Papers by Santanu Bhowmick
Discrete and Computational Geometry, Aug 29, 2019
In this article, we consider the following capacitated covering problem. We are given a set P of ... more In this article, we consider the following capacitated covering problem. We are given a set P of n points and a set B of balls from some metric space, and a positive integer U that represents the capacity of each of the balls in B. We would like to compute a subset B ⊆ B of balls and assign each point in P to some ball in B that contains it, such that the number of points assigned to any ball is at most U. The objective function that we would like to minimize is the cardinality of B. We consider this problem in arbitrary metric spaces as well as Euclidean spaces of constant dimension. In the metric setting, even the uncapacitated version of the problem is hard to approximate to within a logarithmic factor. In the Euclidean setting, the best known approximation guarantee in dimensions 3 and higher is logarithmic in the number of points. Thus we focus on obtaining "bi-criteria" approximations. In particular, we are allowed to expand the balls in our solution by some factor, but optimal solutions do not have that flexibility. Our main result is that allowing constant factor expansion of the input balls suffices to obtain constant approximations for this problem. In fact, in the Euclidean setting, only (1 +) factor expansion is sufficient for any > 0, with the approximation factor being a polynomial in 1/. We obtain these results using a unified scheme for rounding the natural LP relaxation; this scheme may be useful for other capacitated covering problems. We also complement these bi-criteria approximations by obtaining hardness of approximation results that shed light on our understanding of these problems. 2012 ACM Subject Classification Theory of computation → Packing and covering problems, Theory of computation → Rounding techniques, Theory of computation → Computational geometry, Mathematics of computing → Approximation algorithms
arXiv (Cornell University), Apr 14, 2014
We revisit two NP-hard geometric partitioning problems-convex decomposition and surface approxima... more We revisit two NP-hard geometric partitioning problems-convex decomposition and surface approximation. Building on recent developments in geometric separators, we present quasi-polynomial time algorithms for these problems with improved approximation guarantees.
Lecture Notes in Computer Science, 2015
Given an initial placement of a set of rectangles in the plane, we consider the problem of findin... more Given an initial placement of a set of rectangles in the plane, we consider the problem of finding a disjoint placement of the rectangles that minimizes the area of the bounding box and preserves the orthogonal order i.e. maintains the sorted ordering of the rectangle centers along both x-axis and y-axis with respect to the initial placement. This problem is known as Layout Adjustment for Disjoint Rectangles (LADR). It was known that LADR is NP-hard, but only heuristics were known for it. We show that a certain decision version of LADR is APX-hard, and give a constant factor approximation for LADR.
arXiv (Cornell University), Feb 12, 2016
In the metric multi-cover problem (MMC), we are given two point sets Y (servers) and X (clients) ... more In the metric multi-cover problem (MMC), we are given two point sets Y (servers) and X (clients) in an arbitrary metric space (X ∪ Y, d), a positive integer k that represents the coverage demand of each client, and a constant α ≥ 1. Each server can have a single ball of arbitrary radius centered on it. Each client x ∈ X needs to be covered by at least k such balls centered on servers. The objective function that we wish to minimize is the sum of the α-th powers of the radii of the balls. In this article, we consider the MMC problem as well as some non-trivial generalizations, such as (a) the non-uniform MMC, where we allow client-specific demands, and (b) the t-MMC, where we require the number of open servers to be at most some given integer t. For each of these problems, we present an efficient algorithm that reduces the problem to several instances of the corresponding 1-covering problem, where the coverage demand of each client is 1. Our reductions preserve optimality up to a multiplicative constant factor. Applying known constant factor approximation algorithms for 1-covering, we obtain the first constant approximations for the MMC and these generalizations.
arXiv (Cornell University), Oct 26, 2011
For Application Specific Integrated Circuits (ASIC) and System-on-Chip (SOC) designs , Cell-Based... more For Application Specific Integrated Circuits (ASIC) and System-on-Chip (SOC) designs , Cell-Based Design (CBD) is the most prevalent practice as it guarantees a shorter design cycle ,minimizes errors and is easier to maintain. In modern ASIC design, standard cell methodology is practiced with sizeable libraries of cells, each containing multiple implementations of the same logic functionality , in order to give the designer differing options based on area , speed or power consumtion. For such library cells , thorough verification of functionality and timing is crucial for the overall success of the chip , as even a small error can prove fatal due to the repeated use of the cell in the design. Both formal and simulation based methods are being used in the industry for cell verification. We propose a method using the latter approach that generates an optimised set of test vectors for verification of sequential cells, which are guaranteed to give complete Single Input Change transition coverage with minimal redundancy. Knowledge of the cell functionality by means of the State Table is the only prerequisite of this procedure.
