### Citations

1253 |
Approximation algorithms
- Vazirani
- 2004
(Show Context)
Citation Context ...present approximation techniques that are specialized for geometric optimization problems. For a complete description of these techniques we refer the reader to the survey by Arora [3], Chapter 11 in =-=[47]-=-, and Chapters 8 and 9.3.3 in [24]. A typical input to a geometric problem is a set of elements in the4 APPROXIMATION SCHEMES FOR GEOMETRIC PROBLEMS 27 space (such as points in the plane); the goal i... |

678 |
Approximation algorithms for NP-hard problems
- Hochbaum, editor
- 1997
(Show Context)
Citation Context ...hat are specialized for geometric optimization problems. For a complete description of these techniques we refer the reader to the survey by Arora [3], Chapter 11 in [47], and Chapters 8 and 9.3.3 in =-=[24]-=-. A typical input to a geometric problem is a set of elements in the4 APPROXIMATION SCHEMES FOR GEOMETRIC PROBLEMS 27 space (such as points in the plane); the goal is to connect or pack these element... |

427 |
Knapsack problems
- Kellerer, Pferschy, et al.
- 2004
(Show Context)
Citation Context ...17], with the running time improved by factor of n, the number of items. The scheme of [17] is the basis also for PTASs for other variants of the knapsack problem. (A comprehensive survey is given in =-=[31]-=-; see also in [45].) 4 Approximation Schemes for Geometric Problems In this Section we present approximation techniques that are specialized for geometric optimization problems. For a complete descrip... |

390 | Polynomial Time Approximation Schemes for Euclidean Traveling Salesman and Other Geometric Problems
- Arora
- 1998
(Show Context)
Citation Context ...a way that minimizes the resources used (e.g., total length of connecting lines, total number of covering objects). 4.1 Randomized Dissection We present below the techniques used in the PTAS of Arora =-=[2]-=- for the Euclidean Traveling Salesman Problem (TSP). In the classical TSP problem, given non-negative edge weights for the complete graph Kn, the goal is to find a tour of minimum cost, where a tour r... |

380 |
Bounds for certain multiprocessing anomalies
- Graham
- 1966
(Show Context)
Citation Context ...r a Single Subset We demonstrate the first technique for the problem of finding the minimum makespan (completion time) of a schedule of n jobs on m identical machines. The idea in this PTAS of Graham =-=[21]-=- is to schedule first2 PARTIAL ENUMERATION 4 optimally the k longest jobs and then schedule, using some heuristic, the remaining jobs. Formally, the input for the minimum makespan problem consists of... |

370 |
Approximation Algorithms for NP-complete problems in planar graphs
- Baker
- 1994
(Show Context)
Citation Context ...found by exhaustive search. We note that the above shifting technique can be used also to derive PTASs for several problems including minimum vertex-cover and maximum independent-set in planar graphs =-=[6]-=-. The idea is that a planar graph can be decomposed into components of bounded outer-planarity. The solution for each component can be found using dynamic programming. The shifting idea is to remove o... |

243 | Approximation schemes for covering and packing problems in image processing and VLSI
- Hochbaum, Maass
- 1985
(Show Context)
Citation Context ...utions to be considered and the better resulting approximation. We illustrate the shifting technique for the problem of covering n points in the 2-dimensional plane. The complete analysis is given in =-=[24, 25]-=-. Assume that the n points are enclosed in an area I. The goal is to5 CONCLUDING REMARKS 30 cover these points with a minimal number of disks of diameter D. Denote by ℓ the shifting parameter. The ar... |

210 | Using dual approximation algorithms for scheduling problems: theoretical and practical results
- Hochbaum, Shmoys
- 1987
(Show Context)
Citation Context ...bins whose sizes are slightly larger than the original. The objective value is achieved relative to the relaxed resource. This type of approximation algorithm is called a dual approximation algorithm =-=[23]-=- or approximation with resource augmentation [8]. A dual approximation scheme is a family of algorithms Aε that run in polynomial time, such that for any instance I, A(I) ≤ (1 + ε)OP T (I), and A(I) i... |

