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## A new approximation technique for resource-allocation problems (2009)

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1251 |
Approximation Algorithms
- Vazirani
- 2003
(Show Context)
Citation Context ...repancy of column-sparse matrices [12], in the setting of random matrices. § 6 also suggests that there may be deeper connections to iterated rounding, a fruitful approach in approximation algorithms =-=[25, 30, 33, 42, 49]-=-. We view our approach as having broader connections/applications (e.g., to open problems including capacitated facility location [36]), and are studying these directions. 2 Random Matchings with Line... |

368 |
Randomized rounding: A technique for provably good algorithms and algorithmic proofs
- Raghavan, Thompson
- 1987
(Show Context)
Citation Context ...42. Supported in part by NSF ITR Award CNS-0426683 and NSF Award CNS-0626636. in randomized rounding, we use randomization to map x∗ = (x∗ 1, x∗ 2, . . . , x∗ n) back to some x = (x1, x2, . . . , xn) =-=[39]-=-. Typically, we choose a value α that is problem-specific, and, independently for each i, define xi to be 1 with probability αx∗ i , and to be 0 with the complementary probability of 1 − αx∗ i . Indep... |

262 | A factor 2 approximation algorithm for the generalized steiner network problem
- Jain
- 2001
(Show Context)
Citation Context ...at least for various models of random t-bounded matrices. Second, there appears to be a deeper connection between various forms of dependent randomized rounding – such as ours – and iterated rounding =-=[25, 30, 33, 42, 49]-=-. In particular: (i) the result that we improve upon in § 2 is based on iterated rounding [18]; (ii) certain “budgeted” assignment problems that arise in keyword auctions give the same results under i... |

260 | Approximation algorithms for scheduling unrelated parallel machines
- Lenstra, Shmoys, et al.
- 1990
(Show Context)
Citation Context ...undamental scheduling model, which has spurred many advances and applications in combinatorial optimization, including linear-, quadratic- & convex-programming relaxations and new rounding approaches =-=[6, 8, 10, 15, 21, 29, 32, 34, 41, 43]-=-. This model, scheduling with unrelated parallel machines (UPM) – and its relatives – play a key role in this work. Herein, we are given a set J of n jobs, a set M of m machines, 1and non-negative va... |

200 | An approximation algorithm for the generalized assignment problem
- Shmoys, Tardos
- 1993
(Show Context)
Citation Context ...undamental scheduling model, which has spurred many advances and applications in combinatorial optimization, including linear-, quadratic- & convex-programming relaxations and new rounding approaches =-=[6, 8, 10, 15, 21, 29, 32, 34, 41, 43]-=-. This model, scheduling with unrelated parallel machines (UPM) – and its relatives – play a key role in this work. Herein, we are given a set J of n jobs, a set M of m machines, 1and non-negative va... |

140 | On the approximability of trade-offs and optimal access of web sources
- Papadimitriou, Yannakakis
- 2000
(Show Context)
Citation Context ...E) of N vertices and a collection of k linear functions {fi} of x, many works have considered the problem of constructing (b-)matchings X such that fi(X) is “close” to fi(x) simultaneously for each i =-=[3, 27, 28, 38]-=-. The works [28, 38] focus on the case of constant k; those of [3, 27] consider general k, and require the usual “discrepancy” term of Ω( √ fi(x) log N) in |fi(X) − fi(x)| for most/all i; in a few cas... |

113 | Improved approximation algorithms for the vertex cover problem in graphs and hypergraphs
- Halperin
- 2000
(Show Context)
Citation Context ..., random matchings with sharp tail bounds. Handling “hard capacities” – those that cannot be violated – is generally tricky in various settings, including facilitylocation and other covering problems =-=[19, 26, 36]-=-. Motivated by problems in crew-scheduling [22, 40] and by the fact that servers have a limit on how many jobs can be assigned to them, the natural question of scheduling with a hard capacityconstrain... |

