## Strengthening Integrality Gaps for Capacitated Network Design and Covering Problems

Citations: | 63 - 1 self |

### BibTeX

@MISC{Carr_strengtheningintegrality,

author = {Robert D. Carr and et al.},

title = {Strengthening Integrality Gaps for Capacitated Network Design and Covering Problems},

year = {}

}

### Years of Citing Articles

### OpenURL

### Abstract

A capacitated covering IP is an integer program of the form min{cx|Ux ≥ d, 0 ≤ x ≤ b, x ∈ Z +}, where all entries of c, U, and d are nonnegative. Given such a formulation, the ratio between the optimal integer solution and the optimal solution to the linear program relaxation can be as bad as ||d||∞, even when U consists of a single row. We show that by adding additional inequalities, this ratio can be improved significantly. In the general case, we show that the improved ratio is bounded by the maximum number of non-zero coefficients in a row of U, and provide a polynomial-time approximation algorithm to achieve this bound. This improves the previous best approximation algorithm which guaranteed a solution within the maximum row sum times optimum. We also show that for particular instances of capacitated covering problems, including the minimum knapsack problem and the capacitated network design problem, these additional inequalities yield even stronger improvements in the IP/LP ratio. For the minimum knapsack, we show that this improved ratio is at most 2. This is the first non-trivial IP/LP ratio for this basic problem. Capacitated network design generalizes the classical network design problem by introducing capacities on the edges, whereas previous work only considers the case when all capacities equal 1. For capacitated network design problems, we show that this improved ratio depends on a parameter of the graph, and we also provide polynomial-time approximation algorithms to match this bound. This improves on the best previous m-approximation, where m is the number of edges in the graph. We also discuss improvements for some other special capacitated covering problems, including the fixed charge network flow problem. Finally, for the capacitated network design problem, we give some stronger results and algorithms for series parallel graphs and strengthen these further for outerplanar graphs. Most of our approximation algorithms rely on solving a single LP. When the original LP (before adding our strengthening inequalities) has a polynomial number of constraints, we describe a combinatorial FPTAS for the LP with our (exponentially-many) inequalities added. Our contribution here is to describe an appropriate

### Citations

1500 |
Reducibility among combinatorial problems
- Karp
- 1972
(Show Context)
Citation Context ...nected by the removal . Rdce andhNi f ofn andm verticesl (IP1) oGpq r@Q ce fR ce f s nt^ g4uvo LNMO i w ce fR ce f0UxhNi s e ^ ytuzRdce f ^{P Y X| b } pq Since the minimum knapsack problem is NP-hard =-=[22]-=-, all of the abovementioned problems are NP-hard problems. In this paper, we focus on obtaining improved approximation algorithms for these problems. -approximation algorithm is a polynomial-time algo... |

1177 |
Geometric Algorithms and Combinatorial Optimization, volume 2 of Algorithms and Combinatorics
- Grötschel, Lovász, et al.
- 1988
(Show Context)
Citation Context ...oximation algorithms depend on solving a single LP with an exponential number of constraints. Given a polynomial-time separation oracle, this can be done in polynomial time using the ellipsoid method =-=[15]-=-. When the original LP (before adding our exponentially-many KC inequalities) has a polynomial number of constraints, we describe a combinatorial FPTAS for solving the strengthened LP. We do this by d... |

599 | Approximation algorithms for NP-hard problems - Hochbaum - 1996 |

570 |
Knapsack problems. Algorithms and computer implementation
- Martello, Toth
- 1990
(Show Context)
Citation Context ...cant research on developing the techniques of integer programming and polyhedral combinatorics to attack these problems. For example, see [3, 7, 25]. The knapsack problem has been studied extensively =-=[27]-=-, and is one of the original NP-complete problems [22]. While the knapsack problem and the minimum knapsack problems are equivalent if an exact solution is sought, they are not equivalent for approxim... |

277 | Faster and simpler algorithms for multicommodity flow and other fractional packing problems
- Garg, Konemann
- 1998
(Show Context)
Citation Context ...ombinatorial FPTAS for solving the strengthened LP. We do this by describing an appropriate separation algorithm required by the FPTAS for positive packing and covering described by Garg and Könemann =-=[10]-=-. For solving the LP using the ellipsoid method, or simplex method, we describe simpler separation routines.Ô When there is only one demand pair, Schwarz and Krumke [32] describe an FPTAS on series p... |

