Previous Bound Research Articles

Abundance: Asymmetric graph removal lemmas and integer solutions to linear equations

AbstractWe prove that a large family of pairs of graphs satisfy a polynomial dependence in asymmetric graph removal lemmas. In particular, we give an unexpected answer to a question of Gishboliner, Shapira and Wigderson by showing that for every , there are ‐abundant graphs of chromatic number . Using similar methods, we also extend work of Ruzsa by proving that a set which avoids solutions with distinct integers to an equation of genus at least two has size . The best previous bound was and the exponent of is best possible in such a result. Finally, we investigate the relationship between polynomial dependencies in asymmetric removal lemmas and the problem of avoiding integer solutions to equations. The results suggest a potentially deep correspondence. Many open questions remain.

Reducing the space used by the sieve of Eratosthenes when factoring

We present a version of the sieve of Eratosthenes that can factor all integers ≤x in O(xlog⁡log⁡x) arithmetic operations using at most O(x/log⁡log⁡x) bits of space. Among algorithms that take the optimal O(xlog⁡log⁡x) time, this new space bound is an improvement of a factor proportional to log⁡xlog⁡log⁡x over the implied previous bound of O(xlog⁡x). We also show our algorithm performs well in practice.

A lower bound on the mean value of the Erdős–Hooley Delta function

AbstractWe give an improved lower bound for the average of the Erdős–Hooley function , namely for all and any fixed , where is an exponent previously appearing in work of Green and the first two authors. This improves on a previous lower bound of of Hall and Tenenbaum, and can be compared to the recent upper bound of of the second and third authors.

Separating path systems of almost linear size

A separating path system for a graph G G is a collection P \mathcal {P} of paths in G G such that for every two edges e e and f f , there is a path in P \mathcal {P} that contains e e but not f f . We show that every n n -vertex graph has a separating path system of size O ( n log ∗ ⁡ n ) O(n \log ^* n) . This improves upon the previous best upper bound of O ( n log ⁡ n ) O(n \log n) , and makes progress towards a conjecture of Falgas-Ravry–Kittipassorn–Korándi–Letzter–Narayanan and Balogh–Csaba–Martin–Pluhár, according to which an O ( n ) O(n) bound should hold.

Reconstructing S-matrix Phases with Machine Learning

An important element of the S-matrix bootstrap program is the relationship between the modulus of an S-matrix element and its phase. Unitarity relates them by an integral equation. Even in the simplest case of elastic scattering, this integral equation cannot be solved analytically and numerical approaches are required. We apply modern machine learning techniques to studying the unitarity constraint. We find that for a given modulus, when a phase exists it can generally be reconstructed to good accuracy with machine learning. Moreover, the loss of the reconstruction algorithm provides a good proxy for whether a given modulus can be consistent with unitarity at all. In addition, we study the question of whether multiple phases can be consistent with a single modulus, finding novel phase-ambiguous solutions. In particular, we find a new phase-ambiguous solution which pushes the known limit on such solutions significantly beyond the previous bound.

Simple & Optimal Quantile Sketch: Combining Greenwald-Khanna with Khanna-Greenwald

Estimating the ε-approximate quantiles or ranks of a stream is a fundamental task in data monitoring. Given a stream x_1,..., x_n from a universe \mathcalU with total order, an additive-error quantile sketch \mathcalM allows us to approximate the rank of any query y\in \mathcalU up to additive ε n error. In 2001, Greenwald and Khanna gave a deterministic algorithm (GK sketch) that solves the ε-approximate quantiles estimation problem using O(ε^-1 łog(ε n)) space \citegreenwald2001space ; recently, this algorithm was shown to be optimal by Cormode and Vesleý in 2020 \citecormode2020tight. However, due to the intricacy of the GK sketch and its analysis, over-simplified versions of the algorithm are implemented in practical applications, often without any known theoretical guarantees. In fact, it has remained an open question whether the GK sketch can be simplified while maintaining the optimal space bound. In this paper, we resolve this open question by giving a simplified deterministic algorithm that stores at most (2 + o(1))ε^-1 łog (ε n) elements and solves the additive-error quantile estimation problem; as a side benefit, our algorithm achieves a smaller constant factor than the \frac11 2 ε^-1 łog(ε n) space bound in the original GK sketch~\citegreenwald2001space. Our algorithm features an easier analysis and still achieves the same optimal asymptotic space complexity as the original GK sketch. Lastly, our simplification enables an efficient data structure implementation, with a worst-case runtime of O(łog(1/ε) + łog łog (ε n)) per-element for the ordinary ε-approximate quantile estimation problem. Also, for the related "weighted'' quantile estimation problem, we give efficient data structures for our simplified algorithm which guarantee a worst-case per-element runtime of O(łog(1/ε) + łog łog (ε W_n/w_\textrmmin )), achieving an improvement over the previous upper bound of \citeassadi2023generalizing.

