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## A Threshold of ln n for Approximating Set Cover (1998)

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Venue: | JOURNAL OF THE ACM |

Citations: | 767 - 5 self |

### Citations

1233 | The Knowledge Complexity of Interactive Proof-Systems
- Goldwasser, Micali, et al.
- 1989
(Show Context)
Citation Context ...s.) The first hardness of approximation results for set cover followed from work on probabilistically checkable proof systems (PCPs). The notion of PCPs grew out of the theory 2 of interactive proofs =-=[14, 4, 6, 13]-=- (parts of which we will review shortly) and from major breakthroughs in understanding their power [25, 32, 3]. The relevance of interactive proofs for proving hardness of approximation results was de... |

821 |
Approximation algorithms for combinatorial problems
- Johnson
- 1974
(Show Context)
Citation Context ...2), designed specifically for this purpose. Our proof technique extends that of [26]. 1.1 Related work Set cover was among the first problems for which approximation algorithms were analysed. Johnson =-=[23]-=- showed that the greedy algorithm gives an approximation ratio of ln n. (This was extended by Chvatal [7] to the weighted version of set cover.) Lovasz [24] showed that a linear programming relaxation... |

792 | Proof verification and hardness of approximation problems
- Arora, Lund, et al.
- 1992
(Show Context)
Citation Context ...over can be approximated within a ratio of ln n, where ln denotes the natural logarithm, and that max k-cover can be approximated within a ratio of 1 \Gamma 1=e ' 0:632. The results and techniques in =-=[1, 28]-=- imply that there is a constant ffi ! 1 such that it is NPhard to approximate max k-cover within a ration better than ffi . Lund and Yannakakis [26] showed (under a complexity assumption that will be ... |

748 | Some optimal inapproximability results
- Håstad
- 2001
(Show Context)
Citation Context ...of approximation results where obtained for clique and independent set [17], and for chromatic number [11]. Tight constant factor hardness of approximation results were obtain for several problems in =-=[18]-=-, including a threshold of 7=8 for MAX 3SAT. As for set cover, Raz and Safra [31] constructed new low error constant prover proof systems and used them to show that for some constant c ? 0, it is NP-h... |

678 | Approximation algorithms for NP-hard problems - Hochbaum, editor - 1997 |

606 |
Optimization, approximation, and complexity classes
- Papadimitriou, Yannakakis
- 1991
(Show Context)
Citation Context ...over can be approximated within a ratio of ln n, where ln denotes the natural logarithm, and that max k-cover can be approximated within a ratio of 1 \Gamma 1=e ' 0:632. The results and techniques in =-=[1, 28]-=- imply that there is a constant ffi ! 1 such that it is NPhard to approximate max k-cover within a ration better than ffi . Lund and Yannakakis [26] showed (under a complexity assumption that will be ... |

424 |
On the hardness of approximating minimization problems
- Lund, Yannakakis
- 1994
(Show Context)
Citation Context ...Gamma 1=e ' 0:632. The results and techniques in [1, 28] imply that there is a constant ffi ! 1 such that it is NPhard to approximate max k-cover within a ration better than ffi . Lund and Yannakakis =-=[26]-=- showed (under a complexity assumption that will be presented in Section 1.1) that it is hard to approximate set cover within a ratio of (log n)=2, where log denotes logarithms in base 2. We extend th... |

417 | Nondeterministic exponential time has two-prover interactive protocols
- Babai, Fortnow, et al.
- 1991
(Show Context)
Citation Context ...roof systems (PCPs). The notion of PCPs grew out of the theory 2 of interactive proofs [14, 4, 6, 13] (parts of which we will review shortly) and from major breakthroughs in understanding their power =-=[25, 32, 3]-=-. The relevance of interactive proofs for proving hardness of approximation results was demonstrated in [9], and further developments in [2, 1] led to the PCP notion as stated below. Informally, a PCP... |

413 | Probabilistic checking of proofs: A new characterization of NP
- Arora, Safra
- 1998
(Show Context)
Citation Context ...om major breakthroughs in understanding their power [25, 32, 3]. The relevance of interactive proofs for proving hardness of approximation results was demonstrated in [9], and further developments in =-=[2, 1]-=- led to the PCP notion as stated below. Informally, a PCP for an NP language is a method of encoding NP witnesses, coupled with a verifier -- a very efficient randomized method for verifying the valid... |

