arXiv subjects

Justin Dallant

Publications and source records attributed to Justin Dallant.

17 recordsLinked to original sources

arXiv2026

An optimal deterministic algorithm for finding a strict saddlepoint

Given an $n\times n$ matrix $A$, a saddlepoint of $A$ is an entry that is the maximum in its row and the minimum in its column. It is a strict saddlepoint if no other entry in its row or column has the same value. Finding a non-strict saddlepoint requires $\Theta(n^2)$ matrix queries in the worst case. In contrast, a strict saddlepoint can be found with only $O(n)$ queries. In 1991, Bienstock, Chung, Fredman, Sch\"affer, Shor, and Suri---and, independently, Byrne and Vaserstein---showed that one can find a strict saddlepoint (or certify that none exists) in $O(n\log n)$ time using $O(n)$ matrix queries. In 2024, Dallant, Haagensen, Jacob, Kozma, and Wild gave an $O(n\log^* n)$-time algorithm, followed shortly after by an optimal randomized algorithm running in $O(n)$ time with high probability. Whether $O(n)$ time could also be achieved deterministically was left open by these works. Here we resolve this question by presenting a simple deterministic algorithm that finds a strict saddlepoint, or reports that none exists, in optimal $O(n)$ time. Our algorithm combines elementary ingredients from previous approaches with linear-time selection from a collection of sorted lists.

cs.DS↗

arXiv2026

Quantum Space-Time Tradeoffs for TSP via Extremal Set Systems

Recent work of Ameli, Nederlof and Wang and of Dallant and Kozma introduced a framework for improving classical space--time tradeoffs for the Traveling Salesman Problem (TSP) and related permutation problems via extremal set systems with many maximal chains. In this note we observe that, for so called permutation problems whose outer aggregation is a minimum (such as TSP), the same framework admits a simple quantum analogue: instead of iterating over the covering family of set systems, we apply quantum minimum finding over the family. More precisely, let $P_S$ denote the optimal inverse normalized chain density among set systems of normalized size at most $S$. Then TSP admits a bounded-error quantum algorithm using $\widetilde O(S^n)$ QRAM space and \[ \widetilde O\!\left((S\sqrt{P_S})^n\right) \] time. The same argument applies to other minimization problems over permutations with a similar structure to TSP. Combining this observation with improved extremal set-system constructions of Andoni, Dallant, Kozma and Yu gives an explicit quantum space--time tradeoff curve, which beats the known quantum tradeoff by Caroppo et al. for all $1<S \leq 1.657$.

cs.DS↗

arXiv2026

Optimal chain density, entropy, and space-time tradeoffs for the TSP

We nearly settle a natural extremal question about set systems over $[n]$: the tradeoff between the {size} (number of sets) and the number of {full chains}. This question was initially raised by Johnson, Leader, and Russell [Combin.~Probab.~Comp., 2015] as a counterpart to Sperner-type results in combinatorics. Recently, a framework introduced by Ameli, Nederlof, and Wang, and independently by Dallant and Kozma [FOCS 2026] linked this question to the space- and time-complexity of Bellman-Held-Karp-style dynamic programming algorithms for permutation problems such as the traveling salesman (TSP). Precisely, they showed that a space-time product $\gamma^{n+o(n)}$ is feasible for the TSP, whenever a set system of (normalized) size $S$ and chain density $D$ exists, with $ \gamma = S^2/D$. In this paper we show an essentially {optimal} bound of $\gamma \approx 3.1819$ for this quantity, closing the gap between the previous best lower and upper bounds of $\gamma \geq 3.015$ and $ \gamma \leq 3.572$ respectively. This implies a TSP algorithm with space-time product $O(3.1819^n)$ for input size $n$, as well as a limit to further improvements in this broad framework. More generally, we can obtain close to optimal values $D$ for any feasible value $S$, effectively settling the question of the number of full chains at every size. The crucial step towards our results is casting the extremal combinatorics question as an {information~vs.~entropy} tradeoff involving two random variables. This reformulation {exactly} captures the optimal tradeoff for the combinatorial problem, leading to a framework in which primal-dual certificates can be derived, proving rigorous upper and lower bounds on $\gamma$. We also give a further application of our techniques, improving a bound of Duffus, Sands, and Winkler on the minimum size of fibres in the Boolean lattice.

