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## Proving Integrality Gaps Without Knowing the Linear Program (2002)

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Venue: | THEORY OF COMPUTING |

Citations: | 61 - 2 self |

### Citations

1971 | The probabilistic method
- Alon, Spencer
- 2004
(Show Context)
Citation Context ...nd-project method of Sherali-Adams [28] that was contemporaneous to Lovász-Schrijver. Acknowledgements The first author would like to thank Noga Alon for drawing his attention to the Erdős results of =-=[2, 13]-=- and their tightenings, and to Subhash Khot for useful discussions. We would also like to thank Joseph Cheriyan and Fei Qian for letting us know about the error in the proof of the LS lower bound in t... |

1404 | Geometric Algorithms and Combinatorial Optimization - Grotschel, Lovasz, et al. - 1993 |

1278 | Approximation Algorithms - Vazirani - 2003 |

679 | Approximation Algorithms for NP-Hard Problems - Hochbaum - 1997 |

347 | Cones of matrices and setfunctions and 0--1 optimization - Lovasz, Schrijver - 1991 |

252 |
A hierarchy of relaxations between the continuous and convex hull representations for 0--1 programming problems
- Sherali, Adams
- 1990
(Show Context)
Citation Context ...uestions introduced here. However, the techniques in all the above papers do not seem to apply to graph VERTEX COVER. Furthermore, they also do not apply to a lift-and-project method of Sherali-Adams =-=[28]-=- that was contemporaneous to Lovász-Schrijver. Acknowledgements The first author would like to thank Noga Alon for drawing his attention to the Erdős results of [2, 13] and their tightenings, and to S... |

242 | Modern graph theory, Graduate Texts - Bollobás - 1998 |

221 | Graph theory and probability - Erdős - 1959 |

169 | Edmonds polytopes and a hierarchy of combinatorial problems, - Chvatal - 1973 |

158 | Lower bounds on the monotone complexity of some Boolean function. - Razborov - 1985 |

146 | Expressing combinatorial optimization problems by linear programs. - Yannakakis - 1991 |

121 | The probabilistic method, 2nd ed., - Alon, Spencer - 2000 |

117 | Hardness of approximations. - Arora, Lund - 1997 |

88 | The importance of being biased. In: - Dinur, Safra - 2002 |

81 | Cut problems and their applications to divide-and-conquer. In - Shmoys - 1997 |

76 | Lower bounds for cutting planes proofs with small coefficients. - Bonet, Pitassi, et al. - 1997 |

76 |
0.879-Approximation Algorithms for MAX CUT and MAX 2SAT.
- Goemans, Williamson
- 1994
(Show Context)
Citation Context ...lift-and-project” construction of Lovász and Schrijver [24]. This is a method that underlies semidefinite relaxations used in many recent approximation algorithms starting with Goemans and Williamson =-=[19]-=-. The LS procedure over many rounds generates tighter and tighter linear relaxations for 0/1 optimization problems. It is more round-efficient than classical cutting planes procedures such as Gomory-C... |

55 |
Heuristic analysis, linear programming and branch
- WOLSEY
- 1980
(Show Context)
Citation Context ...ation is NP-hard. The best hardness result for metric TSP only shows that 1.01-approximation is NP-hard [25], yet decades of work has failed to yield a relaxation with integrality gap better than 1.5 =-=[32]-=- (or 4/3, if one believes a well-known conjecture [17]). For graph expansion and related graph problems essentially no hardness results exist yet we only know relaxations with integrality gap Ω(logn) ... |

41 | On the Approximability of the Traveling Salesman Problem, - Papadimitriou, Vempala - 2000 |

39 | Worst-case comparison of valid inequalities for the TSP. - GOEMANS - 1995 |

28 | When does the positive semidefiniteness constraint help in lifting procedures, - Goemans, Tuncel - 2000 |

27 | The probable value of the Lovász–Schrijver relaxations for maximum independent set - Feige, Krauthgamer - 2003 |

14 |
Iannis Tourlakis. Towards strong nonapproximability results in the Lovasz-Schrijver hierarchy
- Alekhnovich, Arora
- 2005
(Show Context)
Citation Context ...hree families of LPs is unbounded, even though the graphs witnessing these gaps have independent sets of size Ω(n). In the first two families of relaxations we allow only the variables x1,x2,...,xn ∈ =-=[0,1]-=- for the vertices and no auxiliary variables. Some such restriction seems necessary because auxiliary variables would give the LP the power of arbitrary polynomial-time computations. The third family ... |