85 | Pipage rounding: a new method of constructing algorithms with proven performance guarantee
- Ageev, Sviridenko
(Show Context)
Citation Context ...of random variables are held fixed with probability one, while still retaining randomness in the individual variables and guaranteeing certain types of negative-correlation properties among them. See =-=[1]-=- for a related deterministic approach that precedes these works. These dependent-rounding approaches lead to numerous improved approximation algorithms in scheduling and packet-routing [1, 27, 32, 45]... |

80 | Randomized distributed edge coloring via an extension of the Chernoff-Hoeffding bounds
- Panconesi, Srinivasan
- 1997
(Show Context)
Citation Context ... − w ′ v,j uv,jfj T/λ ɛ1k ) (8) (1 − ɛ1)λ Now we will show in Lemma 14 that Yv,j’s 12are negatively correlated. Therefore, applying Chernoff-Hoeffding bound for negativelycorrelated random variables =-=[37]-=-, we get Pr [ Gv ≤ (1 − ɛ)E [ ]] Gv ≤ exp(−E[Gv]ɛ 2 /3) Hh ∑ j = i yh i,j , where yh i,j denote the value of yi,j at the beginning of iteration h. We will show, ∀i E [∏ ] [∏ ≤ E (9) j∈J H h j j∈J H (h... |

78 | A new rounding procedure for the assignment problem with applications to dense graph assignment problems
- Arora, Frieze, et al.
- 2002
(Show Context)
Citation Context ...E) of N vertices and a collection of k linear functions {fi} of x, many works have considered the problem of constructing (b-)matchings X such that fi(X) is “close” to fi(x) simultaneously for each i =-=[3, 27, 28, 38]-=-. The works [28, 38] focus on the case of constant k; those of [3, 27] consider general k, and require the usual “discrepancy” term of Ω( √ fi(x) log N) in |fi(X) − fi(x)| for most/all i; in a few cas... |

69 |
Six standard deviations suffice
- Spencer
- 1985
(Show Context)
Citation Context ...bounded matrix A; this, if true, would be bestpossible. Ingenious melding of randomized rounding, entropy-based arguments and the pigeonhole principle have helped show that lindisc(A) ≤ O( √ t log n) =-=[11, 35, 44]-=-, improved further to O( √ t log n) in [7]. However, the number of columns n may not be bounded as a function of t, and it would be very interesting to even get some o(t) bound on lindisc(A), to start... |

64 |
The Santa Claus problem
- Bansal, Sviridenko
(Show Context)
Citation Context ...undamental scheduling model, which has spurred many advances and applications in combinatorial optimization, including linear-, quadratic- & convex-programming relaxations and new rounding approaches =-=[6, 8, 10, 15, 21, 29, 32, 34, 41, 43]-=-. This model, scheduling with unrelated parallel machines (UPM) – and its relatives – play a key role in this work. Herein, we are given a set J of n jobs, a set M of m machines, 1and non-negative va... |

62 | L.C.: Approximating minimum bounded degree spanning trees to within one of optimal
- Singh, Lau
- 2007
(Show Context)
Citation Context ...repancy of column-sparse matrices [12], in the setting of random matrices. § 6 also suggests that there may be deeper connections to iterated rounding, a fruitful approach in approximation algorithms =-=[25, 30, 33, 42, 49]-=-. We view our approach as having broader connections/applications (e.g., to open problems including capacitated facility location [36]), and are studying these directions. 2 Random Matchings with Line... |

61 | Survivable Network Design with Degree or Order Constraints
- LAU, NAOR, et al.
- 2009
(Show Context)
Citation Context ...repancy of column-sparse matrices [12], in the setting of random matrices. § 6 also suggests that there may be deeper connections to iterated rounding, a fruitful approach in approximation algorithms =-=[25, 30, 33, 42, 49]-=-. We view our approach as having broader connections/applications (e.g., to open problems including capacitated facility location [36]), and are studying these directions. 2 Random Matchings with Line... |