cs.DS↗

arXiv2026

Hamilton paths and cycles in flip graphs of (almost-)perfect matchings

We consider the set of matchings of a graph and a local change operation, called a flip, between them. In the combinatorial setting, the base graphs are either complete graphs or complete bipartite graphs, and in the geometric setting, the graphs are embedded on point sets in the plane, with the requirement that edges must be drawn as straight lines and must not cross. For base graphs with an even number of vertices, we consider perfect matchings, i.e., all vertices are matched, and for base graphs with an odd number of vertices, we consider almost-perfect matchings, i.e., all but one vertex of the graph are matched. A 2-flip between two perfect matchings exchanges two edges, and a 1-flip between two almost-perfect matchings exchanges one edge. The corresponding flip graph has the set of perfect or almost-perfect matchings as vertices, with pairs of them connected by an edge if they differ in a 2-flip or 1-flip, respectively. In this work, we provide a comprehensive picture of Hamiltonicity properties of these flip graphs. We prove that the flip graphs in the combinatorial setting are Hamilton-connected, i.e., they admit a Hamilton path between any two vertices, or, if the flip graphs are bipartite, we prove that they are Hamilton-laceable, i.e., they admit a Hamilton path between any two vertices from different partition classes. In the geometric setting, we prove that any path in them misses exponentially many vertices, in particular, they have no Hamilton paths or cycles. For points in convex position and almost-perfect matchings under 1-flips, we complement this by constructing a cycle in the flip graph that visits almost all vertices.

math.CO↗

arXiv2026

Improved space-time tradeoff for TSP via extremal set systems

The traveling salesman problem (TSP) is a cornerstone of combinatorial optimization and has deeply influenced the development of algorithmic techniques in both exact and approximate settings. Yet, improving on the decades-old bounds for solving TSP exactly remains elusive: the dynamic program of Bellman, Held, and Karp from 1962 uses $2^{n+O(\log{n})}$ time and space, and the divide-and-conquer approach of Gurevich and Shelah from 1987 uses $4^{n + O(\log^2{n})}$ time and polynomial space. A straightforward combination of the two algorithms trades off $T^{n+o(n)}$ time and $S^{n+o(n)}$ space at various points of the curve $ST = 4$. An improvement to this tradeoff when $2 < T < 2\sqrt{2}$ was found by Koivisto and Parviainen (SODA 2010), yielding a minimum of $ST \approx 3.93$. Koivisto and Parviainen show their method to be optimal among a broad class of partial-order-based approaches, and to date, no improvement or alternative method has been found. In this paper we give a tradeoff that strictly improves all previous ones for all $2 < T < 4$, achieving a minimum of $ST < 3.572$. A key ingredient is the construction of sparse set systems (hypergraphs) that admit a large number of maximal chains. The existence of such objects is of independent interest in extremal combinatorics, likely to see further applications. Along the way we disprove a combinatorial conjecture of Johnson, Leader, and Russell from 2013, relating it with the optimality of the previous tradeoff schemes for TSP. Our techniques extend to a broad class of permutation problems over arbitrary semirings, yielding improved space-time tradeoffs in these settings as well.

