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## Approximability of sparse integer programs (2009)

Venue: | In Proc. 17th ESA |

Citations: | 9 - 1 self |

### Citations

748 | Some optimal inapproximability results - HÅSTAD |

427 |
Knapsack problems
- Kellerer, Pferschy, et al.
- 2004
(Show Context)
Citation Context ... pseudopolynomial-time algorithm and a PTAS (both still hold even with multiplicity constraints), but no FPTAS for any constant k ≥ 2 unless P=NP (this was originally shown for d = 1 around 1980 (see =-=[25]-=-) and subsequently even for infinite or arbitrary d in [37]). One simple PTAS is as follows: guess the t most profitable items (counting multiplicity) in the knapsack, then solve the natural LP relaxa... |

313 | Integer programming with a fixed number of variables
- LENSTRA
- 1983
(Show Context)
Citation Context ...nvestigate the following problem: what is the best possible approximation ratio for integer programs where the constraint matrix is sparse? To put this in context we recall a famous result of Lenstra =-=[35]-=-: integer programs with a constant number of variables or a constant number of constraints can be solved in polynomial time. Our investigations analogously ask what is possible if the constraints each... |

149 | Vertex cover might be hard to approximate to within 2-epsilon
- Khot, Regev
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Citation Context ... k-Bounded Hypergraph Min Vertex Cover (a.k.a. Set Cover with maximum frequency k) which is not approximable to k − 1 − ǫ for any fixed ǫ > 0 unless P=NP [12] (k − ǫ under the unique games conjecture =-=[27]-=-). This special case is known to admit a matching positive result: set cover with maximum frequency k can be k-approximated by direct rounding of the naive LP [20] or local ratio/primal-dual methods [... |

146 |
Primal-dual approximation algorithms for integral flow and multicuts in trees
- Garg, Yannakakis, et al.
- 1997
(Show Context)
Citation Context ... To make Theorem 6 more concrete, notice this implies a 1+O(k/W)-approximation for W > 1.01k. This is tight in the sense that for any fixed k ≥ 4, 1 + o(1/W)-approximation is NP-hard, by results from =-=[15, 30]-=- on approximating multicommodity flows in trees. The previous general frameworks give approximation guarantees — [11] gives 1 + O(k/ √ W) for W = Ω(k 2 ) and [30] gives (W + k)/(W + k − k 2 ) — but bo... |

142 |
Approximation Algorithms for the Set Covering and Vertex Cover Problems
- Hochbaum
- 1982
(Show Context)
Citation Context ... ǫ under the unique games conjecture [27]). This special case is known to admit a matching positive result: set cover with maximum frequency k can be k-approximated by direct rounding of the naive LP =-=[20]-=- or local ratio/primal-dual methods [4]. The following results are known for other special cases of k-RS CIP with multiplicity constraints: Hochbaum [17] gave a k-approximation in the special case tha... |

115 |
An analysis of the greedy algorithm for the submodular set covering problem
- Wolsey
- 1982
(Show Context)
Citation Context ...gave a O(k)-approximation algorithm. It is not clear whether anyone has considered the problem of submodular k-CS CIPs. Some work has been done already on submodular set cover, for example see Wolsey =-=[45]-=-. One might hope to use results of Kolliopoulos & Young [29] (see also the references therein including work by Srinivasan) to get an O(ln k) approximation for this problem. 1.4 Summary We summarize t... |

95 |
On the size of systems of sets every t of which have an SDR, with an application to the worst-case ratio of heuristics for packing problems
- Hurkens, Schrijver
- 1989
(Show Context)
Citation Context ...ges of size ≤ k, and strong independent sets in hypergraphs with degree at most k. The best approximation ratio known for this problem is (k + 1)/2 + ǫ [6] for general weights, and k/2 + ǫ when c = 1 =-=[23]-=-. For k = 3 a 2-approximation in the weighted case is known due to Lau and Chan citeCL09. The best lower bound is due to Hazan et al. [19], who showed Ω(k/ lnk)-inapproximability unless P=NP, even for... |

