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Quasi-PTAS for Scheduling with Precedences using LP Hierarchies
[article]
2017
arXiv
pre-print
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A central problem in scheduling is to schedule n unit size jobs with precedence constraints on m identical machines so as to minimize the makespan. For m=3, it is not even known if the problem is NP-hard and this is one of the last open problems from the book of Garey and Johnson. We show that for fixed m and ϵ, ( n)^O(1) rounds of Sherali-Adams hierarchy applied to a natural LP of the problem provides a (1+ϵ)-approximation algorithm running in quasi-polynomial time. This improves over the
arXiv:1708.04369v1
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... t result of Levey and Rothvoss, who used r=( n)^O( n) rounds of Sherali-Adams in order to get a (1+ϵ)-approximation algorithm with a running time of n^O(r).
Algorithmic Discrepancy Beyond Partial Coloring
[article]
2017
arXiv
pre-print
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The partial coloring method is one of the most powerful and widely used method in combinatorial discrepancy problems. However, in many cases it leads to sub-optimal bounds as the partial coloring step must be iterated a logarithmic number of times, and the errors can add up in an adversarial way. We give a new and general algorithmic framework that overcomes the limitations of the partial coloring method and can be applied in a black-box manner to various problems. Using this framework, we give
arXiv:1611.01805v2
fatcat:tljb7i37prad5gq43lp2vemple
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... new improved bounds and algorithms for several classic problems in discrepancy. In particular, for Tusnady's problem, we give an improved O(^2 n) bound for discrepancy of axis-parallel rectangles and more generally an O_d(^dn) bound for d-dimensional boxes in R^d. Previously, even non-constructively, the best bounds were O(^2.5 n) and O_d(^d+0.5n) respectively. Similarly, for the Steinitz problem we give the first algorithm that matches the best known non-constructive bounds due to Banaszczyk [Banaszczyk 2012] in the ℓ_∞ case, and improves the previous algorithmic bounds substantially in the ℓ_2 case. Our framework is based upon a substantial generalization of the techniques developed recently in the context of the Komlós discrepancy problem [BDG16].
Towards a Constructive Version of Banaszczyk's Vector Balancing Theorem
[article]
2016
arXiv
pre-print
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An important theorem of Banaszczyk (Random Structures & Algorithms '98) states that for any sequence of vectors of ℓ_2 norm at most 1/5 and any convex body K of Gaussian measure 1/2 in R^n, there exists a signed combination of these vectors which lands inside K. A major open problem is to devise a constructive version of Banaszczyk's vector balancing theorem, i.e. to find an efficient algorithm which constructs the signed combination. We make progress towards this goal along several fronts. As
arXiv:1612.04304v1
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... ur first contribution, we show an equivalence between Banaszczyk's theorem and the existence of O(1)-subgaussian distributions over signed combinations. For the case of symmetric convex bodies, our equivalence implies the existence of a universal signing algorithm (i.e. independent of the body), which simply samples from the subgaussian sign distribution and checks to see if the associated combination lands inside the body. For asymmetric convex bodies, we provide a novel recentering procedure, which allows us to reduce to the case where the body is symmetric. As our second main contribution, we show that the above framework can be efficiently implemented when the vectors have length O(1/√( n)), recovering Banaszczyk's results under this stronger assumption. More precisely, we use random walk techniques to produce the required O(1)-subgaussian signing distributions when the vectors have length O(1/√( n)), and use a stochastic gradient ascent method to implement the recentering procedure for asymmetric bodies.
Tighter Estimates for epsilon-nets for Disks
[article]
2015
arXiv
pre-print
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The geometric hitting set problem is one of the basic geometric combinatorial optimization problems: given a set P of points, and a set D of geometric objects in the plane, the goal is to compute a small-sized subset of P that hits all objects in D. In 1994, Bronniman and Goodrich made an important connection of this problem to the size of fundamental combinatorial structures called ϵ-nets, showing that small-sized ϵ-nets imply approximation algorithms with correspondingly small approximation
arXiv:1501.03246v1
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... tios. 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 Õ(n) time for hitting sets for disks in the plane. This constant factor depends on the sizes of ϵ-nets for disks; unfortunately, the current state-of-the-art bounds are large -- at least 24/ϵ and most likely larger than 40/ϵ. Thus the approximation factor of the Agarwal and Pan algorithm ends up being more than 40. The best lower-bound is 2/ϵ, 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/ϵ 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 ϵ-nets for a variety of data-sets is lower, around 9/ϵ.
