arXiv subjects

Saurabh Ray

Publications and source records attributed to Saurabh Ray.

15 recordsLinked to original sources

arXiv2026

Random Garbage Separates XOR from Forward-Only Queries

We give exponential quantum query separations between the standard XOR interface and two forward-only interfaces that supply neither an adjoint nor an inverse oracle. Let $X=\F_2^n$, $N=|X|$, and $f_{h,r}(x)=(h(x),x,r_x)$, where $h:X\to X$ is promised to be either a permutation or a Simon two-to-one function, and $r$ is a fixed table of $n$-bit tags, unrestricted by the promise and reused on every query. The resulting problem is solvable with at most $n+2$ standard XOR queries, but has forward-erasing query complexity $\Theta(\sqrt N)$. This answers affirmatively open question 11 in [Scott Aaronson. Open problems related to quantum query complexity. ACM Transactions on Quantum Computing, 2(4):14:1-14:9, 2021] . We also embed these instances into permutations. The detailed construction retains the copy of $x$ in each prescribed output, but for these promises that copy can be replaced by one bit that distinguishes the two inputs in every Simon pair. This gives a permutation domain of size $L=4N^2$ and a permutation problem with the same standard-query upper bound and forward-only in-place query complexity $\Theta(\sqrt N)=\Theta(L^{1/4})$. Both lower bounds remain valid with a clean coherent bypass. The common lower bound uses an analysis-only recording replacement. In the replacement computation, tracing out the fixed random tag table after $T$ calls gives a sum of positive-semidefinite operator contributions, each depending on $h$ at no more than $T$ addresses. On such a set, the restrictions induced by random permutations and random Simon functions differ only if the set contains a hidden Simon pair, an event of probability $O(T^2/N)$.

cs.CC↗

arXiv2026

Tight UGC Thresholds for Geometric Stabbing Problems

Many geometric stabbing problems admit natural covering LPs in which each constraint is a union of consecutive traces on ordered candidate sets. We prove a transfer theorem showing that every fixed finite, bounded-arity integrality-gap instance of this form yields a matching hardness ratio under the Unique Games Conjecture. Using the strict-CSP framework of Kumar, Manokaran, Tulsiani, and Vishnoi [SODA 2011], we construct the required connected local distributions by randomized rounding and a full-support perturbation. Given a fractional vector $x$ on a block, the rounding selects candidate $i$ with marginal probability $x_i$ and hits each consecutive trace $T$ with probability $\min\{1,x(T)\}$. We obtain three tight UGC thresholds. First, for every fixed $d\ge 2$, stabbing arbitrary-size axis-parallel $d$-cubes with coordinate hyperplanes has threshold $d$. For $d=2$, the hardness holds for arbitrary-size squares and establishes threshold $2$ for rectangle and square stabbing, matching the $2$-approximation of Gaur, Ibaraki, and Krishnamurti [ESA 2000]. Second, stabbing horizontal segments with horizontal and vertical lines has threshold $e/(e-1)$, matching the $e/(e-1)$-approximation of Kovaleva and Spieksma [ESA 2004]. Third, separated $d$-interval transversal has threshold $d$ for every fixed $d\ge 2$, closing under UGC the gap left by the $d$-approximation of Ben-David, Grant, Ma, and Sharpe [CCCG 2012].

cs.CG↗

arXiv2026

Counterexamples to Wegner's Conjecture for Rectangles

Wegner conjectured in 1965 that every finite family $\mathcal R$ of axis-parallel rectangles satisfies $\tau(\mathcal R)\le 2\nu(\mathcal R)-1$, where $\tau(\mathcal R)$ is the minimum number of piercing points and $\nu(\mathcal R)$ is the maximum size of a pairwise-disjoint subfamily. We disprove the conjecture by an explicit triangle-free family of $64$ rectangles with $\nu=16$ and $\tau\ge 32$. More generally, for every $\varepsilon>0$, we construct triangle-free rectangle families for which the standard clique-LP relaxation for maximum independent set of rectangles has integrality gap at least $5/2-\varepsilon$. The same families satisfy $\tau(\mathcal R)\ge (5/2-\varepsilon)\nu(\mathcal R)$. We also prove that, on triangle-free rectangle families, this LP has gap at most $3$. Our approach gives an example with axis-parallel segments instead of rectangles with integrality gap tending to $2$. We also give a relatively small $4092$-rectangle triangle-free family with chromatic number $6$ improving the construction of Asplund and Gr\"unbaum (On a coloring problem, Mathematica Scandinavica, 1960) that required more than $10^8$ rectangles.