188 | Polynomial-time approximation schemes for dense instances of NP-hard optimization problems
- Arora, Karger, et al.
(Show Context)
Citation Context ...riodic scheduling to obtain a PTAS for data broadcast. Some techniques are specialized for certain types of problems. For graph problems, some PTASs exploit the density of the input graph (see, e.g., =-=[5]-=-). There are PTASs which build on the properties of planar graphs (see, e.g., [22, 14]). Finally, we have mentioned in Sections 2.3 and 4 some techniques used in randomized approximation schemes. A de... |

110 | A PTAS for the multiple knapsack problem
- Chekuri, Khanna
- 2000
(Show Context)
Citation Context ...ty of the FPTAS for fractional covering, as given in Lemma 1.7, we get the statement of the theorem. ✷ Enumeration is combined with LP rounding in the PTAS of Chekuri and Khanna for Multiple Knapsack =-=[12]-=-, as well as in the PTAS of Caprara et al. [9] for the knapsack problem with cardinalities constraints. The scheme of [9] is based on the scheme of Frieze and Clarke [17], with the running time improv... |

92 | On multidimensional packing problems
- Chekuri, Khanna
(Show Context)
Citation Context ...ent requirement of the problem, imply that only few variables can get fractional values. This essential property is used, e.g. in the PTAS of Chekuri and Khanna for the vector scheduling (VS) problem =-=[11]-=-. The VS problem is to schedule d-dimensional jobs on m identical machines, such that the maximum load over all dimensions and over all machines is minimized. Formally, an instance I of VS consists of... |

89 |
Algorithms for scheduling independent tasks
- Sahni
- 1976
(Show Context)
Citation Context ...et of elements with a small number of distinct values. This approach is described and demonstrated in Section 2.2.2. 2.2.1 Grouping Subsets of Elements We demonstrate this technique with Sahni’s PTAS =-=[41]-=- for the minimum makespan problem on two identical machines. The input consists of n jobs with processing times p1, . . . , pn. The goal is to schedule the jobs on two identical parallel machines in a... |

87 | Bounds for multiprocessor scheduling with resource constraints
- Garey, Graham
- 1975
(Show Context)
Citation Context ...mal schedule can be guessed, within factor 1 + ε, by obtaining first a (d + 1)-approximate solution. This can be done by applying an approximation algorithm for resource constrained scheduling due to =-=[20]-=-. ✷ 3.2 LP Rounding Combined with Enumeration As described in Section 2.1, a common technique for obtaining a PTAS is to extend all possible solutions for small subsets of elements. This technique can... |

85 |
Approximate algorithms for the 0/1 knapsack problem
- Sahni
- 1975
(Show Context)
Citation Context ...ty of approximation. 2.1.2 Extend All Possible Solutions for Small Subsets The second technique, of considering all possible subsets, is illustrated in an early PTAS of Sahni for the knapsack problem =-=[40]-=-. An instance of the knapsack problem consists of n items, each having a specified size and a profit, and a single knapsack, having size B. Denote by si ≥ 0, pi ≥ 0 the size and profit associated with... |

79 | Scheduling algorithms
- Karger, Stein, et al.
- 1997
(Show Context)
Citation Context ... 3 We refer the reader also to the comprehensive survey on Approximation Algorithms by Motwani [37], a tutorial by Schuurman and Woeginger [42], and the survey on scheduling by Karger, Stein and Wein =-=[29]-=-, from which we borrowed some of the examples in this chapter. 2 Partial Enumeration 2.1 Extending Partial Small-Size Solutions There are two main techniques that are based on extending partial small-... |

77 | Approximation schemes for minimizing average weighted completion time with release dates
- Afrati, Bampis, et al.
(Show Context)
Citation Context ... )⌈ ε2 ⌉) = O n The technique of applying enumeration to a compacted instance through grouping/rounding has been 1 ⌈ ε 2 ⌉) ✷ extensively used in PTASs for scheduling problems (see, e.g., [43], [28], =-=[1]-=-). A common approach for compacting the instance is to reduce the input parameters to poly-bounded, i.e., parameters whose values can be bounded as function of the input size. This approach is used, e... |