New lower bounds on scattering amplitudes: non-locality constraints

Under reasonable working assumptions including the polynomial boundedness, one proves the well-known Cerulus-Martin lower bound on how fast an elastic scattering amplitude can decrease in the hard-scattering regime. In this paper we consider two non-trivial extensions of the previous bound. (i) We generalize the assumption of polynomial boundedness by allowing amplitudes to exponentially grow for some complex momenta and prove a more general lower bound in the hard-scattering regime. (ii) We prove a new lower bound on elastic scattering amplitudes in the Regge regime, in both cases of polynomial and exponential boundedness. A bound on the Regge trajectory for negative momentum transfer squared is also derived. We discuss the relevance of our results for understanding gravitational scattering at the non-perturbative level and for constraining ultraviolet completions. In particular, we use the new bounds as probes of non-locality in black-hole formation, perturbative string theory, classicalization, Galileons, and infinite-derivative field theories, where both the polynomial boundedness and the Cerulus-Martin bound are violated.

Ramsey numbers of cliques versus monotone paths

One formulation of the Erdős-Szekeres monotone subsequence theorem states that for any red/blue coloring of the edge set of the complete graph on {1,2,…,N}, there exists a monochromatic red s-clique or a monochromatic blue increasing path Pn with n vertices, provided N>(s−1)(n−1). Here, we prove a similar statement as above in the off-diagonal case for triple systems, with the quasipolynomial bound N>2c(logn)s−1. For the tth power Pnt of the ordered increasing graph path with n vertices, we prove a near linear bound cn(logn)s−2 which improves the previous bound that applied to a more general class of graphs than Pnt due to Conlon-Fox-Lee-Sudakov.

Improved upper bound on the Frank number of 3-edge-connected graphs

In an orientation O of the graph G, an arc e is deletable if and only if O−e is strongly connected. For a 3-edge-connected graph G, the Frank number is the minimum k for which G admits k strongly connected orientations such that for every edge e of G the corresponding arc is deletable in at least one of the k orientations. Hörsch and Szigeti conjectured the Frank number is at most 3 for every 3-edge-connected graph G. We prove an upper bound of 5, which improves the previous bound of 7.

On the Turán number of the hypercube

Abstract In 1964, Erdős proposed the problem of estimating the Turán number of the d-dimensional hypercube $Q_d$ . Since $Q_d$ is a bipartite graph with maximum degree d, it follows from results of Füredi and Alon, Krivelevich, Sudakov that $\mathrm {ex}(n,Q_d)=O_d(n^{2-1/d})$ . A recent general result of Sudakov and Tomon implies the slightly stronger bound $\mathrm {ex}(n,Q_d)=o(n^{2-1/d})$ . We obtain the first power-improvement for this old problem by showing that $\mathrm {ex}(n,Q_d)=O_d\left (n^{2-\frac {1}{d-1}+\frac {1}{(d-1)2^{d-1}}}\right )$ . This answers a question of Liu. Moreover, our techniques give a power improvement for a larger class of graphs than cubes. We use a similar method to prove that any n-vertex, properly edge-coloured graph without a rainbow cycle has at most $O(n(\log n)^2)$ edges, improving the previous best bound of $n(\log n)^{2+o(1)}$ by Tomon. Furthermore, we show that any properly edge-coloured n-vertex graph with $\omega (n\log n)$ edges contains a cycle which is almost rainbow: that is, almost all edges in it have a unique colour. This latter result is tight.