arXiv (Cornell University), Feb 12, 2015
Given an initial placement of a set of rectangles in the plane, we consider the problem of findin... more Given an initial placement of a set of rectangles in the plane, we consider the problem of finding a disjoint placement of the rectangles that minimizes the area of the bounding box and preserves the orthogonal order i.e. maintains the sorted ordering of the rectangle centers along both x-axis and y-axis with respect to the initial placement. This problem is known as Layout Adjustment for Disjoint Rectangles (LADR). It was known that LADR is NP-hard, but only heuristics were known for it. We show that a certain decision version of LADR is APX-hard, and give a constant factor approximation for LADR.
Algorithmica, Sep 29, 2020
In this article, we study some fault-tolerant covering problems in metric spaces. In the metric m... more In this article, we study some fault-tolerant covering problems in metric spaces. In the metric multi-cover problem (MMC), we are given two point sets Y (servers) and X (clients) in an arbitrary metric space (X ∪ Y , d), a positive integer k that represents the coverage demand of each client, and a constant α ≥ 1. Each server can host a single ball of arbitrary radius centered on it. Each client x ∈ X needs to be covered by at least k such balls centered on servers. The objective function that we wish to minimize is the sum of the α-th powers of the radii of the balls. We also study some non-trivial generalizations of the MMC, such as (a) the non-uniform MMC, where we allow client-specific demands, and (b) the t-MMC, where we require the number of open servers to be at most some given integer t. We present the first constant approximations for these fault-tolerant covering problems. Our algorithms are based on the following paradigm: for each of the three problems, we present an efficient algorithm that reduces the problem to several instances of the corresponding 1-covering problem, where the coverage demand of each client is 1. The reductions preserve optimality up to a multiplicative constant factor. Applying known constant factor approximation algorithms for 1-covering, we obtain our results for the MMC and these generalizations.
arXiv (Cornell University), Jul 17, 2017
In this article, we consider the following capacitated covering problem. We are given a set P of ... more In this article, we consider the following capacitated covering problem. We are given a set P of n points and a set B of balls from some metric space, and a positive integer U that represents the capacity of each of the balls in B. We would like to compute a subset B ⊆ B of balls and assign each point in P to some ball in B that contains it, such that the number of points assigned to any ball is at most U. The objective function that we would like to minimize is the cardinality of B. We consider this problem in arbitrary metric spaces as well as Euclidean spaces of constant dimension. In the metric setting, even the uncapacitated version of the problem is hard to approximate to within a logarithmic factor. In the Euclidean setting, the best known approximation guarantee in dimensions 3 and higher is logarithmic in the number of points. Thus we focus on obtaining "bi-criteria" approximations. In particular, we are allowed to expand the balls in our solution by some factor, but optimal solutions do not have that flexibility. Our main result is that allowing constant factor expansion of the input balls suffices to obtain constant approximations for these problems. In fact, in the Euclidean setting, only (1 +) factor expansion is sufficient for any > 0, with the approximation factor being a polynomial in 1/. We obtain these results using a unified scheme for rounding the natural LP relaxation; this scheme may be useful for other capacitated covering problems. We also complement these bi-criteria approximations by obtaining hardness of approximation results that shed light on our understanding of these problems.
arXiv (Cornell University), Jul 17, 2017
In this article, we consider the following capacitated covering problem. We are given a set P of ... more In this article, we consider the following capacitated covering problem. We are given a set P of n points and a set B of balls from some metric space, and a positive integer U that represents the capacity of each of the balls in B. We would like to compute a subset B ⊆ B of balls and assign each point in P to some ball in B that contains it, such that the number of points assigned to any ball is at most U. The objective function that we would like to minimize is the cardinality of B. We consider this problem in arbitrary metric spaces as well as Euclidean spaces of constant dimension. In the metric setting, even the uncapacitated version of the problem is hard to approximate to within a logarithmic factor. In the Euclidean setting, the best known approximation guarantee in dimensions 3 and higher is logarithmic in the number of points. Thus we focus on obtaining "bi-criteria" approximations. In particular, we are allowed to expand the balls in our solution by some factor, but optimal solutions do not have that flexibility. Our main result is that allowing constant factor expansion of the input balls suffices to obtain constant approximations for these problems. In fact, in the Euclidean setting, only (1 +) factor expansion is sufficient for any > 0, with the approximation factor being a polynomial in 1/. We obtain these results using a unified scheme for rounding the natural LP relaxation; this scheme may be useful for other capacitated covering problems. We also complement these bi-criteria approximations by obtaining hardness of approximation results that shed light on our understanding of these problems.