247 |
Multiterminal Network Flows
- Gomory, Hu
- 1961
(Show Context)
Citation Context ... build the set of candidate integer solutions. We need onlycheck the lowest-capacity integer solution (last bucket) for feasibility. This can be done by using the polynomial-time Gomory-Hu algorithm =-=[14]-=- to determine the value of the minimum cut separating each pair of vertices, and comparing these values with the demand values. If some € cut has insufficient capacity, the set † ‚ ‡6ˆŠ‰ ‹ Œ Ž ‰ yield... |

237 | Fast approximation algorithms for fractional packing and covering problems
- Plotkin, Shmoys, et al.
- 1995
(Show Context)
Citation Context ...d, exact solution procedure. There exist polynomial-time, combinatorial methods for approximately solving LPs with special structure using separation oracles. For example, Plotkin, Shmoys, and Tardos =-=[30]-=- describe such a method for linear programs with all coefficients non-negative, and all inequalities (a packing LP), or all inequalities (a covering LP). Recently, Garg and Könemann [10] describe a si... |

205 | A factor 2 approximation algorithm for the generalized steiner network problem
- Jain
(Show Context)
Citation Context ... network design problem, wherew ce f …5| for all edgese , and multiple copies of each edge are allowed (although some do handle upper bounds on the number of copies of each edge). In particular, Jain =-=[21]-=- describes a 2-approximation for precisely this problem. Before [21], the best approximation guarantees obtained by polynomial-time algorithms for the uncapacitated network design problem were all log... |

191 |
Fast approximation algorithms for the knapsack and sum of subset problem
- Ibarra, Kim
- 1984
(Show Context)
Citation Context ...ught, they are not equivalent for approximation purposes in that -approximation algorithm for one problem does not imply the existence of a comparable guarantee for the second. The FPTAS for knapsack =-=[24, 19]-=- can be easily modified to work for min knapsack. However, the bound on the IP/LP ratios for the two problems is vastly different: 2 for a~ versush knapsack for minimum knapsack. 1.2 Our results. Almo... |

122 | The primal-dual method for approximation algorithms and its application to network design problems
- Goemans, Williamson
- 1995
(Show Context)
Citation Context ...imation guarantees obtained by polynomial-time algorithms for the uncapacitated network design problem were all logarithmic in†‡u…$Sˆ S . For references, and a survey of related work, see for example =-=[13]-=-. When all edge costs are also uniform, the problem remains NP-hard, even when all demands are also uniform. The best known approximation in this case is‰2Š[| ‚ ‹ when connectivity requirement is [23]... |

98 | Buy-at-bulk network design
- Awerbuch, Azar
- 1997
(Show Context)
Citation Context ...capacitated network design problem when the objective is to design a network with enough capacity to route all demands simultaneously, without any restriction on the number of copies of edges allowed =-=[1, 6, 26, 31]-=-. There has also been significant research on developing the techniques of integer programming and polyhedral combinatorics to attack these problems. For example, see [3, 7, 25]. The knapsack problem ... |

89 |
Fast approximation algorithms for knapsack problems
- Lawler
- 1979
(Show Context)
Citation Context ...ught, they are not equivalent for approximation purposes in that -approximation algorithm for one problem does not imply the existence of a comparable guarantee for the second. The FPTAS for knapsack =-=[24, 19]-=- can be easily modified to work for min knapsack. However, the bound on the IP/LP ratios for the two problems is vastly different: 2 for a~ versush knapsack for minimum knapsack. 1.2 Our results. Almo... |

85 | Approximating a finite metric by a small number of tree metrics
- Charikar, Chekuri, et al.
- 1998
(Show Context)
Citation Context ...capacitated network design problem when the objective is to design a network with enough capacity to route all demands simultaneously, without any restriction on the number of copies of edges allowed =-=[1, 6, 26, 31]-=-. There has also been significant research on developing the techniques of integer programming and polyhedral combinatorics to attack these problems. For example, see [3, 7, 25]. The knapsack problem ... |

81 | Improved approximation algorithms for network design problems
- Goemans, Goldberg, et al.
- 1994
(Show Context)
Citation Context ...m for the capacitated network design problem is the algorithm that greedily removes the unnecessary edges in order of decreasing cost. This finds a solution within a factor ofƒ„u…tSy S of the optimum =-=[12]-=-. Many of the approximation algorithms for network design problems that achieve approximation guarantees that are better than linear consider the uncapacitated network design problem, wherew ce f …5| ... |