360 | A parallel repetition theorem
- Raz
- 1998
(Show Context)
Citation Context ... be approximated within any constant ratio, and unless NP ae TIME(n O(loglog n) ) then set cover cannot be approximated within a ratio of log n=8. Improved analysis of two prover proof systems by Raz =-=[30]-=- implies that unless NP ae TIME(n O(loglogn) ) then set cover cannot be approximated within a ratio of log n=4, and that unless NP ae ZTIME(n O(loglogn) ) then set cover cannot be approximated within ... |

338 |
Arthur-Merlin games: a randomized proof system, and a hierarchy of complexity classes
- Babai, Shlomo
- 1988
(Show Context)
Citation Context ...s.) The first hardness of approximation results for set cover followed from work on probabilistically checkable proof systems (PCPs). The notion of PCPs grew out of the theory 2 of interactive proofs =-=[14, 4, 6, 13]-=- (parts of which we will review shortly) and from major breakthroughs in understanding their power [25, 32, 3]. The relevance of interactive proofs for proving hardness of approximation results was de... |

336 | Algebraic methods for interactive proof systems
- Lund, Fortnow, et al.
- 1992
(Show Context)
Citation Context ...roof systems (PCPs). The notion of PCPs grew out of the theory 2 of interactive proofs [14, 4, 6, 13] (parts of which we will review shortly) and from major breakthroughs in understanding their power =-=[25, 32, 3]-=-. The relevance of interactive proofs for proving hardness of approximation results was demonstrated in [9], and further developments in [2, 1] led to the PCP notion as stated below. Informally, a PCP... |

328 | A sub-constant error-probability low-degree test, and a sub-constant error-probability PCP characterization of NP
- Raz, Safra
- 1997
(Show Context)
Citation Context ...for chromatic number [11]. Tight constant factor hardness of approximation results were obtain for several problems in [18], including a threshold of 7=8 for MAX 3SAT. As for set cover, Raz and Safra =-=[31]-=- constructed new low error constant prover proof systems and used them to show that for some constant c ? 0, it is NP-hard to approximate set cover within a ratio of c log n. It is not known whether h... |

317 | On the ratio of optimal integral and fractional covers - Lovász - 1975 |

211 | Greedy strikes back: improved facility location algorithms
- GUHA, KHULLER
- 1999
(Show Context)
Citation Context ...mma 1=e) 2 Using a Turing reduction, Theorem 10 can be used to show that the performance guarantee of the greedy algorithm is optimal up to low order terms. This has also been observed by others (see =-=[15]-=-, for example). 16 Proposition 12 If max k-cover can be constructively approximated in polynomial time within a ratio of (1 \Gamma 1=e + ffl) for some ffl ? 0, then NP ae TIME(n O(loglog n) ). Proof: ... |

194 | Zero Knowledge and the Chromatic number
- Feige, Kilian
- 1998
(Show Context)
Citation Context ...l other thresholds for approximation were discovered. Essentially tight O(n 1\Gammaffl ) hardness of approximation results where obtained for clique and independent set [17], and for chromatic number =-=[11]-=-. Tight constant factor hardness of approximation results were obtain for several problems in [18], including a threshold of 7=8 for MAX 3SAT. As for set cover, Raz and Safra [31] constructed new low ... |

171 | Interactive proofs and the hardness of approximating cliques
- Feige, Goldwasser, et al.
- 1996
(Show Context)
Citation Context ...ch we will review shortly) and from major breakthroughs in understanding their power [25, 32, 3]. The relevance of interactive proofs for proving hardness of approximation results was demonstrated in =-=[9]-=-, and further developments in [2, 1] led to the PCP notion as stated below. Informally, a PCP for an NP language is a method of encoding NP witnesses, coupled with a verifier -- a very efficient rando... |

169 |
A Greedy Heuristic for the Set Covering Problem
- Chvátal
- 1979
(Show Context)
Citation Context ... cover was among the first problems for which approximation algorithms were analysed. Johnson [23] showed that the greedy algorithm gives an approximation ratio of ln n. (This was extended by Chvatal =-=[7]-=- to the weighted version of set cover.) Lovasz [24] showed that a linear programming relaxation approximates set cover within a ratio of ln n. In both cases, the authors were interested mainly in the ... |

162 | Efficient probabilistically checkable proofs and applications to approximation
- BELLARE, GOLDWASSER, et al.
- 1993
(Show Context)
Citation Context ...he proof system), while maintaining small values for other parameters such as the number of provers, the cardinality of the alphabet and the number of random bits used by the verifier. Bellare et.al. =-=[5]-=- constructed four prover proof systems that implied that unless P=NP set cover cannot be approximated within any constant ratio, and unless NP ae TIME(n O(loglog n) ) then set cover cannot be approxim... |

148 | On the power of multi-prover interactive protocols - Fortnow, Rompel, et al. - 1988 |