64 |
Approximation schemes for clustering problems
- Vega, Karpinski, et al.
- 2003
(Show Context)
Citation Context ... in the PTAS of Fernandez de la Vega and Kenyon’s for metric max-cut [19], and in Indyk’s work for metric 2-clustering [26]. For more details on this technique and its usage the reader is referred to =-=[18]-=-. 4.2 Shifted Plane Partitions The next technique is called shifting: it is based on selecting the best solution over a (polynomial size) set of feasible solutions. Each candidate feasible solution is... |

50 | Approximation algorithms for knapsack problems with cardinality constraints
- Caprara, Kellerer, et al.
(Show Context)
Citation Context ...en in Lemma 1.7, we get the statement of the theorem. ✷ Enumeration is combined with LP rounding in the PTAS of Chekuri and Khanna for Multiple Knapsack [12], as well as in the PTAS of Caprara et al. =-=[9]-=- for the knapsack problem with cardinalities constraints. The scheme of [9] is based on the scheme of Frieze and Clarke [17], with the running time improved by factor of n, the number of items. The sc... |

45 |
Approximation algorithms for the m-dimensional 0-1 knapsack problem: worst-case and probabilistic analyses
- Frieze, Clarke
- 1984
(Show Context)
Citation Context ...ee also in [36]). We now describe a PTAS for CIP in fixed dimension. The scheme presented in [44] builds on the classic LP-based scheme due to Frieze and Clarke for the R-dimensional knapsack problem =-=[17]-=-. Consider an instance of CIP in fixed dimension, R. We want to minimize ∑ n i=1 cixi subject to the constraints ∑ n i=1 aijxi ≥ bj for j = 1, . . . , R, and xi ∈ {0, 1, . . . di} for i = 1 . . . , n.... |

44 | Approximation schemes for NP-hard geometric optimization problems: A survey
- Arora
(Show Context)
Citation Context ...In this Section we present approximation techniques that are specialized for geometric optimization problems. For a complete description of these techniques we refer the reader to the survey by Arora =-=[3]-=-, Chapter 11 in [47], and Chapters 8 and 9.3.3 in [24]. A typical input to a geometric problem is a set of elements in the4 APPROXIMATION SCHEMES FOR GEOMETRIC PROBLEMS 27 space (such as points in th... |

43 | Bidimensionality: New connections between FPT algorithms and PTASs
- Demaine, Hajiaghayi
- 2005
(Show Context)
Citation Context ...lized for certain types of problems. For graph problems, some PTASs exploit the density of the input graph (see, e.g., [5]). There are PTASs which build on the properties of planar graphs (see, e.g., =-=[22, 14]-=-). Finally, we have mentioned in Sections 2.3 and 4 some techniques used in randomized approximation schemes. A detailed exposition of randomized approximation schemes for counting problems is given i... |

42 | A sublinear time approximation scheme for clustering in metric spaces
- Indyk
- 1999
(Show Context)
Citation Context ... distribution that depends on the pairwise distances. This technique is used e.g. in the PTAS of Fernandez de la Vega and Kenyon’s for metric max-cut [19], and in Indyk’s work for metric 2-clustering =-=[26]-=-. For more details on this technique and its usage the reader is referred to [18]. 4.2 Shifted Plane Partitions The next technique is called shifting: it is based on selecting the best solution over a... |

39 | Polynomial-time approximation scheme for data broadcast
- Kenyon, C, et al.
- 2000
(Show Context)
Citation Context ...placement on disks (see also in [30]). The technique of extending solutions for small subsets is applied by Khuller at al. [34] to the problem of broadcasting in heterogeneous networks. Kenyon et al. =-=[32]-=- used a non-trivial combination of grouping with periodic scheduling to obtain a PTAS for data broadcast. Some techniques are specialized for certain types of problems. For graph problems, some PTASs ... |

38 |
Approximate algorithms for some generalized knapsack problems, Theoretical Computer Science 3
- Chandra, Hirschberg, et al.
- 1976
(Show Context)
Citation Context ...e. For this special case, we can use a linear programming formulation in which the number of constraints is R, which is fixed. A PTAS for this problem can be derived from the scheme of Chandra et al. =-=[10]-=- for integer multidimensional knapsack. drawing from recent results on solutions of CIPs, we describe below the PTAS of [44], that achieves a better running time, by using a fast approximation scheme ... |