77 |
Facets of the Knapsack Polytope
- Balas
- 1975
(Show Context)
Citation Context ...nt is enforced by a knapsack cover (KC) inequality: ¿Ò¸ÊÈ=¿Ó º Ñ ¿NÍÀtÎ@ÏÐ Ì(º (2.1) Under certain conditions, these inequalities are facet defining, see [33] for example. Previous researchers =-=[2, 17, 33]-=- have considered an uncapacitated form of inequality (2.1) that forces the choice of at least one edge in . We can show that if only NÇÈ ÕÖ ×dØÌ Ì(º ¿Æ º ¿aÙ[¸ È=¿Ú these weaker constraints are ... |

74 | The Steiner problem with edge lengths 1 and 2
- Bern, Plassmann
- 1989
(Show Context)
Citation Context ...mes the optimal solution in time polynomial in the size of the and| problem, . One special case of the capacitated network design problem is the Steiner tree problem, which is known to be MAXSNP-hard =-=[4]-=-. Thus we cannot hope to find an FPTAS for this problem. | Most of our approximation algorithms depend on ‚ strengthening the LP relaxation of the given IP. (The LP relaxation is the problem obtained ... |

62 | Approximation algorithms for finding highly connected subgraphs
- Khuller
- 1996
(Show Context)
Citation Context ...[13]. When all edge costs are also uniform, the problem remains NP-hard, even when all demands are also uniform. The best known approximation in this case is‰2Š[| ‚ ‹ when connectivity requirement is =-=[23]-=-. If, in addition, the underlying graph is the complete graph and multiple edges are allowed, then the problem is solvable in polynomial time [8, 28]. Other researchers have considered approximation a... |

62 |
Modeling and solving the two–facility capacitated network loading problem
- Magnanti, Mirchandani, et al.
- 1995
(Show Context)
Citation Context ...s of edges allowed [1, 6, 26, 31]. There has also been significant research on developing the techniques of integer programming and polyhedral combinatorics to attack these problems. For example, see =-=[3, 7, 25]-=-. The knapsack problem has been studied extensively [27], and is one of the original NP-complete problems [22]. While the knapsack problem and the minimum knapsack problems are equivalent if an exact ... |

57 | The Steiner problem with edge lengths 1 - Bern, Plassmann - 1989 |

53 |
Faces for a linear inequality in 0-1 variables
- WOLSEY
- 1975
(Show Context)
Citation Context ...e demand. Let . The subproblem constraint is enforced by a knapsack cover (KC) inequality: ¿Ò¸ÊÈ=¿Ó º Ñ ¿NÍÀtÎ@ÏÐ Ì(º (2.1) Under certain conditions, these inequalities are facet defining, see =-=[33]-=- for example. Previous researchers [2, 17, 33] have considered an uncapacitated form of inequality (2.1) that forces the choice of at least one edge in . We can show that if only NÇÈ ÕÖ ×dØÌ Ì(º ¿... |

50 | Valid linear inequalities for fixed charge problems - Padberg, Roy, et al. - 1985 |

45 |
Network design using cut inequalities
- Barahona
- 1996
(Show Context)
Citation Context ...s of edges allowed [1, 6, 26, 31]. There has also been significant research on developing the techniques of integer programming and polyhedral combinatorics to attack these problems. For example, see =-=[3, 7, 25]-=-. The knapsack problem has been studied extensively [27], and is one of the original NP-complete problems [22]. While the knapsack problem and the minimum knapsack problems are equivalent if an exact ... |

39 | A cutting plane algorithm for multicommodity survivable network design problems
- Dahl, Stoer
- 1998
(Show Context)
Citation Context ...s of edges allowed [1, 6, 26, 31]. There has also been significant research on developing the techniques of integer programming and polyhedral combinatorics to attack these problems. For example, see =-=[3, 7, 25]-=-. The knapsack problem has been studied extensively [27], and is one of the original NP-complete problems [22]. While the knapsack problem and the minimum knapsack problems are equivalent if an exact ... |

34 |
Buy-at-bulk network design: Approximating the single-sink edge installation problem
- Salman, Cheriyan, et al.
- 1997
(Show Context)
Citation Context ...capacitated network design problem when the objective is to design a network with enough capacity to route all demands simultaneously, without any restriction on the number of copies of edges allowed =-=[1, 6, 26, 31]-=-. There has also been significant research on developing the techniques of integer programming and polyhedral combinatorics to attack these problems. For example, see [3, 7, 25]. The knapsack problem ... |