13 | On circuits and subgraphs of chromatic graphs - Erdős - 1962 |

10 |
Feige. A threshold of lnn for approximating set cover
- Uriel
- 1998
(Show Context)
Citation Context ... obtain relaxations for SET-COVER and rankk hypergraph VERTEX COVER with integrality gaps less than (1−ε)lnn and k−1−ε, respectively. Note that PCP-based results (such as those of H˚astad [20], Feige =-=[15]-=- and Dinur et al. [10]) already ruled out non-trivial polynomial-time approximation algorithms for these problems (assuming P = NP). However, they did not rule out slightly subexponential approximati... |

9 |
Béla Bollobás, and László Lovász. Proving integrality gaps without knowing the linear program
- Arora
- 2002
(Show Context)
Citation Context ...g PCPs that (2 − ε)-approximation to VERTEX COVER is NP-hard, the proof would probably involve even more complex reductions than those in [11]. Thus it 2 Erratum: The conference version of this paper =-=[3]-=- argued Ω( √ logn) rounds of the LS procedure were needed to reduce the integrality gap for VERTEX COVER below 2−o(1). However, Cheriyan and Qian [7] observed that the argument in [3] was incomplete. ... |

9 |
HÅSTAD: Some optimal inapproximability results
- JOHAN
(Show Context)
Citation Context ...re needed to obtain relaxations for SET-COVER and rankk hypergraph VERTEX COVER with integrality gaps less than (1−ε)lnn and k−1−ε, respectively. Note that PCP-based results (such as those of H˚astad =-=[20]-=-, Feige [15] and Dinur et al. [10]) already ruled out non-trivial polynomial-time approximation algorithms for these problems (assuming P = NP). However, they did not rule out slightly subexponential... |

8 | Approximating Covering and Packing - Hochbaum - 1997 |

5 |
Goemans and Levent Tunçel. When does the positive semidefiniteness constraint help in lifting procedures
- Michel
(Show Context)
Citation Context ...oaches. Related work A few authors have viewed the Lovász-Schrijver procedure as a proof system and shown that Ω(n) rounds are required to derive certain simple inequalities (e.g., Goemans and Tunçel =-=[18]-=-, Cook and Dash [9]). However, these papers do not consider the issue of how the integrality gap improves (or fails to improve) after a few rounds of the LS procedure. A recent (and independent) paper... |

4 |
A Razborov. Lower bounds on monotone complexity of the logical permanent
- Alexander
- 1985
(Show Context)
Citation Context ...laxations. An integrality gap result for a large subfamily of relaxations may then be viewed as a lower bound for a restricted computational model, analogous say to lower bounds for monotone circuits =-=[27]-=- and for proof systems [5]. An example is Yannakakis’s result [33] that representing TSP (the exact version) using a symmetric linear program requires exponentially many constraints—this ruled out som... |

4 |
Approximation Algorithms
- Vijay
- 2001
(Show Context)
Citation Context ...ect, vertex cover 1 Introduction Approximation algorithms for NP-hard problems—metric TSP, VERTEX COVER, graph expansion, cut problems, etc.—often use a linear relaxation of the problem (see Vazirani =-=[31]-=-, Hochbaum [22]). For instance, a simple 2-approximation algorithm for VERTEX COVER solves the following relaxation: minimize ∑i∈V xi such that xi + x j ≥ 1 for all {i, j} ∈ E. One can show that in th... |

2 |
Bollobás. Modern Graph Theory, volume 184 of Graduate Texts in Mathematics
- Béla
- 2002
(Show Context)
Citation Context ...ε. The proof of Theorem 4.3 relies on the following two theorems. The first (which also follows as a subcase from the arguments used to prove Lemma 2.8) is essentially due to Erdős [12]; see Bollobás =-=[4]-=-, Theorem 4, Ch VII. The second, Theorem 4.5, will be proved in Section 4.2 with an overview of the argument first given in Section 4.1. Theorem 4.4. For any α > 0 there is an n0(α) such that for ever... |