36 |
Analysis of a linear programming heuristic for scheduling unrelated parallel machines
- Potts
- 1985
(Show Context)
Citation Context ...s assigned to at least one machine, the number of vectors which are fractionally assigned to more than one machine is at most d · m. ✷ The above type of argument was first made and exploited by Potts =-=[39]-=- in the context of parallel machine scheduling. It was later applied for other problems, such as job shop scheduling (see e.g., [27]).3 ROUNDING LINEAR PROGRAMS 20 Thus, we solve the above program an... |

31 | Approximation schemes for minimum latency problems
- Arora, Karakostas
- 1999
(Show Context)
Citation Context ... the running time of the scheme, some schemes are classified as quasi-polynomial and others as fully polynomial. In particular, when the running time is O(npolylog(n) ) we get a quasi PTAS (see, e.g. =-=[4]-=-, [13]); when the running time is polynomial in both |I| and 1/ε we get a fully polynomial time approximation scheme (FPTAS). Such schemes are studied in detail in Chapter R-11. There is a wide litera... |

31 | C.: A randomized approximation scheme for metric max-cut
- Vega, Kenyon
- 2001
(Show Context)
Citation Context ...se sampling of data points at random from a biased distribution that depends on the pairwise distances. This technique is used e.g. in the PTAS of Fernandez de la Vega and Kenyon’s for metric max-cut =-=[19]-=-, and in Indyk’s work for metric 2-clustering [26]. For more details on this technique and its usage the reader is referred to [18]. 4.2 Shifted Plane Partitions The next technique is called shifting:... |

29 |
Randomized Algorithms,’’ Cambridge Univ
- Motwani, Raghavan
- 1995
(Show Context)
Citation Context ...ave mentioned in Sections 2.3 and 4 some techniques used in randomized approximation schemes. A detailed exposition of randomized approximation schemes for counting problems is given in Chapter 11 in =-=[38]-=- (see also Chapter R-8 in this book). Benczúr and D.R. Karger present in [7] randomized approximation schemes for cuts and flows in capacitated graphs. Efraimidis and Spirakis used in [15] the techniq... |

27 |
Lecture Notes on Approximation Algorithms
- Motwani
- 1992
(Show Context)
Citation Context ...We focus here on the techniques that have been repeatedly used in developing PTASs.2 PARTIAL ENUMERATION 3 We refer the reader also to the comprehensive survey on Approximation Algorithms by Motwani =-=[37]-=-, a tutorial by Schuurman and Woeginger [42], and the survey on scheduling by Karger, Stein and Wein [29], from which we borrowed some of the examples in this chapter. 2 Partial Enumeration 2.1 Extend... |

25 | There is no asymptotic PTAS for two-dimensional vector packing
- Woeginger
- 1997
(Show Context)
Citation Context ... inefficient in practice, these works essentially help identify the class of problems that admit PTAS. There have been also some studies towards characterizing this class of problems (see, e.g. [33], =-=[48]-=- and Chapter R-14 in this book). We focus here on the techniques that have been repeatedly used in developing PTASs.2 PARTIAL ENUMERATION 3 We refer the reader also to the comprehensive survey on App... |

23 | Tools for multicoloring with applications to planar graphs and partial k-trees
- Halldórsson, Kortsarz
(Show Context)
Citation Context ...lized for certain types of problems. For graph problems, some PTASs exploit the density of the input graph (see, e.g., [5]). There are PTASs which build on the properties of planar graphs (see, e.g., =-=[22, 14]-=-). Finally, we have mentioned in Sections 2.3 and 4 some techniques used in randomized approximation schemes. A detailed exposition of randomized approximation schemes for counting problems is given i... |

23 |
Makespan minimization in open shops: A polynomial time approximation scheme, Mathematical Programming 82
- Sevastianov, Woeginger
- 1998
(Show Context)
Citation Context ... O ( n ε ( 1 )⌈ ε2 ⌉) = O n The technique of applying enumeration to a compacted instance through grouping/rounding has been 1 ⌈ ε 2 ⌉) ✷ extensively used in PTASs for scheduling problems (see, e.g., =-=[43]-=-, [28], [1]). A common approach for compacting the instance is to reduce the input parameters to poly-bounded, i.e., parameters whose values can be bounded as function of the input size. This approach... |