90 | Non-approximability results for optimization problems on bounded degree instances
- Trevisan
- 2001
(Show Context)
Citation Context ...dle multiplicity constraints. There is a matching hardness result: it is NP-hard to approximate k-Set Cover, which is the special case where A, b, c are 0-1, better than lnk − O(ln lnk) for any k ≥ 3 =-=[44]-=-. Hence for k-CS CIP the best possible approximation ratio is Θ(log k). A (k + ǫ)approximation algorithm can be obtained by separately applying an approximation scheme to the knapsack problem correspo... |

84 | Strengthening integrality gaps for capacitated network design and covering problems
- Carr, Fleischer, et al.
- 2000
(Show Context)
Citation Context ...ion algorithms for the case that k = 2 and d is finite. For the special case d = 1, Carr et al. [8, §2.6] gave a k-approximation, and Fujito & Yabuta [14] gave a primal-dual k-approximation. Moreover =-=[8, 14]-=- claim a k-approximation for general d, but there seems to have been some oversights as the papers do not provide full proofs and their methods alone seem to be insufficient for general d. Briefly, [8... |

70 | Better inapproximability results for maxclique, chromatic number and min-3lin-deletion
- Khot, Ponnuswami
- 2006
(Show Context)
Citation Context ...∞. See Appendix A for details. The special case of 2-RS PIP where A, b, c are 0-1 is the same as Max Independent Set, which is not approximable within n/2 log3/4+ǫ n unless NP ⊂ BPTIME(2 log O(1) n ) =-=[26]-=-. On the other hand, n-approximation of any packing problem is easy to accomplish by looking at the best singleton-support solution. A slightly better n/t-approximation, for any fixed t, can be accomp... |

55 | A new multilayered PCP and the hardness of hypergraph vertex cover
- Dinur, Guruswami, et al.
(Show Context)
Citation Context ...d. More generally, 0-1 k-RS CIP is the same as k-Bounded Hypergraph Min Vertex Cover (a.k.a. Set Cover with maximum frequency k) which is not approximable to k − 1 − ǫ for any fixed ǫ > 0 unless P=NP =-=[12]-=- (k − ǫ under the unique games conjecture [27]). This special case is known to admit a matching positive result: set cover with maximum frequency k can be k-approximated by direct rounding of the naiv... |

54 |
On the Advantage of Network Coding for Improving Network Throughput
- Agarwal, Charikar
- 2004
(Show Context)
Citation Context ...9] for an alternate characterization of the integrality gap. For packing integer programs there are analogous definitions of the integrality gap. Integrality gaps are studied in their own right (e.g. =-=[1]-=- where the integrality gap of the bidirected cut relaxation turns out to exactly equal a type of coding advantage) but also because they are connected to the following common concept in the literature... |

48 | Improved approximation guarantees for packing and covering integer programs
- Srinivasan
- 1999
(Show Context)
Citation Context ...es: we get a 4-approximation when k = 2, and we get a (W +k)/(W − k)-approximation when the program’s width, defined as W := mini,j:Aij =0 bi Aij satisfies W > k. 21.3 Other Related Work Srinivasan =-=[25, 26]-=- showed that k-CS CIPs admit a O(log k)-approximation. Kolliopoulos and Young [19] extended this result to handle multiplicity constraints. There is a matching hardness result: it is NP-hard to approx... |

47 |
A d/2 approximation for maximum weight independent set in d-claw free graphs. Algorithm Theory-SWAT 2000
- Berman
- 2000
(Show Context)
Citation Context ...eight k-set packing, hypergraph matching with edges of size ≤ k, and strong independent sets in hypergraphs with degree at most k. The best approximation ratio known for this problem is (k + 1)/2 + ǫ =-=[6]-=- for general weights, and k/2 + ǫ when c = 1 [23]. For k = 3 a 2-approximation in the weighted case is known due to Lau and Chan citeCL09. The best lower bound is due to Hazan et al. [19], who showed ... |