The Gram-Schmidt Walk: A Cure for the Banaszczyk Blues
[article]
2017
arXiv
pre-print
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An important result in discrepancy due to Banaszczyk states that for any set of n vectors in R^m of ℓ_2 norm at most 1 and any convex body K in R^m of Gaussian measure at least half, there exists a ± 1 combination of these vectors which lies in 5K. This result implies the best known bounds for several problems in discrepancy. Banaszczyk's proof of this result is non-constructive and a major open problem has been to give an efficient algorithm to find such a ± 1 combination of the vectors. In
arXiv:1708.01079v1
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... s paper, we resolve this question and give an efficient randomized algorithm to find a ± 1 combination of the vectors which lies in cK for c>0 an absolute constant. This leads to new efficient algorithms for several problems in discrepancy theory.
Tighter estimates for ϵ -nets for disks
2016
Computational geometry
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The geometric hitting set problem is one of the basic geometric combinatorial optimization problems: given a set P of points, and a set D of geometric objects in the plane, the goal is to compute a smallsized subset of P that hits all objects in D. In 1994, Bronniman and Goodrich [5] made an important connection of this problem to the size of fundamental combinatorial structures called -nets, showing that small-sized -nets imply approximation algorithms with correspondingly small approximation
doi:10.1016/j.comgeo.2015.12.002
fatcat:qu6wisunijf2fpdkxypamakkyq
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... atios. Very recently, Agarwal and Pan [2] showed that their scheme can be implemented in near-linear time for disks in the plane. Altogether this gives O(1)-factor approximation algorithms inÕ(n) time for hitting sets for disks in the plane. This constant factor depends on the sizes of -nets for disks; unfortunately, the current state-of-theart bounds are large -at least 24/ and most likely larger than 40/ . Thus the approximation factor of the Agarwal and Pan algorithm ends up being more than 40. The best lower-bound is 2/ , which follows from the Pach-Woeginger construction [32] for halfplanes 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/ 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 -nets for a variety of data-sets is lower, around 9/ .
Improved Algorithmic Bounds for Discrepancy of Sparse Set Systems
[article]
2016
arXiv
pre-print
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We consider the problem of finding a low discrepancy coloring for sparse set systems where each element lies in at most t sets. We give an algorithm that finds a coloring with discrepancy O((t n s)^1/2) where s is the maximum cardinality of a set. This improves upon the previous constructive bound of O(t^1/2 n) based on algorithmic variants of the partial coloring method, and for small s (e.g.s=poly(t)) comes close to the non-constructive O((t n)^1/2) bound due to Banaszczyk. Previously, no
arXiv:1601.03311v2
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... rithmic results better than O(t^1/2 n) were known even for s = O(t^2). Our method is quite robust and we give several refinements and extensions. For example, the coloring we obtain satisfies the stronger size-sensitive property that each set S in the set system incurs an O((t n |S|)^1/2) discrepancy. Another variant can be used to essentially match Banaszczyk's bound for a wide class of instances even where s is arbitrarily large. Finally, these results also extend directly to the more general Komlós setting.
Limits of Local Search: Quality and Efficiency
2016
Discrete & Computational Geometry
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Over the past several decades there has been steady progress towards the goal of polynomial-time approximation schemes (PTAS) for fundamental geometric combinatorial optimization problems. A foremost example is the geometric hitting set problem: given a set P of points and a set D of geometric objects, compute the minimum-sized subset of P that hits all objects in D. For the case where D is a set of disks in the plane, the 30-year quest for a PTAS, starting from the seminal work of Hochbaum
doi:10.1007/s00454-016-9819-x
fatcat:yqrdr5vhhjcyhpzwsco4fwklie
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... , was finally achieved in 2010 [26] . Surprisingly, the algorithm to achieve the PTAS is simple: local-search. In particular, the algorithm starts with any hitting set, and iteratively tries to decrease its size by trying to replace some k points by k − 1 points; call such an algorithm a (k, k − 1)-local search algorithm. Since then, local-search has turned out to be a powerful algorithmic approach towards achieving good approximation ratios for geometric problems (for geometric independent-set problem, for dominating sets, for the terrain guarding problem and several others). Unfortunately all these algorithms have the same limitation: local search is able to give a PTAS, but with large running times. In particular, the current best work shows that if k ≥ 30, then local-search is able to give a constant factor (as a function of k) approximation ratio [17] . Unfortunately this then implies that the running time for local-search to provably work at all is Ω(n 30 ) using the current framework. As currently local search is the only known method that gives approximation factors that could be useful in practice, it becomes important to explore the limits -in both efficiency and quality -of local search. Simple examples show that (1, 0) and (2, 1) local search cannot give constant factor approximations. In this paper, we show that, surprisingly, just (3, 2) local search is able to give a constant-factor approximation; in fact we are able to get the precise quality limit of (3, 2)-local search: factor 8 approximation. This simplest working instance of local search already gives an approximation factor that is better than all known other methods! In fact, our improvement applies to all algorithms that use local-search for geometric independentset problem, for dominating sets, for the terrain guarding problem and several others. Finding efficient (3, 2)-local search algorithms then becomes the key bottleneck in efficient and good-quality algorithms. In this paper we present such improved algorithms.