19 | A fast approximation scheme for fractional covering problems with variable upper bounds
- Fleischer
(Show Context)
Citation Context ...e done in polynomial time since R is fixed. Now, for each subset of R variables we have an instance of the fractional covering problem, for which we can use a fast approximation scheme (see, e.g., in =-=[16]-=-) to obtain a (1+ε)-approximate solution. Proof of Theorem 1.8: For showing (i), assume that the optimal (integral) solution for the CIP ∞ instance is obtained by the vector y = (y1, . . . , yn). If ∑... |

19 | Tight Approximation Results for General Covering Integer Programs
- Kolliopoulos, Young
- 2001
(Show Context)
Citation Context ...roximation results for set cover carry over to CIP. Comprehensive surveys of known results for CIP and CIP∞, where the multiplicity constraints are omitted, are given in [44] and in [35] (see also in =-=[36]-=-). We now describe a PTAS for CIP in fixed dimension. The scheme presented in [44] builds on the classic LP-based scheme due to Frieze and Clarke for the R-dimensional knapsack problem [17]. Consider ... |

18 | Approximation schemes for preemptive weighted flow time
- Chekuri, Khanna
- 2002
(Show Context)
Citation Context ...running time of the scheme, some schemes are classified as quasi-polynomial and others as fully polynomial. In particular, when the running time is O(npolylog(n) ) we get a quasi PTAS (see, e.g. [4], =-=[13]-=-); when the running time is polynomial in both |I| and 1/ε we get a fully polynomial time approximation scheme (FPTAS). Such schemes are studied in detail in Chapter R-11. There is a wide literature p... |

17 | Randomized approximation schemes for cuts and flows in capacitated graphs
- Benczúr, Karger
(Show Context)
Citation Context ...imation schemes. A detailed exposition of randomized approximation schemes for counting problems is given in Chapter 11 in [38] (see also Chapter R-8 in this book). Benczúr and D.R. Karger present in =-=[7]-=- randomized approximation schemes for cuts and flows in capacitated graphs. Efraimidis and Spirakis used in [15] the technique of filtered randomized rounding in developing randomized approximation sc... |

17 | Algorithms for non-uniform size data placement on parallel disks
- Kashyap, Khuller
(Show Context)
Citation Context ...echniques described in this chapter. We mention a few of them. Golubchik et al. apply enumeration to a structured instance in solving the problem of dataREFERENCES 31 placement on disks (see also in =-=[30]-=-). The technique of extending solutions for small subsets is applied by Khuller at al. [34] to the problem of broadcasting in heterogeneous networks. Kenyon et al. [32] used a non-trivial combination ... |

13 | Approximation schemes for packing with item fragmentation
- Shachnai, Tamir, et al.
- 2006
(Show Context)
Citation Context ... from a partial solution for the problem, which can then enable to obtain a complete solution for the problem efficiently. For example, the problem of bin packing with item fragmentation is solved in =-=[46]-=- in two steps. Given the input, we need to determine the set of items that will be fragmented, as well as the fragment sizes in a feasible approximate solution. Since the possible number of fragment s... |

12 | Makespan Minimization in Job Shops: A Linear Time Approximation Scheme
- Jansen, Solis-Oba, et al.
(Show Context)
Citation Context ... ✷ The above type of argument was first made and exploited by Potts [39] in the context of parallel machine scheduling. It was later applied for other problems, such as job shop scheduling (see e.g., =-=[27]-=-).3 ROUNDING LINEAR PROGRAMS 20 Thus, we solve the above program and obtain a basic solution. Denote by S ′ the set of vectors which are assigned fractionally to two machines or more. Since |S ′ | ≤ ... |

11 | Approximating covering integer programs with multiplicity constraints
- Kolliopoulos
- 2003
(Show Context)
Citation Context ...he hardness of approximation results for set cover carry over to CIP. Comprehensive surveys of known results for CIP and CIP∞, where the multiplicity constraints are omitted, are given in [44] and in =-=[35]-=- (see also in [36]). We now describe a PTAS for CIP in fixed dimension. The scheme presented in [44] builds on the classic LP-based scheme due to Frieze and Clarke for the R-dimensional knapsack probl... |