arXiv (Cornell University), Feb 12, 2016
In the metric multi-cover problem (MMC), we are given two point sets Y (servers) and X (clients) ... more In the metric multi-cover problem (MMC), we are given two point sets Y (servers) and X (clients) in an arbitrary metric space (X ∪ Y, d), a positive integer k that represents the coverage demand of each client, and a constant α ≥ 1. Each server can have a single ball of arbitrary radius centered on it. Each client x ∈ X needs to be covered by at least k such balls centered on servers. The objective function that we wish to minimize is the sum of the α-th powers of the radii of the balls. In this article, we consider the MMC problem as well as some non-trivial generalizations, such as (a) the non-uniform MMC, where we allow client-specific demands, and (b) the t-MMC, where we require the number of open servers to be at most some given integer t. For each of these problems, we present an efficient algorithm that reduces the problem to several instances of the corresponding 1-covering problem, where the coverage demand of each client is 1. Our reductions preserve optimality up to a multiplicative constant factor. Applying known constant factor approximation algorithms for 1-covering, we obtain the first constant approximations for the MMC and these generalizations.
Symposium on Discrete Algorithms, Jan 4, 2015
Recently, Adamaszek and Wiese [1, 2] presented a quasipolynomial time approximation scheme (QPTAS... more Recently, Adamaszek and Wiese [1, 2] presented a quasipolynomial time approximation scheme (QPTAS) for the problem of computing a maximum weight independent set for certain families of planar objects. This major advance on the problem was based on their proof that a certain type of separator exists for any independent set. Subsequently, Har-Peled [22] simplified and generalized their result. Mustafa et al. [36] also described a simplification, and somewhat surprisingly, showed that QPTAS's can be obtained for certain, albeit special, type of covering problems. Building on these developments, we revisit two NPhard geometric partitioning problems-convex decomposition and surface approximation. Partitioning problems combine the features of packing and covering. In particular, since the optimal solution does form a packing, the separator theorems are potentially applicable. Nevertheless, the two partitioning problems we study bring up additional difficulties that are worth examining in the context of the wider applicability of the separator methodology. We show how these issues can be handled in presenting quasi-polynomial time algorithms for these two problems with improved approximation guarantees.
We consider the following multi-covering problem with disks. We are given two point sets Y (serve... more We consider the following multi-covering problem with disks. We are given two point sets Y (servers) and X (clients) in the plane, a coverage function κ : X → N, and a constant α ≥ 1. Centered at each server is a single disk whose radius we are free to set. The requirement is that each client x ∈ X be covered by at least κ(x) of the server disks. The objective function we wish to minimize is the sum of the α-th powers of the disk radii. We present a polynomial-time algorithm for this problem achieving an O(1) approximation.
arXiv: Computational Geometry, Feb 12, 2016
In the metric multi-cover problem (MMC), we are given two point sets Y (servers) and X (clients) ... more In the metric multi-cover problem (MMC), we are given two point sets Y (servers) and X (clients) in an arbitrary metric space (X ∪ Y, d), a positive integer k that represents the coverage demand of each client, and a constant α ≥ 1. Each server can have a single ball of arbitrary radius centered on it. Each client x ∈ X needs to be covered by at least k such balls centered on servers. The objective function that we wish to minimize is the sum of the α-th powers of the radii of the balls. In this article, we consider the MMC problem as well as some non-trivial generalizations, such as (a) the non-uniform MMC, where we allow client-specific demands, and (b) the t-MMC, where we require the number of open servers to be at most some given integer t. For each of these problems, we present an efficient algorithm that reduces the problem to several instances of the corresponding 1-covering problem, where the coverage demand of each client is 1. Our reductions preserve optimality up to a multiplicative constant factor. Applying known constant factor approximation algorithms for 1-covering, we obtain the first constant approximations for the MMC and these generalizations.
Given an initial placement of a set of rectangles in the plane, we consider the problem of findin... more Given an initial placement of a set of rectangles in the plane, we consider the problem of finding a disjoint placement of the rectangles that minimizes the area of the bounding box and preserves the orthogonal order i.e. maintains the sorted ordering of the rectangle centers along both x-axis and y-axis with respect to the initial placement. This problem is known as Layout Adjustment for Disjoint Rectangles(LADR). It was known that LADR is NP-hard, but only heuristics were known for it. We show that a certain decision version of LADR is APX-hard, and give a constant factor approximation for LADR.