cs.DS↗

arXiv2025

The Price of Connectivity Augmentation on Planar Graphs

Given two classes of graphs, $\mathcal{G}_1\subseteq \mathcal{G}_2$, and a $c$-connected graph $G\in \mathcal{G}_1$, we wish to augment $G$ with a smallest cardinality set of new edges $F$ to obtain a $k$-connected graph $G'=(V,E\cup F) \in \mathcal{G}_2$. In general, this is the $c\to k$ connectivity augmentation problem. Previous research considered variants where $\mathcal{G}_1=\mathcal{G}_2$ is the class of planar graphs, plane graphs, or planar straight-line graphs. In all three settings, we prove that the $c\to k$ augmentation problem is NP-complete when $2\leq c<k\leq 5$. However, the connectivity of the augmented graph $G'$ is at most $5$ if $\mathcal{G}_2$ is limited to planar graphs. We initiate the study of the $c\to k$ connectivity augmentation problem for arbitrary $k\in \mathbb{N}$, where $\mathcal{G}_1$ is the class of planar graphs, plane graphs, or planar straight-line graphs, and $\mathcal{G}_2$ is a beyond-planar class of graphs: $\ell$-planar, $\ell$-plane topological, or $\ell$-plane geometric graphs. We obtain tight bounds on the tradeoffs between the desired connectivity $k$ and the local crossing number $\ell$ of the augmented graph $G'$. We also show that our hardness results apply to this setting. The connectivity augmentation problem for triangulations is intimately related to edge flips; and the minimum augmentation problem to the flip distance between triangulations. We prove that it is NP-complete to find the minimum flip distance between a given triangulation and a 4-connected triangulation, settling an open problem posed in 2014, and present an EPTAS for this problem.

cs.CG↗

arXiv2025

Improved Bound on the Number of Pseudoline Arrangements via the Zone Theorem

Pseudoline arrangements are fundamental objects in discrete and computational geometry, and different works have tackled the problem of improving the known bounds on the number of simple arrangements of $n$ pseudolines over the past decades. The lower bound in particular has seen two successive improvements in recent years (Dumitrescu and Mandal in 2020 and Cort\'es K\"uhnast et al. in 2024). Here we focus on the upper bound, and show that for large enough $n$, there are at most $2^{0.6496n^2}$ different simple arrangements of $n$ pseudolines. This follows a series of incremental improvements starting with work by Knuth in 1992 showing a bound of roughly $2^{0.7925n^2},$ then a bound of $2^{0.6975n^2}$ by Felsner in 1997, and finally the previous best known bound of $2^{0.6572n^2}$ by Felsner and Valtr in 2011. The improved bound presented here follows from a simple argument to combine the approach of this latter work with the use of the Zone Theorem.

cs.CG↗

arXiv2024

Improved Lower Bound on the Number of Pseudoline Arrangements

We show that for large enough $n$, the number of non-isomorphic pseudoline arrangements of order $n$ is greater than $2^{c\cdot n^2}$ for some constant $c > 0.2604$, improving the previous best bound of $c>0.2083$ by Dumitrescu and Mandal (2020). Arrangements of pseudolines (and in particular arrangements of lines) are important objects appearing in many forms in discrete and computational geometry. They have strong ties for example with oriented matroids, sorting networks and point configurations. Let $B_n$ be the number of non-isomorphic pseudoline arrangements of order $n$ and let $b_n := \log_2(B_n)$. The problem of estimating $b_n$ dates back to Knuth, who conjectured that $b_n \leq 0.5n^2 + o(n^2)$ and derived the first bounds $n^2/6-O(n) \leq b_n \leq 0.7924(n^2+n)$. Both the upper and the lower bound have been improved a couple of times since. For the upper bound, it was first improved to $b_n < 0.6988n^2$ (Felsner, 1997), then $b_n < 0.6571 n^2$ by Felsner and Valtr (2011), for large enough $n$. In the same paper, Felsner and Valtr improved the constant in the lower bound to $c> 0.1887$, which was subsequently improved by Dumitrescu and Mandal to $c>0.2083$. Our new bound is based on a construction which starts with one of the constructions of Dumitrescu and Mandal and breaks it into constant sized pieces. We then use software to compute the contribution of each piece to the overall number of pseudoline arrangements. This method adds a lot of flexibility to the construction and thus offers many avenues for future tweaks and improvements which could lead to further tightening of the lower bound.