cs.CG↗

arXiv2026

W-state graphs: Structure and Algorithms

We study the class of edge-coloured graphs arising from the graph-theoretic representation of quantum photonic experiments that generate multipartite W-states. Abstracting away physical amplitudes and phases, we introduce W-state graphs: matching-covered graphs equipped with a half-edge 2-colouring such that every perfect matching contains exactly one bichromatic edge and every vertex is incident with a red half-edge. Our main contribution is a complete structural characterization of W-state graphs. We show that a graph is a W-state graph if and only if each of its 3-connected components is a W-cone, a simple and rigid building block defined by a universal vertex and a factor-critical base. This characterization implies that no W-state graph is simple and yields a recognition algorithm running as fast as verifying whether a graph is matching-covered. We also show that the natural generalization to Dicke states encounters a complexity barrier: verifying one of the two Dicke state conditions is itself coNP-complete, resolving an open problem of Vardi and Zhang [IJCAI 2023]. Our results place W-state graphs firmly within classical matching theory and precisely delineate the combinatorial structures capable of realizing idealized W-states in the experiment-graph framework.

quant-ph↗

arXiv2024

A fast algorithm for computing a planar support for non-piercing rectangles

For a hypergraph $\mathcal{H}=(X,\mathcal{E})$ a \emph{support} is a graph $G$ on $X$ such that for each $E\in\mathcal{E}$, the induced subgraph of $G$ on the elements in $E$ is connected. If $G$ is planar, we call it a planar support. A set of axis parallel rectangles $\mathcal{R}$ forms a non-piercing family if for any $R_1, R_2 \in \mathcal{R}$, $R_1 \setminus R_2$ is connected. Given a set $P$ of $n$ points in $\mathbb{R}^2$ and a set $\mathcal{R}$ of $m$ \emph{non-piercing} axis-aligned rectangles, we give an algorithm for computing a planar support for the hypergraph $(P,\mathcal{R})$ in $O(n\log^2 n + (n+m)\log m)$ time, where each $R\in\mathcal{R}$ defines a hyperedge consisting of all points of $P$ contained in~$R$. We use this result to show that if for a family of axis-parallel rectangles, any point in the plane is contained in at most $k$ pairwise \emph{crossing} rectangles (a pair of intersecting rectangles such that neither contains a corner of the other is called a crossing pair of rectangles), then we can obtain a support as the union of $k$ planar graphs.

cs.CG↗

arXiv2024

Sweeping Arrangements of Non-Piercing Curves in Plane

Let $\Gamma$ be an arrangement of Jordan curves in the plane, i.e., simple closed curves in the plane. For any curve $\gamma \in \Gamma$, we denote the bounded region enclosed by $\gamma$ as $\tilde{\gamma}$. We say that $\Gamma$ is non-piercing if for any two curves $\alpha , \beta \in \Gamma$, $\tilde{\alpha} \,\setminus\, \tilde{\beta}$ is connected. A non-piercing arrangement of curves generalizes a set of $2$-intersecting curves in which each pair of curves intersect in at most two points. Snoeyink and Hershberger (``Sweeping Arrangements of Curves'', SoCG '89) proved that if we are given an arrangement $\Gamma$ of $2$-intersecting curves and a {\em sweep} curve $\gamma\in{\Gamma}$, then the arrangement can be \emph{swept} by $\gamma$ while always maintaining the $2$-intersecting property of the curves in $\Gamma$. We generalize the result of Snoeyink and Hershberger to the setting of non-piercing arrangements. Given an arrangement $\Gamma$ of non-piercing curves, a sweep curve $\gamma\in \Gamma$, and a point $P$ in $\tilde{\gamma}$, we show that we can continuously shrink $\gamma$ to $P$ so that throughout the process, the arrangement remains non-piercing (except at a finite set of points in time where $\gamma$ crosses other curves), and $P$ lies in $\tilde{\gamma}$. We show that our arguments can be modified if $P$ lies outside $\tilde{\gamma}$, and we want to sweep $\gamma$ \emph{outwards} so that $P$ lies outside $\tilde{\gamma}$, and the arrangement remains non-piercing. As a second contribution, we give an alternate proof of the result of Snoeyink and Hershberger, and give several applications of our results to combinatorial and algorithmic questions including to the \emph{multi-hitting set} problem involving points and non-piercing regions.