47 | Submodular function minimization under covering constraints
- Iwata, Nagano
- 2009
(Show Context)
Citation Context ...odular objective subject to k-row sparse covering constraints, three k-approximations were recently published, one in the framework of Koufogiannakis & Young [31, 34, 32, 33], one by Iwata and Nagano =-=[24]-=-, and one by Goel et al. [16]. For maximizing a monotone submodular function subject to k-column sparse packing constraints, Bansal et al. [3] gave a O(k)-approximation algorithm. It is not clear whet... |

45 | Tight bounds and 2-approximation algorithms for integer programs with two variables per inequality
- Hochbaum, Megiddo, et al.
- 1993
(Show Context)
Citation Context ...2]. The following results are known for other special cases of k-RS CIP with multiplicity constraints: Hall and Hochbaum [10] gave a k-approximation in the special case that A is 0-1; Hochbaum et al. =-=[14]-=- and Bar-Yehuda & Rawitz [1] gave pseudopolynomial 2-approximation algorithms for the case that k = 2 and d is finite. For the special case d = 1, Carr et al. [3, §2.6] gave a k-approximation, and Fuj... |

44 | Improved approximation algorithms for unsplittable flow problems
- Kolliopoulos, Stein
- 1997
(Show Context)
Citation Context ... k/W)-approximation algorithm. Srinivasan [42, 43] gave a (1 + ln(1 + k)/W)-approximation algorithm for unbounded k-CS CIPs. Using grouping and scaling techniques introduced by Kolliopoulos and Stein =-=[28]-=-, Chekuri et al. [11] showed that no-bottleneck demand multicommodity flow in a tree admits a (1+O(1/ √ W))-approximation algorithm, and gave general sufficient conditions for a problem to admit a (1 ... |

39 |
Two np-complete problems in non-negative integer programming
- Lueker
- 1975
(Show Context)
Citation Context ...om 3-Partition; but for IPs where each variable appears at most twice including in upper/lower bounds, it appears all that is known is NP-completeness (for example, via the unbounded knapsack problem =-=[36]-=-). Acknowledgement. We would like to thank Glencora Borradaile, Christina Boucher, Deeparnab Chakrabarty, Stephane Durocher, Jochen Könemann and Christos Koufogiannakis for helpful discussions, and th... |

38 | An Extension of the Lovász Local Lemma and its applications to integer programming
- Srinivasan
- 1996
(Show Context)
Citation Context ....2. Finally, Bansal, Korula, and Nagarajan [3] gave a simple and elegant O(k)-approximation algorithm based on randomized rounding with a careful sort of alteration. 1.3 Other Related Work Srinivasan =-=[42, 43]-=- showed that k-CS CIPs admit a O(log k)-approximation. Kolliopoulos and Young [29] extended this result to handle multiplicity constraints. There is a matching hardness result: it is NP-hard to approx... |

37 |
Multicommodity demand flow in a tree and packing integer programs
- Chekuri, Mydlarz, et al.
(Show Context)
Citation Context ...mand matching is APX-hard but admits a ( 11 2 − √ 5)-approximation algorithm when d = 1; their approach also gives a 7 2-approximation for 2-CS PIP instances satisfying (i). Results of Chekuri et al. =-=[11]-=- yield a 11.542k-approximation algorithm for k-CS PIP instances satisfying (i) and such that the maximum entry of A is less than the minimum entry of b. The special case of k-CS PIP where A, b are 0-1... |

35 | O.: On the complexity of approximating k-set packing
- Hazan, Safra, et al.
- 2006
(Show Context)
Citation Context ...(k + 1)/2 + ǫ [6] for general weights, and k/2 + ǫ when c = 1 [23]. For k = 3 a 2-approximation in the weighted case is known due to Lau and Chan citeCL09. The best lower bound is due to Hazan et al. =-=[19]-=-, who showed Ω(k/ lnk)-inapproximability unless P=NP, even for c = 1. Our second main result, given in Section 3, is the following result. Theorem 2. There is a polynomial time 2 k (k 2 − 2k + 1) + 1-... |