On RAC Drawings of Graphs with Two Bends per Edge

It is shown that every $n$-vertex graph that admits a 2-bend RAC drawing in the plane, where the edges are polylines with two bends per edge and any pair of edges can only cross at a right angle, has at most $20n-24$ edges for $n\geq 3$. This improves upon the previous upper bound of $74.2n$; this is the first improvement in more than 12 years. A crucial ingredient of the proof is an upper bound on the size of plane multigraphs with polyline edges in which the first and last segments are either parallel or orthogonal.

Discrepancy of arithmetic progressions in grids

AbstractWe prove that the discrepancy of arithmetic progressions in the d‐dimensional grid is within a constant factor depending only on d of . This extends the case , which is a celebrated result of Roth and of Matoušek and Spencer, and removes the polylogarithmic factor from the previous upper bound of Valkó from about two decades ago. We further prove similarly tight bounds for grids of differing side lengths in many cases.

Improved maximin guarantees for subadditive and fractionally subadditive fair allocation problem

In this work, we study the maximin share fairness notion (MMS) for allocation of indivisible goods in the subadditive and fractionally subadditive settings. While previous work refutes the possibility of obtaining an allocation which is better than 1/2-MMS, the only positive result for the subadditive setting states that when the number of items is equal to m, there always exists an Ω(1/log⁡m)-MMS allocation. Since the number of items may be larger than the number of agents (n), such a bound can only imply a weak bound of Ω(1nlog⁡n)-MMS allocation in general.In this work, we improve this bound exponentially to Ω(1log⁡nlog⁡log⁡n)-MMS guarantee. In addition to this, we prove that when the valuation functions are fractionally subadditive, a 0.2192235-MMS allocation is guaranteed to exist. This also improves upon the previous bound of 1/5-MMS guarantee for the fractionally subadditive setting.

Towards the Erdős-Gallai cycle decomposition conjecture

In the 1960's, Erdős and Gallai conjectured that the edges of any n-vertex graph can be decomposed into O(n) cycles and edges. We improve upon the previous best bound of O(nlog⁡log⁡n) cycles and edges due to Conlon, Fox and Sudakov, by showing an n-vertex graph can always be decomposed into O(nlog⋆⁡n) cycles and edges, where log⋆⁡n is the iterated logarithm function.

STATIONARY REFLECTION AND THE FAILURE OF THE SCH

AbstractIn this paper we prove that from large cardinals it is consistent that there is a singular strong limit cardinal $\nu $ such that the singular cardinal hypothesis fails at $\nu $ and every collection of fewer than $\operatorname {\mathrm {cf}}(\nu )$ stationary subsets of $\nu ^{+}$ reflects simultaneously. For $\operatorname {\mathrm {cf}}(\nu )> \omega $ , this situation was not previously known to be consistent. Using different methods, we reduce the upper bound on the consistency strength of this situation for $\operatorname {\mathrm {cf}}(\nu ) = \omega $ to below a single partially supercompact cardinal. The previous upper bound of infinitely many supercompact cardinals was due to Sharon.

An improved upper bound on the domination number of a tree

A set D of vertices in a graph G is a dominating set of G if every vertex in V(G)∖D is adjacent to at least one vertex in D. The domination number of G, denoted by γ(G), is the minimum cardinality among all dominating sets of G. In this article, we first provide an alternative proof for the bound γ(T)≤(n(T)+|S(T)|)/3 obtained recently by Cabrera-Martínez et al. (2023). Moreover, we improve this previous upper bound. In particular, we show that if T is a tree of order n(T)≥2, then γ(T)≤n(T)+|S(T)|−|SL(T)|3−|Ls(T)|−|Ss(T)|3,where S(T), SL(T), Ls(T) and Ss(T) are the sets of support vertices, support link vertices, strong leaves and strong support vertices of T, respectively. Finally, we characterize the trees attaining each of these upper bounds above.