60 | Dependent rounding and its applications to approximation algorithms
- Gandhi, Khuller, et al.
- 2006
(Show Context)
Citation Context ...x∗ i . Independence can, however, lead to noticeable deviations from the mean for random variables that are required to be very close to (or even be equal to) their mean. A fruitful idea developed in =-=[27, 32, 45]-=- is to carefully introduce dependencies into the rounding process: in particular, some sums of random variables are held fixed with probability one, while still retaining randomness in the individual ... |

58 | An approximation algorithm for max-min fair allocation of indivisible goods
- Asadpour, Saberi
- 2007
(Show Context)
Citation Context ...also known as the Santa Claus problem, is the maxmin version of UPM, where we aim to maximize the minimum “load” (viewed as utility) on the machines; it has received a good deal of attention recently =-=[4, 5, 8, 10, 15, 24]-=-. We are able to employ dependent randomized rounding to near-optimally determine the integrality gap of a well-studied LP relaxation. Also, Theorem 1 lets us generalize a result of [14] on max-min fa... |

54 | A constant-factor approximation algorithm for packet routing, and balancing local vs. global criteria
- Srinivasan, Teo
- 1997
(Show Context)
Citation Context ...very column has at most t nonzeroes), then lindisc(A) ≤ t; see [31] for a closely-related result. These results have also helped in the development of improved rounding-based approximation algorithms =-=[9, 47]-=-. A major open question from [12] is whether lindisc(A) ≤ O( √ t) for any tbounded matrix A; this, if true, would be bestpossible. Ingenious melding of randomized rounding, entropy-based arguments and... |

53 | Multicasting in heterogeneous networks
- Bar-Noy, Guha, et al.
- 1998
(Show Context)
Citation Context ...very column has at most t nonzeroes), then lindisc(A) ≤ t; see [31] for a closely-related result. These results have also helped in the development of improved rounding-based approximation algorithms =-=[9, 47]-=-. A major open question from [12] is whether lindisc(A) ≤ O( √ t) for any tbounded matrix A; this, if true, would be bestpossible. Ingenious melding of randomized rounding, entropy-based arguments and... |

52 | Facility Location with Nonuniform Hard Capacities
- Pál, Tardos, et al.
- 2001
(Show Context)
Citation Context ..., random matchings with sharp tail bounds. Handling “hard capacities” – those that cannot be violated – is generally tricky in various settings, including facilitylocation and other covering problems =-=[19, 26, 36]-=-. Motivated by problems in crew-scheduling [22, 40] and by the fact that servers have a limit on how many jobs can be assigned to them, the natural question of scheduling with a hard capacityconstrain... |

50 |
Distributions on level-sets with applications to approximation algorithms
- Srinivasan
(Show Context)
Citation Context ...x∗ i . Independence can, however, lead to noticeable deviations from the mean for random variables that are required to be very close to (or even be equal to) their mean. A fruitful idea developed in =-=[27, 32, 45]-=- is to carefully introduce dependencies into the rounding process: in particular, some sums of random variables are held fixed with probability one, while still retaining randomness in the individual ... |

43 | Structural and algorithmic aspects of massive social networks
- Eubank, Kumar, et al.
- 2004
(Show Context)
Citation Context ...P, and present one such in Section 5. Theorem 1 generalizes such capacitated problems to random bipartite (b-)matchings with target degree bounds and sharp tail bounds for given linear functions; see =-=[23]-=- to applications to models for complex networks. Recall that a (b)-matching is a subgraph in which every vertex v has degree at most b(v). Given a fractional (b)-matching x in a bipartite graph G = (J... |

39 |
Roth’s estimate of the discrepancy of integer sequences is nearly sharp
- BECK
- 1981
(Show Context)
Citation Context ...bounded matrix A; this, if true, would be bestpossible. Ingenious melding of randomized rounding, entropy-based arguments and the pigeonhole principle have helped show that lindisc(A) ≤ O( √ t log n) =-=[11, 35, 44]-=-, improved further to O( √ t log n) in [7]. However, the number of columns n may not be bounded as a function of t, and it would be very interesting to even get some o(t) bound on lindisc(A), to start... |