2 |
AND RAN RAZ: Lower bounds for cutting planes proofs with small coefficients
- BONET, PITASSI
- 1997
(Show Context)
Citation Context ...ap result for a large subfamily of relaxations may then be viewed as a lower bound for a restricted computational model, analogous say to lower bounds for monotone circuits [27] and for proof systems =-=[5]-=-. An example is Yannakakis’s result [33] that representing TSP (the exact version) using a symmetric linear program requires exponentially many constraints—this ruled out some approaches to P = NP tha... |

2 |
ERDÖS: Graph theory and probability
- PAUL
- 1959
(Show Context)
Citation Context ...gn is at least 2 − ε. The proof of Theorem 4.3 relies on the following two theorems. The first (which also follows as a subcase from the arguments used to prove Lemma 2.8) is essentially due to Erdős =-=[12]-=-; see Bollobás [4], Theorem 4, Ch VII. The second, Theorem 4.5, will be proved in Section 4.2 with an overview of the argument first given in Section 4.1. Theorem 4.4. For any α > 0 there is an n0(α) ... |

2 |
Feige. Randomized graph products, chromatic numbers, and the lovász vartheta-funktion
- Uriel
- 1997
(Show Context)
Citation Context ...sulting by taking the k-fold inclusive graph product of G with itself. The key observation is that α(G × H) = α(G) × α(H) and χ f (G × H) = χ f (G)χ f (H) (the former fact is easy; for the latter see =-=[14]-=- for a proof). Moreover, if all sets of size at most βn have fractional chromatic number C in G, then all sets of size at most βn in G k have fractional chromatic number C k . So taking products of a ... |

2 |
Feige and Robert Krauthgamer. The probable value of the lovász–schrijver relaxations for maximum independent set
- Uriel
(Show Context)
Citation Context ...ver, these papers do not consider the issue of how the integrality gap improves (or fails to improve) after a few rounds of the LS procedure. A recent (and independent) paper by Feige and Krauthgamer =-=[16]-=- considers the question of integrality gaps, but for the maximum CLIQUE problem on a random graph with edge probabilities 1/2. They show that Ω(logn) rounds of LS+, the semi-definite version of Lovász... |

2 |
Goemans. Worst-case comparison of valid inequalities for the TSP.Mathematical Programming
- Michel
- 1995
(Show Context)
Citation Context ...TSP only shows that 1.01-approximation is NP-hard [25], yet decades of work has failed to yield a relaxation with integrality gap better than 1.5 [32] (or 4/3, if one believes a well-known conjecture =-=[17]-=-). For graph expansion and related graph problems essentially no hardness results exist yet we only know relaxations with integrality gap Ω(logn) (Shmoys [29]). When decades of work has failed to turn... |

2 | AND SANTOSH VEMPALA: On the approximability of the traveling salesman problem (extended abstract - PAPADIMITRIOU - 2000 |

1 |
AND TONIANN PITASSI: Rank bounds and integrality gaps for cutting planes procedures
- BURESH-OPPENHEIM, GALESI, et al.
- 2003
(Show Context)
Citation Context ...es can clearly approximate INDEPENDENT SET within a factor of n 1−ε . Our techniques seem applicable to problems other than VERTEX COVER (and INDEPENDENT SET) and have been the subject of future work =-=[6, 1, 30]-=-. These developments are discussed in the related work section below. However, several open problems remain. For example, extending our ideas to semidefinite relaxations as well as to the semidefinite... |

1 |
CHVÁTAL: Edmonds polytopes and a hierarchy of combinatorial problems
- VASEK
- 1973
(Show Context)
Citation Context ...S procedure over many rounds generates tighter and tighter linear relaxations for 0/1 optimization problems. It is more round-efficient than classical cutting planes procedures such as Gomory-Chvátal =-=[8]-=- since it generates every valid inequality in at most n rounds. Even in one round it generates nontrivial inequalities for VERTEX COVER. Furthermore, the set of inequalities derivable in O(1) rounds—t... |

1 |
AND SANJEEB DASH: On the matrix-cut rank of polyhedra
- COOK
(Show Context)
Citation Context ... A few authors have viewed the Lovász-Schrijver procedure as a proof system and shown that Ω(n) rounds are required to derive certain simple inequalities (e.g., Goemans and Tunçel [18], Cook and Dash =-=[9]-=-). However, these papers do not consider the issue of how the integrality gap improves (or fails to improve) after a few rounds of the LS procedure. A recent (and independent) paper by Feige and Kraut... |