CNOT circuits need little help to implement arbitrary Hadamard-free Clifford transformations they generate

A Hadamard-free Clifford transformation is a circuit composed of quantum Phase (P), CZ, and CNOT gates. It is known that such a circuit can be written as a three-stage computation, -P-CZ-CNOT-, where each stage consists only of gates of the specified type. In this paper, we focus on the minimization of circuit depth by entangling gates, corresponding to the important time-to-solution metric and the reduction of noise due to decoherence. We consider two popular connectivity maps: Linear Nearest Neighbor (LNN) and all-to-all. First, we show that a Hadamard-free Clifford operation can be implemented over LNN in depth 5n, i.e., in the same depth as the -CNOT- stage alone. This allows us to implement arbitrary Clifford transformation over LNN in depth no more than 7n − 4, improving the best previous upper bound of 9n. Second, we report heuristic evidence that on average a random uniformly distributed Hadamard-free Clifford transformation over n > 6 qubits can be implemented with only a tiny additive overhead over all-to-all connected architecture compared to the best-known depth-optimized implementation of the -CNOT- stage alone. This suggests the reduction of the depth of Clifford circuits from 2n,+,O({log }^{2}(n)) to 1.5n,+,O({log }^{2}(n)) over unrestricted architectures.

Coloring triangle-free L-graphs with [formula omitted] colors

It is proved that triangle-free intersection graphs of n L-shapes in the plane have chromatic number O(loglogn). This improves the previous bound of O(logn) (McGuinness, 1996) and matches the known lower bound construction (Pawlik et al., 2013).

Improved Elekes-Szabó type estimates using proximity

We prove a new Elekes-Szabó type estimate on the size of the intersection of a Cartesian product A×B×C with an algebraic surface {f=0} over the reals. In particular, if A,B,C are sets of N real numbers and f is a trivariate polynomial, then either f has a special form that encodes additive group structure (for example, f(x,y,x)=x+y−z), or A×B×C∩{f=0} has cardinality O(N12/7). This is an improvement over the previous bound O(N11/6). We also prove an asymmetric version of our main result, which yields an Elekes-Ronyai type expanding polynomial estimate with exponent 3/2. This has applications to questions in combinatorial geometry related to the Erdős distinct distances problem.Like previous approaches to the problem, we rephrase the question as an L2 estimate, which can be analyzed by counting additive quadruples. The latter problem can be recast as an incidence problem involving points and curves in the plane. The new idea in our proof is that we use the order structure of the reals to restrict attention to a smaller collection of proximate additive quadruples.

Online Predictions for Online TSP on the Line

With a rapidly growing attention on the development of learning-augmented algorithms, we revisit the classical online traveling salesman problem (OLTSP) from the perspective of such learning approaches. A learning-augmented online algorithm, or simply an online algorithm with predictions, considers how to improve its competitive performance with theoretical guarantees by forecasting the information of future requests, e.g., their release time or locations in the OLTSP. In this study, we investigate the OLTSP on the real line, motivated by the parcel picking problem in a smart warehouse, which aims at scheduling a route for a server (saying, a Kiva robot), starting at the origin, serving all online requests and returning to the origin such that the completion time is minimized. Each online request is released over time on the real line and the server moves back and forth along the line with unit speed. We mainly focus on zealous algorithms, a special type of online algorithms for the OLTSP which never allow the sever to wait if there are still unserved requests, and exploit a specific forecasting strategy, called online predictions, which makes a sequence of predictions one by one in an online manner. In order to ensure better competitive performance, we particularly explore the worst-case scenarios that restrict the power of such predictions. Based on the discussion, we make an assumption in which online predictions are guaranteed to be useful, and devise a learning-augmented algorithm with [Formula: see text]-robustness and [Formula: see text]-consistency, [Formula: see text], comparing to the previous lower bound of [Formula: see text], where [Formula: see text] and [Formula: see text] which indicate prediction errors are sufficiently small constants. Moreover, we also conduct numerical experiments to demonstrate the practical effectiveness of the proposed algorithm.