arXiv subjects

Debajyoti Kar

Publications and source records attributed to Debajyoti Kar.

9 recordsLinked to original sources

arXiv2026

On Linear-Size Guillotine-Separable Subsets of Fat Convex Objects, Disks, and Squares

Let $\mathcal{K}$ be a family of pairwise disjoint objects in the plane. We say that a subset $\mathcal{K}^*\subseteq \mathcal{K}$ is \emph{separable} if it admits a sequence of guillotine cuts that separate all objects in $\mathcal{K}^*$ from each other while not cutting any of them. Urrutia (1996) asked whether any family of $n$ convex objects has a separable subset of size $\Omega(n)$. Pach and Tardos (2000) answered this question negatively for line segments, but established positive results for fat objects of similar size. More recently, it was shown that sets of arbitrarily-sized axis-aligned squares also admit a separable subset of linear size. However, the question whether any set of arbitrarily-sized fat convex objects has a separable subset of linear size has remained open, even for disks. A major obstacle is that the existing technique for arbitrarily-sized squares uses only axis-aligned cuts, while even for disks, axis-aligned cuts alone are insufficient to obtain a separable subset of linear size. We resolve this longstanding open problem by proving that every family of pairwise disjoint fat convex objects has a separable subset of linear size. Our result extends to higher dimensions: any family of pairwise disjoint arbitrarily-sized fat convex objects in $\mathbb{R}^d$, where $d$ is a fixed constant, has a subset of linear size that is recursively separable by a sequence of hyperplane cuts. Our framework also yields improved guarantees for important special cases. For axis-aligned squares with axis-aligned guillotine cuts, we leverage additional structural properties of squares to show that at least $13.46\%$ of the squares are separable, improving the previous best bound of $9/256 \approx 3.51\%$ due to Chalermsook, Kugelmann, Orgo, Uniyal, and Zarsav (2025). For disks, by exploiting Oler's packing inequality, we prove that at least $n/93$ disks can always be separated.

cs.CG↗

arXiv2026

Approximation Schemes and Structural Barriers for the Two-Dimensional Knapsack Problem with Rotations

We study the two-dimensional (geometric) knapsack problem with rotations (2DKR), in which we are given a square knapsack and a set of rectangles with associated profits. The objective is to find a maximum profit subset of rectangles that can be packed without overlap in an axis-aligned manner, possibly by rotating some rectangles by $90^{\circ}$. The best-known polynomial time algorithm for the problem has an approximation ratio of $3/2+\epsilon$ for any constant $\epsilon>0$, with an improvement to $4/3+\epsilon$ in the cardinality case, due to G{\'a}lvez et al. (FOCS 2017, TALG 2021). Obtaining a PTAS for the problem, even in the cardinality case, has remained a major open question in the setting of multidimensional packing problems, as mentioned in the survey by Christensen et al. (Computer Science Review, 2017). In this paper, we present a PTAS for the cardinality case of 2DKR. In contrast to the setting without rotations, we show that there are $(1+\epsilon)$-approximate solutions in which all items are packed greedily inside a constant number of rectangular {\em containers}. Our result is based on a new resource contraction lemma, which might be of independent interest. In contrast, for the general weighted case, we prove that this simple type of packing is not sufficient to obtain a better approximation ratio than $1.5$. However, we break this structural barrier and design a $(1.497+\epsilon)$-approximation algorithm for 2DKR in the weighted case. Our arguments also improve the best-known approximation ratio for the (weighted) case {\em without rotations} to $13/7+\epsilon \approx 1.857+\epsilon$. Finally, we establish a lower bound of $n^{\Omega(1/\epsilon)}$ on the running time of any $(1+\epsilon)$-approximation algorithm for our problem with or without rotations -- even in the cardinality setting, assuming the $k$-\textsc{Sum} Conjecture.

cs.DS↗

arXiv2025

Improved Approximation Algorithms for Three-Dimensional Knapsack

We study the three-dimensional Knapsack (3DK) problem, in which we are given a set of axis-aligned cuboids with associated profits and an axis-aligned cube knapsack. The objective is to find a non-overlapping axis-aligned packing (by translation) of the maximum profit subset of cuboids into the cube. The previous best approximation algorithm is due to Diedrich, Harren, Jansen, Th\"{o}le, and Thomas (2008), who gave a $(7+\varepsilon)$-approximation algorithm for 3DK and a $(5+\varepsilon)$-approximation algorithm for the variant when the items can be rotated by 90 degrees around any axis, for any constant $\varepsilon>0$. Chleb\'{\i}k and Chleb\'{\i}kov\'{a} (2009) showed that the problem does not admit an asymptotic polynomial-time approximation scheme. We provide an improved polynomial-time $(139/29+\varepsilon) \approx 4.794$-approximation algorithm for 3DK and $(30/7+\varepsilon) \approx 4.286$-approximation when rotations by 90 degrees are allowed. We also provide improved approximation algorithms for several variants such as the cardinality case (when all items have the same profit) and uniform profit-density case (when the profit of an item is equal to its volume). Our key technical contribution is container packing -- a structured packing in 3D such that all items are assigned into a constant number of containers, and each container is packed using a specific strategy based on its type. We first show the existence of highly profitable container packings. Thereafter, we show that one can find near-optimal container packing efficiently using a variant of the Generalized Assignment Problem (GAP).