39 | A polynomial time approximation scheme for the multiple knapsack problem
- Chekuri, Khanna
- 2005
(Show Context)
Citation Context ...ere we incur a cost ci,j if we schedule job j on machine i; a simultaneous (2, 1)– approximation for the (makespan, total cost)-pair is developed in [41], leading to numerous applications (see, e.g., =-=[2, 17]-=-). We generalize the methods of [1, 27, 31, 32, 45], via a type of random walk toward a vertex of the underlying polytope that we outline next. We then present several applications in scheduling and b... |

36 | Global wire routing in two-dimensional arrays
- Karp, Leighton, et al.
- 1987
(Show Context)
Citation Context ... job j on machine i; a simultaneous (2, 1)– approximation for the (makespan, total cost)-pair is developed in [41], leading to numerous applications (see, e.g., [2, 17]). We generalize the methods of =-=[1, 27, 31, 32, 45]-=-, via a type of random walk toward a vertex of the underlying polytope that we outline next. We then present several applications in scheduling and bipartite matching through problem-specific speciali... |

35 | Dependent Rounding in Bipartite Graphs
- Gandhi, Khuller, et al.
- 2002
(Show Context)
Citation Context ...ain contribution in this regard is to near-optimally pin-point the integrality gap of a configuration LP previously proposed 16and analyzed in [8, 5]. Our algorithm uses bipartite dependent rounding =-=[28]-=- and its generalization to weighted graphs. Bipartite dependent rounding can be viewed as a specific type of RandMove on bipartite graphs. A crucial ingredient of our analysis is to show certain rando... |

34 | On the approximability of budgeted allocations and improved lower bounds for submodular welfare maximization and GAP
- Chakrabarty, Goel
(Show Context)
Citation Context ...(i) the result that we improve upon in § 2 is based on iterated rounding [18]; (ii) certain “budgeted” assignment problems that arise in keyword auctions give the same results under iterated rounding =-=[16]-=- and weighted dependent rounding [46]; and (iii) our ongoing work suggests that our random-walk ap16proach improves upon the iterated-rounding-based work of [28] on bipartite matchings that are simul... |

31 |
Integer-making theorems
- Beck, Fiala
- 1981
(Show Context)
Citation Context ...parse matrices have received much attention in this regard. Section 6 discusses a concrete approach to use our method for the famous Beck-Fiala conjecture on the discrepancy of column-sparse matrices =-=[12]-=-, in the setting of random matrices. § 6 also suggests that there may be deeper connections to iterated rounding, a fruitful approach in approximation algorithms [25, 30, 33, 42, 49]. We view our appr... |

27 | Convex programming for scheduling unrelated parallel machines - Azar, Epstein |

26 |
Balancing vectors and Gaussian measures of n-dimensional convex bodies. Random Structures and Algorithms
- Banaszczyk
- 1998
(Show Context)
Citation Context ...ble. Ingenious melding of randomized rounding, entropy-based arguments and the pigeonhole principle have helped show that lindisc(A) ≤ O( √ t log n) [11, 35, 44], improved further to O( √ t log n) in =-=[7]-=-. However, the number of columns n may not be bounded as a function of t, and it would be very interesting to even get some o(t) bound on lindisc(A), to start with. We have preliminary ideas about usi... |

26 | Maxmin allocation via degree lower-bounded arborescences
- Bateni, Charikar, et al.
- 2009
(Show Context)
Citation Context |

26 | On allocations that maximize fairness
- Feige
- 2008
(Show Context)
Citation Context ...also known as the Santa Claus problem, is the maxmin version of UPM, where we aim to maximize the minimum “load” (viewed as utility) on the machines; it has received a good deal of attention recently =-=[4, 5, 8, 10, 15, 24]-=-. We are able to employ dependent randomized rounding to near-optimally determine the integrality gap of a well-studied LP relaxation. Also, Theorem 1 lets us generalize a result of [14] on max-min fa... |