1 |
ERDÖS: On circuits and subgraphs of chromatic graphs
- PAUL
- 1962
(Show Context)
Citation Context ...number which we define below. We will construct the required graph by the probabilistic method in Theorem 2.3. This result appears to be new, although it fits in a line of results starting with Erdős =-=[13]-=- showing that the chromatic number of a graph cannot be deduced from “local considerations” (see also Alon and Spencer [2], p.130). Definition 2.1. A fractional γ-coloring of a graph G is a multiset C... |

1 |
HOCHBAUM: Approximating covering and packing problems: Set cover, vertex cover, independent set, and related problems. In Approximation Algorithms for NP-hard Problems
- DORIT
- 1997
(Show Context)
Citation Context ...s the following relaxation: minimize ∑i∈V xi such that xi + x j ≥ 1 for all {i, j} ∈ E. One can show that in the optimum solution, xi ∈ {0,1/2,1}. Thus rounding the 1/2’s up to 1 gives a VERTEX COVER =-=[21]-=-. This also proves an upper bound of 2 on the integrality gap of the relaxation, which is the maximum over all graphs G of the ratio of the size of the minimum VERTEX COVER in G and the cost of the op... |

1 |
LIPTÁK AND LÁSZLÓ LOVÁSZ: Critical facets of the stable set polytope
- LÁSZLÓ
(Show Context)
Citation Context ... small constant ε > 0. The second family consists of linear programs containing inequalities with low defect. Usually one defines “defect” for facets of the INDEPENDENT SET polytope (see for instance =-=[24, 23]-=-); here we will make an analogous definition for the VERTEX COVER polytope (i.e., the convex hull of all integra vertex covers): The defect of a VERTEX COVER polytope facet a T x ≥ b, where a is a vec... |

1 |
LOVÁSZ AND ALEXANDER SCHRIJVER: Cones of matrices and set-functions and 0-1 optimization
- LÁSZLÓ
- 1991
(Show Context)
Citation Context ...lynomial size description (where additional variables are allowed). THEORY OF COMPUTING, Volume 2 (2006), pp. 19–51 20PROVING INTEGRALITY GAPS WITHOUT KNOWING THE LINEAR PROGRAM Lovász and Schrijver =-=[24]-=-. The first family consists of linear programs that can include arbitrary inequalities on any set of εn variables, for some small constant ε > 0. The second family consists of linear programs containi... |

1 |
The integrality gap of the minimum vertex cover problem
- QIAN
- 2003
(Show Context)
Citation Context ...VER below 2−o(1). However, Cheriyan and Qian [7] observed that the argument in [3] was incomplete. In the current paper we give a new (complete) proof of the LS round lower bound. Independently, Qian =-=[26]-=- also provides a fix for the proof in [3]. However, our new proof has the advantage of showing that in fact at least Ω(logn) rounds of LS tightenings are needed to reduce the integrality gap below 2 −... |

1 |
SHMOYS: Cut problems and their application to divide-and-conquer. In Approximation Algorithms for NP-hard Problems
- DAVID
- 1997
(Show Context)
Citation Context ..., if one believes a well-known conjecture [17]). For graph expansion and related graph problems essentially no hardness results exist yet we only know relaxations with integrality gap Ω(logn) (Shmoys =-=[29]-=-). When decades of work has failed to turn up tighter relaxations, one should seriously investigate the possibility that no tighter relaxations exist. However, proving such a statement may be related ... |

1 |
TOURLAKIS: Towards optimal integrality gaps for hypergraph vertex cover in the Lovász-Schrijver hierarchy
- IANNIS
- 2005
(Show Context)
Citation Context ...es can clearly approximate INDEPENDENT SET within a factor of n 1−ε . Our techniques seem applicable to problems other than VERTEX COVER (and INDEPENDENT SET) and have been the subject of future work =-=[6, 1, 30]-=-. These developments are discussed in the related work section below. However, several open problems remain. For example, extending our ideas to semidefinite relaxations as well as to the semidefinite... |

1 |
YANNAKAKIS: Expressing combinatorial optimization problems by linear programs
- A, College, et al.
- 1991
(Show Context)
Citation Context ...xations may then be viewed as a lower bound for a restricted computational model, analogous say to lower bounds for monotone circuits [27] and for proof systems [5]. An example is Yannakakis’s result =-=[33]-=- that representing TSP (the exact version) using a symmetric linear program requires exponentially many constraints—this ruled out some approaches to P = NP that were being tried at the time. In this ... |