An Algorithm for Komlós Conjecture Matching Banaszczyk's Bound
2019
SIAM journal on computing (Print)
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Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php NIKHIL BANSAL, DANIEL DADUSH, AND SHASHWAT GARG 2. Algorithm for the Beck-Fiala problem. ...
Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php NIKHIL BANSAL, DANIEL DADUSH, AND SHASHWAT GARG
. Downloaded 06/05/19 to 131.155.144.47. ...
doi:10.1137/17m1126795
fatcat:ihpcfttzb5bxplqeyef3qmlzyi
CURCUMIN: A PLEIOTROPIC DRUG
2020
Asian Journal of Pharmaceutical Analysis and Medicinal Chemistry
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Curcumin in the form of Turmeric powder along with hot milk is widely consumed in India for its antiinflammatory effects. Curcumin is widely used in traditional Indian ayurvedic medicine to treat hepatic disorders, anorexia, cough, diabetic wounds, rheumatoid arthritis, and sinusitis. Turmeric paste in slaked lime is a popular home remedy for the treatment of inflammation and wounds. Ancient texts of Indian medicine describe the use of curcumin in inflammatory diseases, wound healing, and
doi:10.36673/ajpamc.2020.v08.i01.a05
fatcat:yrlirgozybcrvco7eaoqsr3xki
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... nal problems. Curcumin, exhibits pleiotropic effects such as anti-inflammatory, antioxidant, anticancer, antiviral and neurotropic activity and therefore holds a promise as a therapeutic agent to prevent and treat several diseases. The purpose of this review is to provide a brief overview of the plethora of research regarding the five major pleiotropic effects of curcumin.
Quasi-PTAS for Scheduling with Precedences using LP Hierarchies
2018
International Colloquium on Automata, Languages and Programming
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Garg
59:7 Clearly a schedule σ as in Theorem 3 has makespan at most T . ...
Garg
59:5 The above high-level description skims over a few important issues. ...
Garg 59:9 schedule σ I which are put together to form a partial feasible schedule σ for all the jobs assigned to a level ≥ (C − (4m/ ) 2 )k. Call this step a recursion of type 2. ...
doi:10.4230/lipics.icalp.2018.59
dblp:conf/icalp/Garg18
fatcat:7f2q4xdvgngwnarimvoxzecvda
An Algorithm for Komlós Conjecture Matching Banaszczyk's bound
[article]
2016
arXiv
pre-print
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We consider the problem of finding a low discrepancy coloring for sparse set systems where each element lies in at most t sets. We give an efficient algorithm that finds a coloring with discrepancy O((t log n)^1/2), matching the best known non-constructive bound for the problem due to Banaszczyk. The previous algorithms only achieved an O(t^1/2 log n) bound. The result also extends to the more general Komlós setting and gives an algorithmic O(log^1/2 n) bound.
arXiv:1605.02882v3
fatcat:2bq7xk6nzreuxo4n652fk4jinu
Improved Local Search for Geometric Hitting Set
2015
Symposium on Theoretical Aspects of Computer Science
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An Algorithm for Komlós Conjecture Matching Banaszczyk's Bound
2016
2016 IEEE 57th Annual Symposium on Foundations of Computer Science (FOCS)
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Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php NIKHIL BANSAL, DANIEL DADUSH, AND SHASHWAT GARG 2. Algorithm for the Beck-Fiala problem. ...
Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php NIKHIL BANSAL, DANIEL DADUSH, AND SHASHWAT GARG
. Downloaded 05/21/19 to 192.16.191.140. ...
doi:10.1109/focs.2016.89
dblp:conf/focs/BansalDG16
fatcat:hv26b5xfjzabvkfh52f5ipuig4
Letter to the editor regarding "Selective dorsal rhizotomy for spasticity of genetic etiology"
2020
Child's Nervous System
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