cs.CG↗

arXiv2024

An Improved Lower Bound on the Number of Pseudoline Arrangements

Arrangements of pseudolines are classic objects in discrete and computational geometry. They have been studied with increasing intensity since their introduction almost 100 years ago. The study of the number $B_n$ of non-isomorphic simple arrangements of $n$ pseudolines goes back to Goodman and Pollack, Knuth, and others. It is known that $B_n$ is in the order of $2^{\Theta(n^2)}$ and finding asymptotic bounds on $b_n = \frac{\log_2(B_n)}{n^2}$ remains a challenging task. In 2011, Felsner and Valtr showed that $0.1887 \leq b_n \le 0.6571$ for sufficiently large $n$. The upper bound remains untouched but in 2020 Dumitrescu and Mandal improved the lower bound constant to $0.2083$. Their approach utilizes the known values of $B_n$ for up to $n=12$. We tackle the lower bound by utilizing dynamic programming and the Lindstr\"om-Gessel-Viennot lemma. Our new bound is $b_n \geq 0.2721$ for sufficiently large $n$. The result is based on a delicate interplay of theoretical ideas and computer assistance.

math.CO↗

arXiv2024

An Optimal Randomized Algorithm for Finding the Saddlepoint

A \emph{saddlepoint} of an $n \times n$ matrix is an entry that is the maximum of its row and the minimum of its column. Saddlepoints give the \emph{value} of a two-player zero-sum game, corresponding to its pure-strategy Nash equilibria; efficiently finding a saddlepoint is thus a natural and fundamental algorithmic task. For finding a \emph{strict saddlepoint} (an entry that is the strict maximum of its row and the strict minimum of its column) we recently gave an $O({n\log^*{n}})$-time algorithm, improving the $O({n\log{n}})$ bounds from 1991 of Bienstock, Chung, Fredman, Sch\"affer, Shor, Suri and of Byrne and Vaserstein. In this paper we present an optimal $O({n})$-time algorithm for finding a strict saddlepoint based on random sampling. Our algorithm, like earlier approaches, accesses matrix entries only via unit-cost binary comparisons. For finding a (non-strict) saddlepoint, we extend an existing lower bound to randomized algorithms, showing that the trivial $O(n^2)$ runtime cannot be improved even with the use of randomness.

cs.CC↗

arXiv2023

A General Technique for Searching in Implicit Sets via Function Inversion

In recent years, the Fiat-Naor function inversion scheme has been used to disprove conjectures in fine-grained complexity theory and design state of the art data structures for a number of combinatorial problems. We pursue this line of research by considering its application to data structures for searching in implicit sets, defined as the image of a function. We show that, if $f$ is of the form $[N]\to [2^{w}]^d$ for some $w=polylog(N)$ and is computable in constant time, then, for any $0<\alpha <1$, we can obtain a data structure using $\~O(N^{1-\alpha/3})$ space such that, for a given $d$-dimensional axis-aligned box $B$, we can search for some $x\in [N]$ such that $f(x) \in B$ in time $\~O(N^{\alpha})$. (Here the $\~O(.)$ notation omits polylogarithmic factors.) Using similar techniques, we further obtain - data structures for range counting and reporting, predecessor, selection, ranking queries, and combinations thereof, on the set $f([N])$, - data structures for preimage size and preimage selection queries for a given value of $f$, and - data structures for selection and ranking queries on geometric quantities computed from tuples of points in $d$-space. These results unify and generalize previously known results on 3SUM-indexing and string searching, and are widely applicable as a black box to a variety of problems. In particular, we give a data structure for a generalized version of gapped string indexing, and show how to preprocess a set of points on an integer grid in order to efficiently compute (in sublinear time), for points contained in a given axis-aligned box, their Theil-Sen estimator, the $k$th largest area triangle, or the induced hyperplane that is the $k$th furthest from the origin.