cs.DS↗

arXiv2025

Improved Approximation Algorithms for Three-Dimensional Bin Packing

We study two fundamental three-dimensional (3D) geometric packing problems: 3D (Geometric) Bin Packing (3D-BP), and 3D Minimum Volume Bounding Box (3D-MVBB), where given a set of 3D (rectangular) cuboids, the goal is to find an axis-aligned nonoverlapping packing of all cuboids. In 3D-BP, we need to pack the given cuboids into the minimum number of unit cube bins. In 3D-MVBB, the goal is to pack them into a cuboid box of minimum volume. It is NP-hard to even decide whether a set of rectangles can be packed into a unit square bin -- giving an (absolute) approximation hardness of $2$ for 3D-BP. The previous best (absolute) approximation for both the problems follows from a result of Buchwald and Scheithauer (Int.~Trans.~Oper.~Res., 2016), yielding approximation ratios of $11$, and $5+\varepsilon$, respectively, for 3D-BP and 3D-MVBB. We provide improved approximation ratios of $6$, and $3+\varepsilon$, respectively, for the two problems, for any constant $\varepsilon > 0$. For 3D-BP, in the asymptotic regime, Bansal, Correa, Kenyon, and Sviridenko (Math.~Oper.~Res., 2006) showed that there is no asymptotic polynomial-time approximation scheme (APTAS) even when all items have the same height. Caprara (Math.~Oper.~Res., 2008) gave an asymptotic approximation ratio of $T_{\infty}^2 + \varepsilon\approx 2.86$, where $T_{\infty}$ is the well-known Harmonic constant in Bin Packing. We provide an algorithm with an improved asymptotic approximation ratio of $3T_{\infty}/2 +\varepsilon \approx 2.54$. Further, we show that unlike 3D-BP, 3D-MVBB admits an APTAS.

cs.CG↗

arXiv2024

Random-Order Online Independent Set of Intervals and Hyperrectangles

In the Maximum Independent Set of Hyperrectangles problem, we are given a set of $n$ (possibly overlapping) $d$-dimensional axis-aligned hyperrectangles, and the goal is to find a subset of non-overlapping hyperrectangles of maximum cardinality. For $d=1$, this corresponds to the classical Interval Scheduling problem, where a simple greedy algorithm returns an optimal solution. In the offline setting, for $d$-dimensional hyperrectangles, polynomial time $(\log n)^{O(d)}$-approximation algorithms are known. However, the problem becomes notably challenging in the online setting, where the input objects (hyperrectangles) appear one by one in an adversarial order, and on the arrival of an object, the algorithm needs to make an immediate and irrevocable decision whether or not to select the object while maintaining the feasibility. Even for interval scheduling, an $Ω(n)$ lower bound is known on the competitive ratio. To circumvent these negative results, in this work, we study the online maximum independent set of axis-aligned hyperrectangles in the random-order arrival model, where the adversary specifies the set of input objects which then arrive in a uniformly random order. Starting from the prototypical secretary problem, the random-order model has received significant attention to study algorithms beyond the worst-case competitive analysis. Surprisingly, we show that the problem in the random-order model almost matches the best-known offline approximation guarantees, up to polylogarithmic factors. In particular, we give a simple $(\log n)^{O(d)}$-competitive algorithm for $d$-dimensional hyperrectangles in this model, which runs in $\tilde{O_d}(n)$ time. Our approach also yields $(\log n)^{O(d)}$-competitive algorithms in the random-order model for more general objects such as $d$-dimensional fat objects and ellipsoids. Furthermore, our guarantees hold with high probability.