147 | Multi-Prover Interactive Proofs: How to Remove Intractability
- Ben-Or, Goldwasser, et al.
- 1988
(Show Context)
Citation Context ...s.) The first hardness of approximation results for set cover followed from work on probabilistically checkable proof systems (PCPs). The notion of PCPs grew out of the theory 2 of interactive proofs =-=[14, 4, 6, 13]-=- (parts of which we will review shortly) and from major breakthroughs in understanding their power [25, 32, 3]. The relevance of interactive proofs for proving hardness of approximation results was de... |

120 | A tight analysis of the greedy algorithm for set cover
- Slavík
- 1996
(Show Context)
Citation Context ...d mainly in the leading term of the approximation ratio. Analysis of the low order terms of the approximation ratio was provided by Srinivasan [34] (for the linear programming approach) and by Slavik =-=[33]-=- (for the greedy algorithm). For max k-cover, the greedy algorithm gives an approximation ratio of 1 \Gamma 1=e (up to low order terms). See [20] and references therein, and also Proposition 11. (A si... |

107 | Two-prover one-round proof systems: Their power and their problems - Feige, Lovász - 1992 |

65 |
Clique is hard to approximate within n 1\Gammaffl
- Hastad
(Show Context)
Citation Context ...uting, 1996. Since then, several other thresholds for approximation were discovered. Essentially tight O(n 1\Gammaffl ) hardness of approximation results where obtained for clique and independent set =-=[17]-=-, and for chromatic number [11]. Tight constant factor hardness of approximation results were obtain for several problems in [18], including a threshold of 7=8 for MAX 3SAT. As for set cover, Raz and ... |

60 | Splitters and near-optimal derandomization
- Naor, Schulman, et al.
- 1995
(Show Context)
Citation Context ...o set cover described above, with k sufficiently large so that f(k) in Lemma 7 is smaller than ffl=4, and with m = (5n) 2`=ffl . Using the deterministic construction of partition systems described in =-=[27]-=-, and observing that m, R and Q are bounded by n O(log log n) , the time to perform this reduction is n O(log log n) . Recall that the number of points in the set cover problem is N = mR where R = (5n... |

42 |
Non deterministic polynomial optimization problems and their approximations. Theoret
- Paz, Moran
- 1981
(Show Context)
Citation Context ...ure. This can be extended to other problems as well, as the same threshold of ln n holds for all problems that are equivalent to set cover in terms of approximation ratio, such as dominating set (see =-=[29]-=- and [26] for more details). A survey of hardness of approximation results and the techniques involved is provided by Arora and Lund in [20]. Remark: A preliminary version of this paper, without the r... |

39 | Approximating the minimum maximal independence number
- Halldórsson
- 1993
(Show Context)
Citation Context ...ios below 2, whereas Hochbaum and Shmoys [21], and Dyer 4 and Frieze [8], showed how to approximate minimum p-center within a factor of 2. For the minimum maximal independent set problem, Halldorsson =-=[16]-=- shows that it cannot be approximated within a ratio of n 1\Gammaffl , for any ffl ? 0, which is tight up to multiplicative low order terms. Another example of a well characterized approximation probl... |

39 |
DB.: A best possible approximation algorithm for the k-center problem
- Hochbaum, Shmoys
- 1985
(Show Context)
Citation Context ...ximation ratio 1). A very sharp example is the minimum p-center problem, for which Hsu and Nemhauser [22] showed that it is NP-hard to obtain approximation ratios below 2, whereas Hochbaum and Shmoys =-=[21]-=-, and Dyer 4 and Frieze [8], showed how to approximate minimum p-center within a factor of 2. For the minimum maximal independent set problem, Halldorsson [16] shows that it cannot be approximated wit... |

39 |
Improved approximations of packing and covering problems
- Srinivasan
- 1995
(Show Context)
Citation Context ... ratio of ln n. In both cases, the authors were interested mainly in the leading term of the approximation ratio. Analysis of the low order terms of the approximation ratio was provided by Srinivasan =-=[34]-=- (for the linear programming approach) and by Slavik [33] (for the greedy algorithm). For max k-cover, the greedy algorithm gives an approximation ratio of 1 \Gamma 1=e (up to low order terms). See [2... |

38 |
Easy and hard bottleneck location problems
- Hsu, Nemhauser
- 1979
(Show Context)
Citation Context ...zation problems that are known to have a threshold of nontrivial nature (e.g., not located at approximation ratio 1). A very sharp example is the minimum p-center problem, for which Hsu and Nemhauser =-=[22]-=- showed that it is NP-hard to obtain approximation ratios below 2, whereas Hochbaum and Shmoys [21], and Dyer 4 and Frieze [8], showed how to approximate minimum p-center within a factor of 2. For the... |