cs.CG↗

arXiv2021

Threshold Rounding for the Standard LP Relaxation of some Geometric Stabbing Problems

In the rectangle stabbing problem, we are given a set $\cR$ of axis-aligned rectangles in $\RR^2$, and the objective is to find a minimum-cardinality set of horizontal and/or vertical lines such that each rectangle is intersected by one of these lines. The standard LP relaxation for this problem is known to have an integrality gap of 2, while a better intergality gap of 1.58.. is known for the special case when $\cR$ is a set of horizontal segments. In this paper, we consider two more special cases: when $\cR$ is a set of horizontal and vertical segments, and when $\cR$ is a set of unit squares. We show that the integrality gap of the standard LP relaxation in both cases is stricly less than $2$. Our rounding technique is based on a generalization of the {\it threshold rounding} idea used by Kovaleva and Spieksma (SIAM J. Disc. Math 2006), which may prove useful for rounding the LP relaxations of other geometric covering problems.

cs.CG↗

arXiv2015

Tighter Estimates for epsilon-nets for Disks

The geometric hitting set problem is one of the basic geometric combinatorial optimization problems: given a set $P$ of points, and a set $\mathcal{D}$ of geometric objects in the plane, the goal is to compute a small-sized subset of $P$ that hits all objects in $\mathcal{D}$. In 1994, Bronniman and Goodrich made an important connection of this problem to the size of fundamental combinatorial structures called $\epsilon$-nets, showing that small-sized $\epsilon$-nets imply approximation algorithms with correspondingly small approximation ratios. Very recently, Agarwal and Pan showed that their scheme can be implemented in near-linear time for disks in the plane. Altogether this gives $O(1)$-factor approximation algorithms in $\tilde{O}(n)$ time for hitting sets for disks in the plane. This constant factor depends on the sizes of $\epsilon$-nets for disks; unfortunately, the current state-of-the-art bounds are large -- at least $24/\epsilon$ and most likely larger than $40/\epsilon$. Thus the approximation factor of the Agarwal and Pan algorithm ends up being more than $40$. The best lower-bound is $2/\epsilon$, which follows from the Pach-Woeginger construction for halfspaces in two dimensions. Thus there is a large gap between the best-known upper and lower bounds. Besides being of independent interest, finding precise bounds is important since this immediately implies an improved linear-time algorithm for the hitting-set problem. The main goal of this paper is to improve the upper-bound to $13.4/\epsilon$ for disks in the plane. The proof is constructive, giving a simple algorithm that uses only Delaunay triangulations. We have implemented the algorithm, which is available as a public open-source module. Experimental results show that the sizes of $\epsilon$-nets for a variety of data-sets is lower, around $9/\epsilon$.

cs.CG↗

arXiv2014

Counting Triangulations and other Crossing-Free Structures Approximately

We consider the problem of counting straight-edge triangulations of a given set $P$ of $n$ points in the plane. Until very recently it was not known whether the exact number of triangulations of $P$ can be computed asymptotically faster than by enumerating all triangulations. We now know that the number of triangulations of $P$ can be computed in $O^{*}(2^{n})$ time, which is less than the lower bound of $\Omega(2.43^{n})$ on the number of triangulations of any point set. In this paper we address the question of whether one can approximately count triangulations in sub-exponential time. We present an algorithm with sub-exponential running time and sub-exponential approximation ratio, that is, denoting by $\Lambda$ the output of our algorithm, and by $c^{n}$ the exact number of triangulations of $P$, for some positive constant $c$, we prove that $c^{n}\leq\Lambda\leq c^{n}\cdot 2^{o(n)}$. This is the first algorithm that in sub-exponential time computes a $(1+o(1))$-approximation of the base of the number of triangulations, more precisely, $c\leq\Lambda^{\frac{1}{n}}\leq(1 + o(1))c$. Our algorithm can be adapted to approximately count other crossing-free structures on $P$, keeping the quality of approximation and running time intact. In this paper we show how to do this for matchings and spanning trees.