cs.DS↗

arXiv2023

Finding the saddlepoint faster than sorting

A saddlepoint of an $n \times n$ matrix $A$ is an entry of $A$ that is a maximum in its row and a minimum in its column. Knuth (1968) gave several different algorithms for finding a saddlepoint. The worst-case running time of these algorithms is $\Theta(n^2)$, and Llewellyn, Tovey, and Trick (1988) showed that this cannot be improved, as in the worst case all entries of A may need to be queried. A strict saddlepoint of $A$ is an entry that is the strict maximum in its row and the strict minimum in its column. The strict saddlepoint (if it exists) is unique, and Bienstock, Chung, Fredman, Sch\"affer, Shor, and Suri (1991) showed that it can be found in time $O(n \log{n})$, where a dominant runtime contribution is sorting the diagonal of the matrix. This upper bound has not been improved since 1991. In this paper we show that the strict saddlepoint can be found in $O(n \log^{*}{n})$ time, where $\log^{*}$ denotes the very slowly growing iterated logarithm function, coming close to the lower bound of $\Omega(n)$. In fact, we can also compute, within the same runtime, the value of a non-strict saddlepoint, assuming one exists. Our algorithm is based on a simple recursive approach, a feasibility test inspired by searching in sorted matrices, and a relaxed notion of saddlepoint.

cs.DS↗

arXiv2022

Conditional Lower Bounds for Dynamic Geometric Measure Problems

We give new polynomial lower bounds for a number of dynamic measure problems in computational geometry. These lower bounds hold in the Word-RAM model, conditioned on the hardness of either 3SUM, APSP, or the Online Matrix-Vector Multiplication problem [Henzinger et al., STOC 2015]. In particular we get lower bounds in the incremental and fully-dynamic settings for counting maximal or extremal points in R^3, different variants of Klee's Measure Problem, problems related to finding the largest empty disk in a set of points, and querying the size of the i'th convex layer in a planar set of points. We also answer a question of Chan et al. [SODA 2022] by giving a conditional lower bound for dynamic approximate square set cover. While many conditional lower bounds for dynamic data structures have been proven since the seminal work of Patrascu [STOC 2010], few of them relate to computational geometry problems. This is the first paper focusing on this topic. Most problems we consider can be solved in O(n log n) time in the static case and their dynamic versions have only been approached from the perspective of improving known upper bounds. One exception to this is Klee's measure problem in R^2, for which Chan [CGTA 2010] gave an unconditional $Ω(\sqrt{n})$ lower bound on the worst-case update time. By a similar approach, we show that such a lower bound also holds for an important special case of Klee's measure problem in R^3 known as the Hypervolume Indicator problem, even for amortized runtime in the incremental setting.

cs.CG↗

arXiv2022

How Fast Can We Play Tetris Greedily With Rectangular Pieces?

Consider a variant of Tetris played on a board of width $w$ and infinite height, where the pieces are axis-aligned rectangles of arbitrary integer dimensions, the pieces can only be moved before letting them drop, and a row does not disappear once it is full. Suppose we want to follow a greedy strategy: let each rectangle fall where it will end up the lowest given the current state of the board. To do so, we want a data structure which can always suggest a greedy move. In other words, we want a data structure which maintains a set of $O(n)$ rectangles, supports queries which return where to drop the rectangle, and updates which insert a rectangle dropped at a certain position and return the height of the highest point in the updated set of rectangles. We show via a reduction to the Multiphase problem [Pătraşcu, 2010] that on a board of width $w=Θ(n)$, if the OMv conjecture [Henzinger et al., 2015] is true, then both operations cannot be supported in time $O(n^{1/2-ε})$ simultaneously. The reduction also implies polynomial bounds from the 3-SUM conjecture and the APSP conjecture. On the other hand, we show that there is a data structure supporting both operations in $O(n^{1/2}\log^{3/2}n)$ time on boards of width $n^{O(1)}$, matching the lower bound up to a $n^{o(1)}$ factor.