Journal of Computational Geometry jocg.org A CONSTANT-FACTOR APPROXIMATION FOR MULTI-COVERING WITH DISKS∗
Abstract. We consider the following multi-covering problem with disks. We are given two point set... more Abstract. We consider the following multi-covering problem with disks. We are given two point sets Y (servers) and X (clients) in the plane, a coverage function κ: X → N, and a constant α ≥ 1. Centered at each server is a single disk whose radius we are free to set. The requirement is that each client x ∈ X be covered by at least κ(x) of the server disks. The objective function we wish to minimize is the sum of the α-th powers of the disk radii. We present a polynomial-time algorithm for this problem achieving an O(1) approximation. 1
ArXiv, 2017
In this paper, we consider capacitated version of the set cover problem. We consider several metr... more In this paper, we consider capacitated version of the set cover problem. We consider several metric and geometric versions of the capacitated set cover problem. In one such variant, the elements are points and the sets are the balls in a metric space. We also assume that the capacities of the input balls are monotonic -- for any two balls $B_i,B_j$ with radii $r_i$ and $r_j$, and capacities $U_i$ and $U_j$, respectively: $r_i > r_j$ implies that $U_i \ge U_j$. We refer to this problem as the Metric Monotonic Capacitated Covering (MMCC) problem. We note that this assumption may be reasonable in many applications. Also this model generalizes the uniform capacity model. We also consider several variants of the MMCC problem. Unfortunately, MMCC remains as hard as the set cover problem even when the capacities are uniform, and thus we focus on obtaining "bi-criteria" approximations. In particular, we are allowed to expand the balls in our solution by some factor, but optimal...
We consider the metric multi-cover problem (MMC). The input consists of two point sets $Y$(server... more We consider the metric multi-cover problem (MMC). The input consists of two point sets $Y$(servers) and $X$(clients) in an arbitrary metric space $(X \cup Y, d)$, a positive integer $k$ that represents the coverage demand of each client, and a constant $\alpha \geq 1$. Each server can have a single ball of arbitrary radius centered on it. Each client $x \in X$ needs to be covered by at least $k$ such balls centered on servers. The objective function that we wish to minimize is the sum of the $\alpha$-th powers of the radii of the balls. Bar-Yehuda[2013] gave a $3^{\alpha} \cdot k$-approximation for the MMC problem. Bhowmick et al [SoCG 13, JoCG 15] showed that an $O(1)$ approximation (independent of the coverage demand $k$) exists for the special case in which $X, Y$ are points in a fixed-dimensional space $\mathbb{R}^d$. In this paper, we present an $O(1)$-approximation for the MMC problem in any arbitrary metric space, improving the result of Bar-Yehuda[2013] and matching the guar...
Given an initial placement of a set of rectangles in the plane, we consider the problem of findin... more Given an initial placement of a set of rectangles in the plane, we consider the problem of finding a disjoint placement of the rectangles that minimizes the area of the bounding box and preserves the orthogonal order i.e.\ maintains the sorted ordering of the rectangle centers along both $x$-axis and $y$-axis with respect to the initial placement. This problem is known as Layout Adjustment for Disjoint Rectangles(LADR). It was known that LADR is $\mathbb{NP}$-hard, but only heuristics were known for it. We show that a certain decision version of LADR is $\mathbb{APX}$-hard, and give a constant factor approximation for LADR.
In the metric multi-cover problem (MMC), we are given two point sets $Y$ (servers) and $X$ (clien... more In the metric multi-cover problem (MMC), we are given two point sets $Y$ (servers) and $X$ (clients) in an arbitrary metric space $(X \cup Y, d)$, a positive integer $k$ that represents the coverage demand of each client, and a constant $\alpha \geq 1$. Each server can have a single ball of arbitrary radius centered on it. Each client $x \in X$ needs to be covered by at least $k$ such balls centered on servers. The objective function that we wish to minimize is the sum of the $\alpha$-th powers of the radii of the balls. In this article, we consider the MMC problem as well as some non-trivial generalizations, such as (a) the non-uniform MMC, where we allow client-specific demands, and (b) the $t$-MMC, where we require the number of open servers to be at most some given integer $t$. For each of these problems, we present an efficient algorithm that reduces the problem to several instances of the corresponding $1$-covering problem, where the coverage demand of each client is $1$. Our ...
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Papers by Santanu Bhowmick