cs.CG↗

arXiv2014

QPTAS for Geometric Set-Cover Problems via Optimal Separators

Weighted geometric set-cover problems arise naturally in several geometric and non-geometric settings (e.g. the breakthrough of Bansal-Pruhs (FOCS 2010) reduces a wide class of machine scheduling problems to weighted geometric set-cover). More than two decades of research has succeeded in settling the $(1+\epsilon)$-approximability status for most geometric set-cover problems, except for four basic scenarios which are still lacking. One is that of weighted disks in the plane for which, after a series of papers, Varadarajan (STOC 2010) presented a clever \emph{quasi-sampling} technique, which together with improvements by Chan \etal~(SODA 2012), yielded a $O(1)$-approximation algorithm. Even for the unweighted case, a PTAS for a fundamental class of objects called pseudodisks (which includes disks, unit-height rectangles, translates of convex sets etc.) is currently unknown. Another fundamental case is weighted halfspaces in $\Re^3$, for which a PTAS is currently lacking. In this paper, we present a QPTAS for all of these remaining problems. Our results are based on the separator framework of Adamaszek-Wiese (FOCS 2013, SODA 2014), who recently obtained a QPTAS for weighted independent set of polygonal regions. This rules out the possibility that these problems are APX-hard, assuming $\textbf{NP} \nsubseteq \textbf{DTIME}(2^{polylog(n)})$. Together with the recent work of Chan-Grant (CGTA 2014), this settles the APX-hardness status for all natural geometric set-cover problems.

cs.CG↗

arXiv2013

Counting Triangulations and other Crossing-free Structures via Onion Layers

Let $P$ be a set of $n$ points in the plane. A crossing-free structure on $P$ is a plane graph with vertex set $P$. Examples of crossing-free structures include triangulations of $P$, spanning cycles of $P$, also known as polygonalizations of $P$, among others. In this paper we develop a general technique for computing the number of crossing-free structures of an input set $P$. We apply the technique to obtain algorithms for computing the number of triangulations, matchings, and spanning cycles of $P$. The running time of our algorithms is upper bounded by $n^{O(k)}$, where $k$ is the number of onion layers of $P$. In particular, for $k = O(1)$ our algorithms run in polynomial time. In addition, we show that our algorithm for counting triangulations is never slower than $O^{*}(3.1414^{n})$, even when $k = \Theta(n)$. Given that there are several well-studied configurations of points with at least $\Omega(3.464^{n})$ triangulations, and some even with $\Omega(8^{n})$ triangulations, our algorithm asymptotically outperforms any enumeration algorithm for such instances. In fact, it is widely believed that any set of $n$ points must have at least $\Omega(3.464^{n})$ triangulations. If this is true, then our algorithm is strictly sub-linear in the number of triangulations counted. We also show that our techniques are general enough to solve the "Restricted-Triangulation-Counting-Problem", which we prove to be $W[2]$-hard in the parameter $k$. This implies a "no free lunch" result: In order to be fixed-parameter tractable, our general algorithm must rely on additional properties that are specific to the considered class of structures.

cs.CG↗

arXiv2013

A Simple Sweep Line Algorithm for Counting Triangulations and Pseudo-triangulations

Let $P\subset\mathbb{R}^{2}$ be a set of $n$ points. In this paper we show two new algorithms, one to compute the number of triangulations of $P$, and one to compute the number of pseudo-triangulations of $P$. We show that our algorithms run in time $O^{*}(t(P))$ and $O^{*}(pt(P))$ respectively, where $t(P)$ and $pt(P)$ are the largest number of triangulation paths (T-paths) and pseudo-triangulations paths (PT-paths), respectively, that the algorithms encounter during their execution. Moreover, we show that $t(P) = O^{*}(9^{n})$, which is the first non-trivial bound on $t(P)$ to be known. While there already are algorithms that count triangulations in $O^{*}\left(2^n\right)$, and $O^{*}\left(3.1414^{n}\right)$, there are sets of points where the number of T-paths is $O(2^{n})$. In such cases the algorithm herein presented could potentially be faster. Furthermore, it is not clear whether the already-known algorithms can be modified to count pseudo-triangulations so that their running times remain $O^{*}(c^n)$, for some small constant $c\in\mathbb{R}$. Therefore, for counting pseudo-triangulations (and possibly other similar structures) our approach seems better.