26 | An iterative rounding 2approximation algorithm for the element connectivity problem
- Fleischer, Jain, et al.
- 2001
(Show Context)
Citation Context |

25 | Designing Overlay Multicast Networks for Streaming
- Andreev, Maggs, et al.
- 2003
(Show Context)
Citation Context ...d around its expected value with probability ≥ 1 − exp(− T 2 √ log k k log log k T/λ Therefore we get Theorem 11. ) > 1 − log k k . 145 Designing Overlay Multicast Networks For Streaming The work of =-=[2]-=- studies approximation algorithms for designing a multicast overlay network. We first describe the problem and state the results in [2] (Lemma 15 and Lemma 16). Next, we show our main improvement in L... |

23 | On allocating goods to maximize fairness
- Chakrabarty, Chuzhoy, et al.
(Show Context)
Citation Context |

21 | Allocating indivisible goods
- Bez'akov'a, Dani
- 2003
(Show Context)
Citation Context ... 5, 8, 10, 15, 24]. We are able to employ dependent randomized rounding to near-optimally determine the integrality gap of a well-studied LP relaxation. Also, Theorem 1 lets us generalize a result of =-=[14]-=- on max-min fairness to the setting of equitable partitioning of the jobs; see § 4. Directions for the future: some potential connections and applications. Distributions on structured matchings in bip... |

19 | Santa claus meets hypergraph matchings
- Asadpour, Feige, et al.
- 2008
(Show Context)
Citation Context ...also known as the Santa Claus problem, is the maxmin version of UPM, where we aim to maximize the minimum “load” (viewed as utility) on the machines; it has received a good deal of attention recently =-=[4, 5, 8, 10, 15, 24]-=-. We are able to employ dependent randomized rounding to near-optimally determine the integrality gap of a well-studied LP relaxation. Also, Theorem 1 lets us generalize a result of [14] on max-min fa... |

18 |
Airline scheduling: An overview
- ETSCHMAIER, MATHAISEL
- 1985
(Show Context)
Citation Context ...rd capacities” – those that cannot be violated – is generally tricky in various settings, including facilitylocation and other covering problems [19, 26, 36]. Motivated by problems in crew-scheduling =-=[22, 40]-=- and by the fact that servers have a limit on how many jobs can be assigned to them, the natural question of scheduling with a hard capacityconstraint of “at most bi jobs to be scheduled on each machi... |

17 |
Discrepancy in arithmetic progressions
- Matouˇsek, Spencer
- 1996
(Show Context)
Citation Context ...bounded matrix A; this, if true, would be bestpossible. Ingenious melding of randomized rounding, entropy-based arguments and the pigeonhole principle have helped show that lindisc(A) ≤ O( √ t log n) =-=[11, 35, 44]-=-, improved further to O( √ t log n) in [7]. However, the number of columns n may not be bounded as a function of t, and it would be very interesting to even get some o(t) bound on lindisc(A), to start... |

16 |
Graph balancing: a special case of scheduling unrelated parallel machines. Algorithmica 68
- Ebenlendr, Krčál, et al.
- 2014
(Show Context)
Citation Context |

15 | Discrepancy in arithmetic progressions
- Matousek, Spencer
- 1996
(Show Context)
Citation Context ...ounded matrix A; this, if true, would be best-possible. Ingenious melding of randomized rounding, entropy-based arguments and the pigeonhole principle have helped show that lindisc(A) ≤ O( √ t log n) =-=[11, 35, 44]-=-, improved further to O( √ t log n) in [7]. However, the number of columns n may not be bounded as a function of t, and it would be very interesting to even get some o(t) bound on lindisc(A), to start... |

13 |
1992], Asymptotic analysis of an algorithm for balanced parallel processor scheduling
- TSAI
(Show Context)
Citation Context ...rs have a limit on how many jobs can be assigned to them, the natural question of scheduling with a hard capacity-constraint of “at most bi jobs to be scheduled on each machine i” has been studied in =-=[50, 54, 53, 52, 19]-=-. Most recently, the work of [19] has shown that this problem can be approximated to within a factor of 3 in the special case where the machines are identical (job j has processing time pj on any mach... |