34 | On the approximability of budgeted allocations and improved lower bounds for submodular welfare maximization and GAP
- Chakrabarty, Goel
(Show Context)
Citation Context ...e NP-hard to optimize and gave a bicriteria approximation algorithm. Although 0-1 2-CS CIP is Edge Cover which lies in P, 2-CS CIP in general is NP-hard to (17/16 −ǫ)-approximate, due to methods from =-=[10]-=-, even if A has 2 equal nonzeroes per column and d is 0-1 or d is all-+∞. See Appendix A for details. The special case of 2-RS PIP where A, b, c are 0-1 is the same as Max Independent Set, which is no... |

32 | Approximability of combinatorial problems with multi-agent submodular cost functions
- Goel, Karande, et al.
- 2009
(Show Context)
Citation Context ...-row sparse covering constraints, three k-approximations were recently published, one in the framework of Koufogiannakis & Young [31, 34, 32, 33], one by Iwata and Nagano [24], and one by Goel et al. =-=[16]-=-. For maximizing a monotone submodular function subject to k-column sparse packing constraints, Bansal et al. [3] gave a O(k)-approximation algorithm. It is not clear whether anyone has considered the... |

22 | Approximation algorithms for covering/packing integer programs.Journal of Computer and System
- Kolliopoulos, Young
- 2004
(Show Context)
Citation Context ...ation algorithm based on randomized rounding with a careful sort of alteration. 1.3 Other Related Work Srinivasan [42, 43] showed that k-CS CIPs admit a O(log k)-approximation. Kolliopoulos and Young =-=[29]-=- extended this result to handle multiplicity constraints. There is a matching hardness result: it is NP-hard to approximate k-Set Cover, which is the special case where A, b, c are 0-1, better than ln... |

21 | A logarithmic approximation for unsplittable flow on line graphs
- Bansal, Friggstad, et al.
- 2009
(Show Context)
Citation Context ... for any α < α0. (In such cases one can still, in principle, either use a different LP as was done in [38, 13] for the edge dominating set problem, or abandon LP-relative approximation as was done in =-=[2]-=- for the demand interval packing problem.) Unbounded problems. In discussing integrality gaps for k-RS CIP problems, we say that the naive LP relaxation of {mincx | x integral, Ax ≥ b,0 ≤ x ≤ d} is th... |

17 |
A 2-approximation algorithm for the minimum weight edge dominating set problem
- Fujito, Nagamochi
- 2002
(Show Context)
Citation Context ...L) ≥ α0, this is evidence that naive LP-based approximation methods cannot give an α-approximation for any α < α0. (In such cases one can still, in principle, either use a different LP as was done in =-=[38, 13]-=- for the edge dominating set problem, or abandon LP-relative approximation as was done in [2] for the demand interval packing problem.) Unbounded problems. In discussing integrality gaps for k-RS CIP ... |

16 | On k-column sparse packing programs
- Bansal, Korula, et al.
- 2010
(Show Context)
Citation Context ...nd D. Chakrabarty. The idea (which was the same for both groups) is a natural modification of the iterated rounding framework, and we present it in Section 3.2. Finally, Bansal, Korula, and Nagarajan =-=[3]-=- gave a simple and elegant O(k)-approximation algorithm based on randomized rounding with a careful sort of alteration. 1.3 Other Related Work Srinivasan [42, 43] showed that k-CS CIPs admit a O(log k... |

16 |
A fast approximation algorithm for the multicovering problem, Discrete Applied Mathematics 15
- Hall, Hochbaum
- 1986
(Show Context)
Citation Context ...pproximated by direct rounding of the naive LP [20] or local ratio/primal-dual methods [4]. The following results are known for other special cases of k-RS CIP with multiplicity constraints: Hochbaum =-=[17]-=- gave a k-approximation in the special case that A is 0-1; Hochbaum et al. [22] and BarYehuda & Rawitz [5] gave pseudopolynomial 2-approximation algorithms for the case that k = 2 and d is finite. For... |