cs.CG↗

arXiv2013

On Totally Positive Matrices and Geometric Incidences

A matrix is called totally positive if every minor of it is positive. Such matrices are well studied and have numerous applications in Mathematics and Computer Science. We study how many times the value of a minor can repeat in a totally positive matrix and show interesting connections with incidence problems in combinatorial geometry. We prove that the maximum possible number of repeated $d \times d$-minors in a $d \times n$ totally-positive matrix is $O(n^{d-\frac{d}{d+1}})$. For the case $d=2$ we also show that our bound is optimal. We consider some special families of totally postive matrices to show non-trivial lower bounds on the number of repeated minors. In doing so, we arrive at a new interesting problem: How many unit-area and axis-parallel rectangles can be spanned by two points in a set of $n$ points in the plane? This problem seems to be interesting in its own right especially since it seem to have a flavor of additive combinatorics and relate to interesting incidence problems where considering only the topology of the curves involved is not enough. We prove an upper bound of $O(n^{\frac{4}{3}})$ and provide a lower bound of $n^{1+\frac{1}{O(\log\log n)}}$.

math.CO↗

arXiv2013

Point Line Cover: The Easy Kernel is Essentially Tight

The input to the NP-hard Point Line Cover problem (PLC) consists of a set $P$ of $n$ points on the plane and a positive integer $k$, and the question is whether there exists a set of at most $k$ lines which pass through all points in $P$. A simple polynomial-time reduction reduces any input to one with at most $k^2$ points. We show that this is essentially tight under standard assumptions. More precisely, unless the polynomial hierarchy collapses to its third level, there is no polynomial-time algorithm that reduces every instance $(P,k)$ of PLC to an equivalent instance with $O(k^{2-\epsilon})$ points, for any $\epsilon>0$. This answers, in the negative, an open problem posed by Lokshtanov (PhD Thesis, 2009). Our proof uses the machinery for deriving lower bounds on the size of kernels developed by Dell and van Melkebeek (STOC 2010). It has two main ingredients: We first show, by reduction from Vertex Cover, that PLC---conditionally---has no kernel of total size $O(k^{2-\epsilon})$ bits. This does not directly imply the claimed lower bound on the number of points, since the best known polynomial-time encoding of a PLC instance with $n$ points requires $\omega(n^{2})$ bits. To get around this we build on work of Goodman et al. (STOC 1989) and devise an oracle communication protocol of cost $O(n\log n)$ for PLC; its main building block is a bound of $O(n^{O(n)})$ for the order types of $n$ points that are not necessarily in general position, and an explicit algorithm that enumerates all possible order types of n points. This protocol and the lower bound on total size together yield the stated lower bound on the number of points. While a number of essentially tight polynomial lower bounds on total sizes of kernels are known, our result is---to the best of our knowledge---the first to show a nontrivial lower bound for structural/secondary parameters.

cs.DS↗

arXiv2009

On Profit-Maximizing Pricing for the Highway and Tollbooth Problems

In the \emph{tollbooth problem}, we are given a tree $\bT=(V,E)$ with $n$ edges, and a set of $m$ customers, each of whom is interested in purchasing a path on the tree. Each customer has a fixed budget, and the objective is to price the edges of $\bT$ such that the total revenue made by selling the paths to the customers that can afford them is maximized. An important special case of this problem, known as the \emph{highway problem}, is when $\bT$ is restricted to be a line. For the tollbooth problem, we present a randomized $O(\log n)$-approximation, improving on the current best $O(\log m)$-approximation. We also study a special case of the tollbooth problem, when all the paths that customers are interested in purchasing go towards a fixed root of $\bT$. In this case, we present an algorithm that returns a $(1-\epsilon)$-approximation, for any $\epsilon > 0$, and runs in quasi-polynomial time. On the other hand, we rule out the existence of an FPTAS by showing that even for the line case, the problem is strongly NP-hard. Finally, we show that in the \emph{coupon model}, when we allow some items to be priced below zero to improve the overall profit, the problem becomes even APX-hard.

cs.DS↗