cs.CG↗

arXiv2021

Approximability of (Simultaneous) Class Cover for Boxes

Bereg et al. (2012) introduced the Boxes Class Cover problem, which has its roots in classification and clustering applications: Given a set of n points in the plane, each colored red or blue, find the smallest cardinality set of axis-aligned boxes whose union covers the red points without covering any blue point. In this paper we give an alternative proof of APX-hardness for this problem, which also yields an explicit lower bound on its approximability. Our proof also directly applies when restricted to sets of points in general position and to the case where so-called half-strips are considered instead of boxes, which is a new result. We also introduce a symmetric variant of this problem, which we call Simultaneous Boxes Class Cover and can be stated as follows: Given a set S of n points in the plane, each colored red or blue, find the smallest cardinality set of axis-aligned boxes which together cover S such that all boxes cover only points of the same color and no box covering a red point intersects a box covering a blue point. We show that this problem is also APX-hard and give a polynomial-time constant-factor approximation algorithm.

cs.CG↗

arXiv2021

An Instance-optimal Algorithm for Bichromatic Rectangular Visibility

Afshani, Barbay and Chan (2017) introduced the notion of instance-optimal algorithm in the order-oblivious setting. An algorithm A is instance-optimal in the order-oblivious setting for a certain class of algorithms A* if the following hold: - A takes as input a sequence of objects from some domain; - for any instance $\sigma$ and any algorithm A' in A*, the runtime of A on $\sigma$ is at most a constant factor removed from the runtime of A' on the worst possible permutation of $\sigma$. If we identify permutations of a sequence as representing the same instance, this essentially states that A is optimal on every possible input (and not only in the worst case). We design instance-optimal algorithms for the problem of reporting, given a bichromatic set of points in the plane S, all pairs consisting of points of different color which span an empty axis-aligned rectangle (or reporting all points which appear in such a pair). This problem has applications for training-set reduction in nearest-neighbour classifiers. It is also related to the problem consisting of finding the decision boundaries of a euclidean nearest-neighbour classifier, for which Bremner et al. (2005) gave an optimal output-sensitive algorithm. By showing the existence of an instance-optimal algorithm in the order-oblivious setting for this problem we push the methods of Afshani et al. closer to their limits by adapting and extending them to a setting which exhibits highly non-local features. Previous problems for which instance-optimal algorithms were proven to exist were based solely on local relationships between points in a set.

cs.CG↗

arXiv2021

Efficiently stabbing convex polygons and variants of the Hadwiger-Debrunner $(p, q)$-theorem

Hadwiger and Debrunner showed that for families of convex sets in $\mathbb{R}^d$ with the property that among any $p$ of them some $q$ have a common point, the whole family can be stabbed with $p-q+1$ points if $p \geq q \geq d+1$ and $(d-1)p < d(q-1)$. This generalizes a classical result by Helly. We show how such a stabbing set can be computed for a family of convex polygons in the plane with a total of $n$ vertices in $O((p-q+1)n^{4/3}\log^{8} n(\log\log n)^{1/3} + np^2)$ expected time. For polyhedra in $\mathbb{R}^3$, we get an algorithm running in $O((p-q+1)n^{5/2}\log^{10} n(\log\log n)^{1/6} + np^3)$ expected time. We also investigate other conditions on convex polygons for which our algorithm can find a fixed number of points stabbing them. Finally, we show that analogous results of the Hadwiger and Debrunner $(p,q)$-theorem hold in other settings, such as convex sets in $\mathbb{R}^d\times\mathbb{Z}^k$ or abstract convex geometries.

cs.CG↗