15 | Approximate formulations for 0-1 knapsack sets
- Bienstock
- 2008
(Show Context)
Citation Context ... Ax ≥ b, 0 ≤ x ≤ d} where A has k rows behave similarly. An LP relaxation with 1 + ǫ integrality gap for k-dimensional knapsack can be obtained using disjunctive programming as described by Bienstock =-=[7]-=-. 1.3.2 Semimodular Optimization The results mentioned here all assume d = 1. Interestingly they all developed in mid-2009 (excepting Koufogiannakis and Young’s arXiv report in late 2008). (This secti... |

14 |
A Note on Approximation Schemes for Multidimensional Knapsack Problems
- Magazine, Chern
- 1984
(Show Context)
Citation Context ...d even with multiplicity constraints), but no FPTAS for any constant k ≥ 2 unless P=NP (this was originally shown for d = 1 around 1980 (see [25]) and subsequently even for infinite or arbitrary d in =-=[37]-=-). One simple PTAS is as follows: guess the t most profitable items (counting multiplicity) in the knapsack, then solve the natural LP relaxation of the residual problem and round all variables down t... |

11 | Distributed and parallel algorithms for weighted vertex cover and other covering problems
- Koufogiannakis, Young
- 2009
(Show Context)
Citation Context ...onstraints (see Definition 5) with better rounding properties. The algorithm requires a polynomial-time linear programming subroutine. Independently of our work, recent work of Koufogiannakis & Young =-=[31, 34, 32, 33]-=- also gives a full and correct proof of Theorem 1. Their primal-iterative approach works for a broad generalization of k-RS CIPs and runs in low-degree strongly polynomial time. Our approach has the g... |

11 | The Demand-Matching Problem
- Shepherd, Vetta
- 2007
(Show Context)
Citation Context ...e special case of 2-CS PIP where (i) in each column of A all nonzero values in that column are equal to one another and (ii) no two columns have their nonzeroes in the same two rows. Shepherd & Vetta =-=[40]-=- showed demand matching is APX-hard but admits a ( 11 2 − √ 5)-approximation algorithm when d = 1; their approach also gives a 7 2-approximation for 2-CS PIP instances satisfying (i). Results of Cheku... |

10 |
Monotonizing linear programs with up to two nonzeroes per column
- Hochbaum
(Show Context)
Citation Context ...ble approximation ratio is Θ(log k). A (k + ǫ)approximation algorithm can be obtained by separately applying an approximation scheme to the knapsack problem corresponding to each constraint. Hochbaum =-=[21]-=- showed 2-CS CIPs are NP-hard to optimize and gave a bicriteria approximation algorithm. Although 0-1 2-CS CIP is Edge Cover which lies in P, 2-CS CIP in general is NP-hard to (17/16 −ǫ)-approximate, ... |

9 | Efficient algorithms for integer programs with two variables per constraint 1
- Bar-Yehuda, Rawitz
(Show Context)
Citation Context ...esults are known for other special cases of k-RS CIP with multiplicity constraints: Hochbaum [17] gave a k-approximation in the special case that A is 0-1; Hochbaum et al. [22] and BarYehuda & Rawitz =-=[5]-=- gave pseudopolynomial 2-approximation algorithms for the case that k = 2 and d is finite. For the special case d = 1, Carr et al. [8, §2.6] gave a k-approximation, and Fujito & Yabuta [14] gave a pri... |

9 | Distributed fractional packing and maximum weighted b-matching via tail-recursive duality. DISC
- Koufogiannakis, Young
- 2009
(Show Context)
Citation Context ...onstraints (see Definition 5) with better rounding properties. The algorithm requires a polynomial-time linear programming subroutine. Independently of our work, recent work of Koufogiannakis & Young =-=[31, 34, 32, 33]-=- also gives a full and correct proof of Theorem 1. Their primal-iterative approach works for a broad generalization of k-RS CIPs and runs in low-degree strongly polynomial time. Our approach has the g... |