11 | Approximation schemes – a tutorial
- Schuurman, Woeginger
- 2006
(Show Context)
Citation Context ...en repeatedly used in developing PTASs.2 PARTIAL ENUMERATION 3 We refer the reader also to the comprehensive survey on Approximation Algorithms by Motwani [37], a tutorial by Schuurman and Woeginger =-=[42]-=-, and the survey on scheduling by Karger, Stein and Wein [29], from which we borrowed some of the examples in this chapter. 2 Partial Enumeration 2.1 Extending Partial Small-Size Solutions There are t... |

11 |
Approximation schemes for generalized 2-dimensional vector packing with application to data placement
- Shachnai, Tamir
- 2003
(Show Context)
Citation Context ...ing time improved by factor of n, the number of items. The scheme of [17] is the basis also for PTASs for other variants of the knapsack problem. (A comprehensive survey is given in [31]; see also in =-=[45]-=-.) 4 Approximation Schemes for Geometric Problems In this Section we present approximation techniques that are specialized for geometric optimization problems. For a complete description of these tech... |

3 | Two-dimensional bin packing with one dimensional resource augmentation. Discrete Optimization 4:143–153
- Bansal, Sviridenko
- 2007
(Show Context)
Citation Context ...ginal. The objective value is achieved relative to the relaxed resource. This type of approximation algorithm is called a dual approximation algorithm [23] or approximation with resource augmentation =-=[8]-=-. A dual approximation scheme is a family of algorithms Aε that run in polynomial time, such that for any instance I, A(I) ≤ (1 + ε)OP T (I), and A(I) is achieved for resources augmented by factor of ... |

3 | Randomized Approximation Schemes for Scheduling Unrelated Parallel Machines - Efraimidis, Spirakis - 2000 |

3 | Polynomial Time Approximation Schemes for the Multiprocessor Open and Flow Shop Scheduling Problem
- Jansen, Sviridenko
- 2000
(Show Context)
Citation Context ... ε ( 1 )⌈ ε2 ⌉) = O n The technique of applying enumeration to a compacted instance through grouping/rounding has been 1 ⌈ ε 2 ⌉) ✷ extensively used in PTASs for scheduling problems (see, e.g., [43], =-=[28]-=-, [1]). A common approach for compacting the instance is to reduce the input parameters to poly-bounded, i.e., parameters whose values can be bounded as function of the input size. This approach is us... |

3 |
Approximation schemes for deal splitting and covering integer programs with multiplicity constraints
- Shachnai, Shmueli, et al.
- 2004
(Show Context)
Citation Context ...instances, the hardness of approximation results for set cover carry over to CIP. Comprehensive surveys of known results for CIP and CIP∞, where the multiplicity constraints are omitted, are given in =-=[44]-=- and in [35] (see also in [36]). We now describe a PTAS for CIP in fixed dimension. The scheme presented in [44] builds on the classic LP-based scheme due to Frieze and Clarke for the R-dimensional kn... |

2 |
Towards a Syntactic Characterization of
- Khanna, Motwani
- 1996
(Show Context)
Citation Context ...r them inefficient in practice, these works essentially help identify the class of problems that admit PTAS. There have been also some studies towards characterizing this class of problems (see, e.g. =-=[33]-=-, [48] and Chapter R-14 in this book). We focus here on the techniques that have been repeatedly used in developing PTASs.2 PARTIAL ENUMERATION 3 We refer the reader also to the comprehensive survey ... |

1 |
A polynomial time approximation scheme for broadcasting in heterogeneous networks
- Khuller, Kim, et al.
- 2004
(Show Context)
Citation Context ...ration to a structured instance in solving the problem of dataREFERENCES 31 placement on disks (see also in [30]). The technique of extending solutions for small subsets is applied by Khuller at al. =-=[34]-=- to the problem of broadcasting in heterogeneous networks. Kenyon et al. [32] used a non-trivial combination of grouping with periodic scheduling to obtain a PTAS for data broadcast. Some techniques a... |