28 |
Facets of regular 0-1 polytopes
- Hammer, Johnson, et al.
- 1975
(Show Context)
Citation Context ...nt is enforced by a knapsack cover (KC) inequality: ¿Ò¸ÊÈ=¿Ó º Ñ ¿NÍÀtÎ@ÏÐ Ì(º (2.1) Under certain conditions, these inequalities are facet defining, see [33] for example. Previous researchers =-=[2, 17, 33]-=- have considered an uncapacitated form of inequality (2.1) that forces the choice of at least one edge in . We can show that if only NÇÈ ÕÖ ×dØÌ Ì(º ¿Æ º ¿aÙ[¸ È=¿Ú these weaker constraints are ... |

22 | An approximation algorithm for minimum-cost network design
- Mansour, Peleg
- 1994
(Show Context)
Citation Context |

21 | Faces for a Linear Inequality - Wolsey - 1975 |

19 | Complexity and Security of Distributed Protocols
- Franklin
- 1993
(Show Context)
Citation Context ...xample, suppose we have a communications network and each edge has a weight (capacity) corresponding to the cost an attacker incurs to eavesdrop on that edge. There are communications protocols (e.g. =-=[9]-=-) where messages are 1broken into multiple packets and packets are sent along many different paths. In order for an eavesdropper to glean any information from the message, he must intercept all packe... |

19 |
A fast algorithm for optimally increasing the edge connectivity
- Naor, Gusfield, et al.
- 1997
(Show Context)
Citation Context ...his case is‰2Š[| ‚ ‹ when connectivity requirement is [23]. If, in addition, the underlying graph is the complete graph and multiple edges are allowed, then the problem is solvable in polynomial time =-=[8, 28]-=-. Other researchers have considered approximation algo‹ rithms for the capacitated network design problem when the objective is to design a network with enough capacity to route all demands simultaneo... |

17 | Rounding algorithms for covering problems - Bertsimas, Vohra - 1998 |

17 |
Network Models in Optimization and their Applications in Practice
- Glover, Klingman, et al.
- 1992
(Show Context)
Citation Context ...rastructures are physical networks such as telecommunications, water, natural gas, and transportation. The importance of network design is further evident from the wealth of literature on the subject =-=[11]-=-. The research in this paper is motivated by a capacitated network design problem arising in network security. However, the tools we have devised to understand this problem are applicable not only to ... |

17 |
A fast approximation algorithm for the multicovering problem
- Hall, Hochbaum
- 1986
(Show Context)
Citation Context ... ž ,Ÿ of , and are Let¥ nonnegative. be the maximum number of nonzero entries in a row Ÿ of let¦ and be the maximum row ofŸ sum IfŸ . a¢6§ ” matrix,¥x¨ ¦ is , but in ¥l•w¦ general . Hall and Hochbaum =-=[16]-=- give ¥ a -approximation Ÿ for a 0-1 matrix. Bertsimas and Vohra [5] extend this to give ¦ a -approximation for the general problem. In this section, we describe a ¥ -approximation for general, nonneg... |

16 |
Valid inequalities for fixed charge problems
- Padberg, Roy, et al.
- 1985
(Show Context)
Citation Context ... Û-Õ Ö ×=Ù Ð Ñ Ò-ú ø ù é ù Õ Ö × Ø=Õ Ö × ÝAßlò é=Õ ñ × í Ð ñ × óAå í=ã ô=û'ã õ ü ý/ð ßlòxé=Õ A different form of flow cover inequalities were introduced in a polyhedral study of fixed charge problems =-=[29]-=-. They presented them as packing, not covering, inequalities; and they did not consider their effect at tightening the IP/LP ratio. THEOREM 2.10. The integrality gap for 2-node fixedcharge network flo... |

12 |
Augmenting graphs to meet edge connectivity requirements
- Frank
- 1992
(Show Context)
Citation Context ...his case is‰2Š[| ‚ ‹ when connectivity requirement is [23]. If, in addition, the underlying graph is the complete graph and multiple edges are allowed, then the problem is solvable in polynomial time =-=[8, 28]-=-. Other researchers have considered approximation algo‹ rithms for the capacitated network design problem when the objective is to design a network with enough capacity to route all demands simultaneo... |

2 | On budget constrained flow improvement
- Schwarz, Krumke
- 1998
(Show Context)
Citation Context ...escribed by Garg and Könemann [10]. For solving the LP using the ellipsoid method, or simplex method, we describe simpler separation routines.Ô When there is only one demand pair, Schwarz and Krumke =-=[32]-=- describe an FPTAS on series parallel graphs, if this demand pair corresponds to the defining nodes of the series parallel graph. An outerplanar graph is a planar graph that can be embedded so that al... |

1 | to � � and �¦� � � demand to - demand |