8 | Iterative methods in combinatorial optimization
- SINGH
(Show Context)
Citation Context ...wing result. Theorem 2. There is a polynomial time 2 k (k 2 − 2k + 1) + 1-approximation algorithm for k-CS PIPs with multiplicity constraints. 2Our methodology begins by using iterated LP relaxation =-=[41]-=- to find an integral solution with superoptimal value, but violating some constraints in an additively-bounded way. Then we use a combination of probabilistic and greedy methods to recover a high-weig... |

7 |
S.: Randomized meta-rounding. Random Struct
- Carr, Vempala
- 2002
(Show Context)
Citation Context ...he integrality gap is the ratio of the cost of the optimal integral solution to cost of the optimal LP solution. For a given linear program L let Γ(L) denote its 6integrality gap; note Γ(L) ≥ 1. See =-=[9]-=- for an alternate characterization of the integrality gap. For packing integer programs there are analogous definitions of the integrality gap. Integrality gaps are studied in their own right (e.g. [1... |

6 | N.E.: Greedy δ-approximation algorithm for covering with arbitrary constraints and submodular cost
- Koufogiannakis, Young
- 2009
(Show Context)
Citation Context ...onstraints (see Definition 5) with better rounding properties. The algorithm requires a polynomial-time linear programming subroutine. Independently of our work, recent work of Koufogiannakis & Young =-=[31, 34, 32, 33]-=- also gives a full and correct proof of Theorem 1. Their primal-iterative approach works for a broad generalization of k-RS CIPs and runs in low-degree strongly polynomial time. Our approach has the g... |

6 | Polyhedral techniques for graphic covering problems
- Parekh
- 2002
(Show Context)
Citation Context ...L) ≥ α0, this is evidence that naive LP-based approximation methods cannot give an α-approximation for any α < α0. (In such cases one can still, in principle, either use a different LP as was done in =-=[38, 13]-=- for the edge dominating set problem, or abandon LP-relative approximation as was done in [2] for the demand interval packing problem.) Unbounded problems. In discussing integrality gaps for k-RS CIP ... |

6 | An improved randomized approximation algorithm for maximum triangle packing
- Chen, Wang
- 2009
(Show Context)
Citation Context ... for this problem is (k+1)/2+ǫ [2] for general weights, and k/2+ǫ when c = 1 [15] (improvements exist for specific small values of k, e.g. a 1.902-approximation algorithm for min-weight 3-set packing =-=[6]-=-). The best lower bound is due to Hazan et al. [11], who showed Ω(k/ lnk)-inapproximability unless P=NP. Our second main result, given in Section ??, is the following result. Theorem 2. There is a pol... |

3 | Max-weight integral multicommodity flow in spiders and high-capacity trees
- Könemann, Parekh, et al.
- 2011
(Show Context)
Citation Context .../ √ W))-approximation algorithm, and gave general sufficient conditions for a problem to admit a (1 + O(1/ √ W))-approximation algorithm. Along the same vein, using iterated rounding, Könemann et al. =-=[30]-=- obtained a (1+O(1/W))-approximation algorithm for ordinary multicommodity flow in a tree, and general sufficient conditions for a problem to admit a (1 + O(1/W))-approximation algorithm [30]. Whereas... |

2 |
T.: Submodular integer cover and its application to production planning
- Fujito, Yabuta
- 2004
(Show Context)
Citation Context ...da & Rawitz [5] gave pseudopolynomial 2-approximation algorithms for the case that k = 2 and d is finite. For the special case d = 1, Carr et al. [8, §2.6] gave a k-approximation, and Fujito & Yabuta =-=[14]-=- gave a primal-dual k-approximation. Moreover [8, 14] claim a k-approximation for general d, but there seems to have been some oversights as the papers do not provide full proofs and their methods alo... |

1 |
Flooding overcomes small covering constraints
- Koufogiannakis, Young
- 2008
(Show Context)
Citation Context |