12 | Iterative Rounding for Multi-Objective Optimization Problems
- Grandoni, Ravi, et al.
- 2009
(Show Context)
Citation Context ...the same results under iterated rounding [16] and weighted dependent rounding [48]; and (iii) our ongoing work suggests that our random-walk approach improves upon the iterated-rounding-based work of =-=[30]-=- on bipartite matchings that are simultaneously “good” w.r.t. multiple linear objectives (this is related to, but not implied by, Theorem 2.1). We believe it would be very fruitful to understand possi... |

11 | Budgeted allocations in the full-information setting
- Srinivasan
(Show Context)
Citation Context ... Now we again do a kind of dependent rounding on this subgraph, where we additionally consider the utility of the items while modifying the assignment values on the edges. This is partly motivated by =-=[46]-=-. We remove all (i, j) that have already been rounded to 0 or 1. Let F ′ be the current graph consisting of those w ′ i,j that lie in (0, 1). Choose any maximal path P = (v0, v1, .., vs) or a cycle C ... |

10 | A unified approach to scheduling on unrelated parallel machines
- Kumar, Marathe, et al.
(Show Context)
Citation Context ...x∗ i . Independence can, however, lead to noticeable deviations from the mean for random variables that are required to be very close to (or even be equal to) their mean. A fruitful idea developed in =-=[27, 32, 45]-=- is to carefully introduce dependencies into the rounding process: in particular, some sums of random variables are held fixed with probability one, while still retaining randomness in the individual ... |

8 | Scheduling with outliers
- Gupta, Krishnaswamy, et al.
- 2009
(Show Context)
Citation Context |

8 | Recent advances in exact optimization of airline scheduling problems
- Rushmeier, Hoffman, et al.
- 1995
(Show Context)
Citation Context ...rd capacities” – those that cannot be violated – is generally tricky in various settings, including facilitylocation and other covering problems [19, 26, 36]. Motivated by problems in crew-scheduling =-=[22, 40]-=- and by the fact that servers have a limit on how many jobs can be assigned to them, the natural question of scheduling with a hard capacityconstraint of “at most bi jobs to be scheduled on each machi... |

7 | Resource minimization job scheduling
- Chuzhoy, Codenotti
(Show Context)
Citation Context ...roximation factor of O( √ k log 3 k); our bound is near-optimal since the integrality gap of the configuration LP is Ω( √ k) [8]. However, note that the recent work of Chakrabarty, Chuzhoy and Khanna =-=[20]-=- has improved the bound to mɛ . (Also note that m ≥ k.) Our main point is to show the applicability of our types of rounding approaches to this variation of the problem as well. In the context of fair... |

5 |
Convex quadratic and semidefinite relaxations in scheduling
- Skutella
(Show Context)
Citation Context |

4 |
An approximation algorithm for scheduling two parallel machines with capacity constraints
- Yang, Ye, et al.
(Show Context)
Citation Context ...rs have a limit on how many jobs can be assigned to them, the natural question of scheduling with a hard capacity-constraint of “at most bi jobs to be scheduled on each machine i” has been studied in =-=[50, 54, 53, 52, 19]-=-. Most recently, the work of [19] has shown that this problem can be approximated to within a factor of 3 in the special case where the machines are identical (job j has processing time pj on any mach... |

2 |
A comment on scheduling two parallel machines with capacity constraints
- Woeginger
- 2005
(Show Context)
Citation Context ...rs have a limit on how many jobs can be assigned to them, the natural question of scheduling with a hard capacity-constraint of “at most bi jobs to be scheduled on each machine i” has been studied in =-=[50, 54, 53, 52, 19]-=-. Most recently, the work of [19] has shown that this problem can be approximated to within a factor of 3 in the special case where the machines are identical (job j has processing time pj on any mach... |