35 |
A simple heuristic for the p-center problem
- Dyer, Frieze
- 1985
(Show Context)
Citation Context ...arp example is the minimum p-center problem, for which Hsu and Nemhauser [22] showed that it is NP-hard to obtain approximation ratios below 2, whereas Hochbaum and Shmoys [21], and Dyer 4 and Frieze =-=[8]-=-, showed how to approximate minimum p-center within a factor of 2. For the minimum maximal independent set problem, Halldorsson [16] shows that it cannot be approximated within a ratio of n 1\Gammaffl... |

20 | Impossibility results for recycling random bits in two-prover proof systems - FEIGE, KILIAN - 1995 |

18 | A well-characterized approximation problem
- Hastad, Phillips, et al.
- 1993
(Show Context)
Citation Context ...ot be approximated within a ratio of n 1\Gammaffl , for any ffl ? 0, which is tight up to multiplicative low order terms. Another example of a well characterized approximation problem is presented in =-=[19]-=-. Our work shows that also set cover and max k-cover have a threshold of a nontrivial nature. This can be extended to other problems as well, as the same threshold of ln n holds for all problems that ... |

9 |
On the Power of Multi-Prover Interactive
- Fortnow, Rompel, et al.
- 1988
(Show Context)
Citation Context |

8 | Approximating the minimum maximal independence - Halldorsson - 1993 |

7 |
On the ratio of the optimal integral and fractional covers". Discrete Mathematics 13
- Lovasz
- 1975
(Show Context)
Citation Context ...oximation algorithms were analysed. Johnson [23] showed that the greedy algorithm gives an approximation ratio of ln n. (This was extended by Chvatal [7] to the weighted version of set cover.) Lovasz =-=[24]-=- showed that a linear programming relaxation approximates set cover within a ratio of ln n. In both cases, the authors were interested mainly in the leading term of the approximation ratio. Analysis o... |

7 |
IP=PSPACE
- Shamir
- 1992
(Show Context)
Citation Context ...roof systems (PCPs). The notion of PCPs grew out of the theory 2 of interactive proofs [14, 4, 6, 13] (parts of which we will review shortly) and from major breakthroughs in understanding their power =-=[25, 32, 3]-=-. The relevance of interactive proofs for proving hardness of approximation results was demonstrated in [9], and further developments in [2, 1] led to the PCP notion as stated below. Informally, a PCP... |

5 | Improved performance of the greedy algorithm for the minimum set cover and minimum partial cover problems - Slavik - 1995 |

2 |
Two prover one round proof systems: Their power and their problems
- Feige, Lovasz
- 1992
(Show Context)
Citation Context ... we can invoke a recent theorem of Raz [30] regarding reduction of error by parallel repetition. In contrast, Lund and Yannakakis used the more complicated two prover proof system of Feige and Lovasz =-=[12]-=- (the result of [30] was not available at the time), without actually describing it. In Section 3 we explain how to construct the partition systems mentioned above. In Section 4 we describe the reduct... |

2 | codes, PCPs, and inapproximability - Arora - 1995 |

1 | Proof verification and hardness of approximation problems". 33rd FOCS - Arora, Lund, et al. - 1992 |

1 | A well chaaracterized approximation problem - Hastad, Phillips, et al. - 1993 |

1 | A simple heuristic for the p-cent problem - DYER, FRIEZE - 1985 |

1 | 651A Threshold of ln n for Approximating Set Cover FEIGE - U, KILIAN - 1996 |

1 | Clique is hard to approximate within n12e - HASTAD - 1996 |

1 | Improved approximations of packing and covering problems - SRINISAVAN - 1995 |

1 | Efficie n t pro habilistically checkable proofs and applications to approximation”. F%oc. o.f 25th STOC - Bellare, Golclwasser, et al. - 1993 |

1 | A simple heuristic for the p-center problem - yel |

1 | TWO prover one rouncl proof systems: their power and their problems - Feige, Lovasz - 1992 |

1 | A well characterized approximation problem”. Proc. ?nd Israel Sgrnp. on Theory of Coniputing and Systems - ast, ad, et al. - 1993 |

1 | Easy and hard bottleneck Iocatiol, problems - Hsu - 1979 |

1 | i the ratio of the optimal integral ancl iractlonal cover~”. Da.scrwtrMathematics 1 - Lova, sz - 1975 |

1 | Optimization, approximation, ancl complexity classes - Papadimitriou, Yanuakakis - 1991 |