Bompadre, Dror, and Orlin [Journal of Applied Probability 2007] and 1.915 due to Mathieu and Zhou [Random Structures and Algorithms 2022]. In addition, we conjecture that, for any ε > 0, our algorithm is a (1 + ε)-approximation asymptotically almost surely.

arXiv (Cornell University), Dec 15, 2022

We consider the following game. A deck with m copies of each of n distinct cards is shuffled in a... more We consider the following game. A deck with m copies of each of n distinct cards is shuffled in a perfectly random way. The Guesser sequentially guesses the cards from top to bottom. After each guess, the Guesser is informed whether the guess is correct. The goal is to maximize the expected number of correct guesses. We prove that, if n = Ω(√ m), then at most m + O(√ m) cards can be guessed correctly. Our result matches a lower bound of the maximal expected payoff by Diaconis, Graham and Spiro when n = Ω(m).

arXiv (Cornell University), Jan 13, 2023

Following Janson's method, we prove a conjecture of Knuth: the numbers of forward and back arcs f... more Following Janson's method, we prove a conjecture of Knuth: the numbers of forward and back arcs for the depth-first search (DFS) in a digraph with a geometric outdegree distribution have the same distribution.

Princeton, NJ : Princeton University, 2020

In this thesis, we present some results about (1, 1)-knots and L-space conjecture. In particular,... more In this thesis, we present some results about (1, 1)-knots and L-space conjecture. In particular, we prove that (1) L-space twisted torus knots of form T l,m p,kp±1 are closures of 1-bridge braids; (2) the L-space conjecture holds for the L-spaces obtained from Dehn surgery on closures of iterated 1-bridge braids, and for 3-manifolds obtained from Dehn fillings on the hyperbolic Q-homology solid torus v2503; (3) there are infinitely many (1, 1)-knots which are topologically slice but not smoothly slice. iii I would like to extend my genuine gratitude to my advisor Zoltán Szabó who offered me tremendous freedom to explore problems in the field of low dimensional topology. His kindness and encouragement have made me highly motivated during Ph.D.. I would like to thank David Gabai and Peter Ozsváth for serving my thesis committee, and Steven Boyer for being my thesis reader. I would like to thank my peers Weibo Fu,

Proceedings of the 54th Annual ACM SIGACT Symposium on Theory of Computing

We study by random matrices with jointly Gaussian entries. Assuming a global small-ball probabili... more We study by random matrices with jointly Gaussian entries. Assuming a global small-ball probability bound inf , ∈ −1 P * > − (1) ≥ 1 2 and a polynomial bounded on the norm of , we show that the minimum singular value of has a polynomial lower bound. We also consider the problem with the additional self-adjoint assumption. We establish two matrix anti-concentration inequalities, which lower bound the minimum singular values of the sum of independent positive semidefinite self-adjoint matrices and the linear combination of independent random matrices with independent Gaussian coefficients. Both are under a global small-ball probability assumption. Two applications are discussed. First, we derive a better singular value bound for the Krylov space matrix. This leads to a faster and simpler algorithm for solving sparse linear systems. Our algorithm runs in˜ 3 −4 −1 = (2.2716) time where < 2.37286 is the matrix multiplication exponent, improving on the previous fastest one in˜ 5 −4 +1 = (2.33165) time by Peng and Vempala. Second, in compressed sensing, we relax certain restrictions for constructing measurement matrix by the basis pursuit algorithm. CCS CONCEPTS • Mathematics of computing → Computations on matrices; • Theory of computation → Randomness, geometry and discrete structures.

arXiv: Geometric Topology, Nov 29, 2014

2021 IEEE 62nd Annual Symposium on Foundations of Computer Science (FOCS), 2022

In this paper, we prove that the fundamental group of the manifold obtained by Dehn surgery along... more In this paper, we prove that the fundamental group of the manifold obtained by Dehn surgery along a (−2, 3, 2s + 1)-pretzel knot (s ≥ 3) with slope p q is not left orderable if p q ≥ 2s+ 3, and that it is left orderable if p q is in a neighborhood of zero depending on s.

We prove the non-left-orderability of the fundamental group of the n-th fold cyclic branched cove... more We prove the non-left-orderability of the fundamental group of the n-th fold cyclic branched cover of the pretzel knot P (3,−3,−2k− 1) for all integers k and n ≥ 1. These 3-manifolds are L-spaces discovered by Issa and Turner.

arXiv: Algebraic Geometry, 2015

For a set SSS of ddd points in the nnn-dimensional projective space over a field of characteristi... more For a set SSS of ddd points in the nnn-dimensional projective space over a field of characteristic zero, we prove that the polynomials of degree ddd whose zero sets are cones over SSS do not span the vector space of polynomials of degree ddd vanishing on SSS, if ddd is odd and dge3d\ge 3dge3. Furthermore, they span a subspace of codimension at least two, if n=2n=2n=2, d=1pmod4d=1\pmod 4d=1pmod4 and dge5d\ge 5dge5.

arXiv: Geometric Topology, 2020

We define the property (D) for nontrivial knots. We show that the fundamental group of the manifo... more We define the property (D) for nontrivial knots. We show that the fundamental group of the manifold obtained by Dehn surgery on a knot KKK with property (D) with slope fracpqge2g(K)−1\frac{p}{q}\ge 2g(K)-1fracpqge2g(K)−1 is not left orderable. By making full use of the fixed point method, we prove that (1) nontrivial knots which are closures of positive 111-bridge braids have property (D); (2) L-space satellite knots, with positive 111-bridge braid patterns, and companion with property (D), have property (D); (3) the fundamental group of the manifold obtained by Dehn filling on v2503v2503v2503 is not left orderable. Additionally, we prove that L-space twisted torus knots of form Tp,kppm1l,mT_{p,kp\pm 1}^{l,m}Tp,kppm1l,m are closures of positive 111-bridge braids.

Greene, Lewallen and Vafaee characterized (1, 1) L-space knots in S and lens space in the notatio... more Greene, Lewallen and Vafaee characterized (1, 1) L-space knots in S and lens space in the notation of coherent reduced (1, 1)-diagrams. We analyze these diagrams, and deduce an explicit description of these knots. With the new description, we prove that any L-space obtained by Dehn surgery on a (1, 1)-knot in S has non-left-orderable fundamental group.

We prove that there are infinitely many (1,1)(1,1)(1,1)-knots which are topologically slice, but not smoo... more We prove that there are infinitely many (1,1)(1,1)(1,1)-knots which are topologically slice, but not smoothly slice, which was a conjecture proposed by B\'ela Andr\'as R\'acz.

In this paper we study an incidence problem in finite geometry. Suppose we are given a set X whic... more In this paper we study an incidence problem in finite geometry. Suppose we are given a set X which is the union of some number of lines in Fq . We choose a subset Y of X, such that for each line ` in X, at least half of its points are in Y . We show that |Y | is always at least some fraction of |X|. Using the polynomial method and degree reduction, it was previously known that such a statement holds for large and small |Y | (when |Y | ≥ q2 log q or |Y | ≤ q2/ log q). We close the gap by proving the statement for the remaining cases. We first note that such a statement holds for a set of points such that each point lies on at most two lines. We then show that there cannot be too many points with three or more lines lying on the zero set of a nonlinear polynomial, and use this to prove the statement in the remaining cases.