2 | On the Budgeted MAXCUT problem and its Application to the Capacitated Two-Parallel
- Zhang, Ye
- 2001
(Show Context)
Citation Context |

1 |
Discrepancy theory, volume II, chapter 26
- Beck, Sós
- 1995
(Show Context)
Citation Context ...a q ∈ {0, 1} n such that ‖A · (q − p)‖∞ is “small”; the linear discrepancy of A is defined to be lindisc(A) = max p∈[0,1] n min q∈{0,1} n ‖A · (q − p)‖∞. The field of combinatorial discrepancy theory =-=[13]-=- has developed several classical results that bound lindisc(A) for various matrix families A; column-sparse matrices have received much attention in this regard. Section 6 discusses a concrete approac... |

1 |
Approximating scheduling machines with capacity constraints
- Chi, Wang, et al.
- 2009
(Show Context)
Citation Context ...gned to them, the natural question of scheduling with a hard capacityconstraint of “at most bi jobs to be scheduled on each machine i” has been studied in [18, 48, 50– 52]. Most recently, the work of =-=[18]-=- has shown that this problem can be approximated to within a factor of 3 in the special case where the machines are identical (job j has processing time pj on any machine). In § 2, we use our randomwa... |

1 |
Covering problems with hard constraints
- Chuzhoy, Naor
(Show Context)
Citation Context ..., random matchings with sharp tail bounds. Handling “hard capacities” – those that cannot be violated – is generally tricky in various settings, including facilitylocation and other covering problems =-=[19, 26, 36]-=-. Motivated by problems in crew-scheduling [22, 40] and by the fact that servers have a limit on how many jobs can be assigned to them, the natural question of scheduling with a hard capacityconstrain... |

1 |
Iterative rounding for multi-objective optimization problems
- Grandoni, Ravi, et al.
- 2009
(Show Context)
Citation Context ... same results under iterated rounding [16] and weighted dependent rounding [46]; and (iii) our ongoing work suggests that our random-walk ap16proach improves upon the iterated-rounding-based work of =-=[28]-=- on bipartite matchings that are simultaneously “good” w.r.t. multiple linear objectives (this is related to, but not implied by, Theorem 1). We believe it would be very fruitful to understand possibl... |

1 |
Dependent randomized rounding via exchange 29 of combinatorial structures
- Chekuri, Vondrák, et al.
- 2010
(Show Context)
Citation Context ...h that precedes these works. These dependent-rounding approaches lead to numerous improved approximation algorithms in scheduling, packet-routing and in several problems of combinatorial optimization =-=[1, 47, 29, 34, 18]-=-. We now introduce a fundamental scheduling model, which has spurred many advances and applications in combinatorial optimization, including linear-, quadratic& convex-programming relaxations and new ... |

1 |
Discrepancy theory, volumeII, chapter 26
- Beck, Sós
- 1995
(Show Context)
Citation Context ...wantaq ∈{0, 1} n such that ‖A · (q −p)‖∞ is “small”; the linear discrepancy of A is defined to be lindisc(A) =max p∈[0,1] n min q∈{0,1} n ‖A · (q − p)‖∞. The field of combinatorial discrepancy theory =-=[13]-=- has developed several classical results that bound lindisc(A) for various matrix families A; column-sparse matrices have received much attention in this regard. Section 6 discusses a concrete approac... |

1 |
On the approximability of trade-offs and optimal access of 356
- Papadimitriou, Yannakakis
- 2000
(Show Context)
Citation Context ... E) ofN vertices and a collection of k linear functions {fi} of x, many works have considered the problem of constructing (b-)matchings X such that fi(X) is “close” to fi(x) simultaneously for each i =-=[3,27,28,38]-=-. Theworks[28,38]focuson the case of constant k; those of [3, 27] consider general k, and require the usual “discrepancy” term of Ω( √ fi(x)logN) in|fi(X) − fi(x)| for most/all i; in a few cases, o(N)... |