cs.DS↗

arXiv2023

Parameterized Guarantees for Almost Envy-Free Allocations

We study fair allocation of indivisible goods among agents with additive valuations. We obtain novel approximation guarantees for three of the strongest fairness notions in discrete fair division, namely envy-free up to the removal of any positively-valued good (EFx), pairwise maximin shares (PMMS), and envy-free up to the transfer of any positively-valued good (tEFx). Our approximation guarantees are in terms of an instance-dependent parameter $γ\in (0,1]$ that upper bounds, for each indivisible good in the given instance, the multiplicative range of nonzero values for the good across the agents. First, we consider allocations wherein, between any pair of agents and up to the removal of any positively-valued good, the envy is multiplicatively bounded. Specifically, the current work develops a polynomial-time algorithm that computes a $\left( \frac{2γ}{\sqrt{5+4γ}-1}\right)$-approximately EFx allocation for any given fair division instance with range parameter $γ\in (0,1]$. For instances with $γ\geq 0.511$, the obtained approximation guarantee for EFx surpasses the previously best-known approximation bound of $(ϕ-1) \approx 0.618$, here $ϕ$ denotes the golden ratio. Furthermore, for $γ\in (0,1]$, we develop a polynomial-time algorithm for finding allocations wherein the PMMS requirement is satisfied, between every pair of agents, within a multiplicative factor of $\frac{5}{6} γ$. En route to this result, we obtain novel existential and computational guarantees for $\frac{5}{6}$-approximately PMMS allocations under restricted additive valuations. Finally, we develop an algorithm that efficiently computes a $2γ$-approximately tEFx allocation. Specifically, we obtain existence and efficient computation of exact tEFx allocations for all instances with $γ\in [0.5, 1]$.

cs.GT↗

arXiv2023

Feature-based Individual Fairness in k-Clustering

Ensuring fairness in machine learning algorithms is a challenging and essential task. We consider the problem of clustering a set of points while satisfying fairness constraints. While there have been several attempts to capture group fairness in the $k$-clustering problem, fairness at an individual level is relatively less explored. We introduce a new notion of individual fairness in $k$-clustering based on features not necessarily used for clustering. We show that this problem is NP-hard and does not admit a constant factor approximation. Therefore, we design a randomized algorithm that guarantees approximation both in terms of minimizing the clustering distance objective and individual fairness under natural restrictions on the distance metric and fairness constraints. Finally, our experimental results against six competing baselines validate that our algorithm produces individually fairer clusters than the fairest baseline by 12.5% on average while also being less costly in terms of the clustering objective than the best baseline by 34.5% on average.

cs.LG↗

arXiv2022

Sampling-Based Winner Prediction in District-Based Elections

In a district-based election, we apply a voting rule $r$ to decide the winners in each district, and a candidate who wins in a maximum number of districts is the winner of the election. We present efficient sampling-based algorithms to predict the winner of such district-based election systems in this paper. When $r$ is plurality and the margin of victory is known to be at least $\varepsilon$ fraction of the total population, we present an algorithm to predict the winner. The sample complexity of our algorithm is $\mathcal{O}\left(\frac{1}{\varepsilon^4}\log \frac{1}{\varepsilon}\log\frac{1}δ\right)$. We complement this result by proving that any algorithm, from a natural class of algorithms, for predicting the winner in a district-based election when $r$ is plurality, must sample at least $Ω\left(\frac{1}{\varepsilon^4}\log\frac{1}δ\right)$ votes. We then extend this result to any voting rule $r$. Loosely speaking, we show that we can predict the winner of a district-based election with an extra overhead of $\mathcal{O}\left(\frac{1}{\varepsilon^2}\log\frac{1}δ\right)$ over the sample complexity of predicting the single-district winner under $r$. We further extend our algorithm for the case when the margin of victory is unknown, but we have only two candidates. We then consider the median voting rule when the set of preferences in each district is single-peaked. We show that the winner of a district-based election can be predicted with $\mathcal{O}\left(\frac{1}{\varepsilon^4}\log\frac{1}{\varepsilon}\log\frac{1}δ\right)$ samples even when the harmonious order in different districts can be different and even unknown. Finally, we also show some results for estimating the margin of victory of a district-based election within both additive and multiplicative error bounds.

cs.AI↗

arXiv2022

Approximation Algorithms for ROUND-UFP and ROUND-SAP

We study ROUND-UFP and ROUND-SAP, two generalizations of the classical BIN PACKING problem that correspond to the unsplittable flow problem on a path (UFP) and the storage allocation problem (SAP), respectively. We are given a path with capacities on its edges and a set of tasks where for each task we are given a demand and a subpath. In ROUND-UFP, the goal is to find a packing of all tasks into a minimum number of copies (rounds) of the given path such that for each copy, the total demand of tasks on any edge does not exceed the capacity of the respective edge. In ROUND-SAP, the tasks are considered to be rectangles and the goal is to find a non-overlapping packing of these rectangles into a minimum number of rounds such that all rectangles lie completely below the capacity profile of the edges. We show that in contrast to BIN PACKING, both the problems do not admit an asymptotic polynomial-time approximation scheme (APTAS), even when all edge capacities are equal. However, for this setting, we obtain asymptotic $(2+\varepsilon)$-approximations for both problems. For the general case, we obtain an $O(\log\log n)$-approximation algorithm and an $O(\log\log\frac{1}δ)$-approximation under $(1+δ)$-resource augmentation for both problems. For the intermediate setting of the no bottleneck assumption (i.e., the maximum task demand is at most the minimum edge capacity), we obtain absolute $12$- and asymptotic $(16+\varepsilon)$-approximation algorithms for ROUND-UFP and ROUND-SAP, respectively.

cs.DS↗