## Semidefinite Relaxations for Max-Cut (2001)

Venue: | The Sharpest Cut, Festschrift in Honor of M. Padberg's 60th Birthday. SIAM |

Citations: | 8 - 1 self |

### BibTeX

@INPROCEEDINGS{Laurent01semidefiniterelaxations,

author = {Monique Laurent},

title = {Semidefinite Relaxations for Max-Cut},

booktitle = {The Sharpest Cut, Festschrift in Honor of M. Padberg's 60th Birthday. SIAM},

year = {2001},

pages = {291--327}

}

### OpenURL

### Abstract

We compare several semidefinite relaxations for the cut polytope obtained by applying the lift and project methods of Lov'asz and Schrijver and of Lasserre. We show that the tightest relaxation is obtained when aplying the Lasserre construction to the node formulation of the max-cut problem. This relaxation Q t (G) can be defined as the projection on the edge subspace of the set F t (n), which consists of the matrices indexed by all subsets of [1; n] of cardinality t + 1 with the same parity as t + 1 and having the property that their (I ; J)-th entry depends only on the symmetric difference of the sets I and J . The set F 0 (n) is the basic semidefinite relaxation of max-cut consisting of the semidefinite matrices of order n with an all ones diagonal, while Fn\Gamma2 (n) is the 2 n\Gamma1 -dimensional simplex with the cut matrices as vertices. We show the following geometric properties: If Y 2 F t (n) has rank t + 1, then Y can be written as a convex combination of at most 2 t cut matrices, extending a result of Anjos and Wolkowicz for the case t = 1; any 2 t+1 cut matrices form a face of F t (n) for t = 0; 1; n \Gamma 2. The class L t of the graphs G for which Q t (G) is the cut polytope of G is shown to be closed under taking minors. The graph K 7 is a forbidden minor for membership in L 2 , while K 3 and K 5 are the only minimal forbidden minors for the classes L 0 and L 1 , respectively. 1

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Citation Context ... bound for the stability number of a graph G, obtained by optimizing over the semidefinite relaxation TH(G) of the stable set polytope. This idea was again used successfully by Goemans and Williamson =-=[16] who could-=- prove the first nontrivial "berlinlong" 2001/10/18 page 3 i i i i i i i i 3 approximation algorithm for max-cut using a semidefinite relaxation of the cut polytope. Since then semidefinite ... |

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Citation Context ...for constructing projection representations for 0 \Gamma 1 polytopes; in particular, by Balas, Ceria and Cornu'ejols [3], Sherali and Adams [29], Lov'asz and Schrijver [27] and, recently, by Lasserre =-=[19, 20]-=-. A common feature of these methods is the construction of a hierarchy K ' K 1 ' : : : ' K d ' P of relaxations of P obtained as projections of higher dimensional polyhedra, that finds P in d steps; t... |

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Citation Context ...\Gamma y 36 \Gamma y 27 \Gamma y 47 \Gamma y 67s\Gamma5. (Inequalities (1)-(6) belong to the class of hypermetric inequalities and (7)-(9) to the class of clique-web inequalities; cf. section 30.5 in =-=[9]-=- for details). It is shown in [21] that the inequalities (1),(2),(7),(10) are valid for the Lov'aszSchrijver relaxation N+ (K 7 ) and thus for Q 2 (K 7 ) too (by Corollary 12). Using the computer prog... |

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Citation Context ...\Gamma 1)-dimensional simplex ! Several general purpose methods have been proposed for constructing projection representations for 0 \Gamma 1 polytopes; in particular, by Balas, Ceria and Cornu'ejols =-=[3]-=-, Sherali and Adams [29], Lov'asz and Schrijver [27] and, recently, by Lasserre [19, 20]. A common feature of these methods is the construction of a hierarchy K ' K 1 ' : : : ' K d ' P of relaxations ... |

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Citation Context ...implex ! Several general purpose methods have been proposed for constructing projection representations for 0 \Gamma 1 polytopes; in particular, by Balas, Ceria and Cornu'ejols [3], Sherali and Adams =-=[29]-=-, Lov'asz and Schrijver [27] and, recently, by Lasserre [19, 20]. A common feature of these methods is the construction of a hierarchy K ' K 1 ' : : : ' K d ' P of relaxations of P obtained as project... |

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Citation Context ...rs for the class L 2 since the max-cut problem is known to be NP-hard for the class of graphs having no K 6 minor (in fact, also for the graphs having a node whose deletion results in a planar graph) =-=[5]-=-. One can show that the class L t is closed under taking clique k-sums (k = 0; 1; 2; 3); the same holds for the class G t [21] (the proof for L t is analogous to that for G t ). The next result follow... |

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Citation Context ... E . Obviously, CUT(G) = E(CUT(Kn )) and Barahona [4] shows that MET(G) = E(MET(Kn )). In the linear description of MET(G), it suffices to consider the circuit inequalities (4) for chordless circuits =-=[6]-=-; therefore, MET(Kn ) is defined by the 4 \Gamma n 3 \Delta triangle inequalities: x ij + x ik + x jks\Gamma1; x ij \Gamma x ik \Gamma x jks\Gamma1 (5) for all distinct i; j; k 2 V . As a consequence,... |

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Citation Context ...tion TH(G); moreover, this semidefinite relaxation coincides with the set Q t (FR(G)) obtained by applying the Lasserre construction to the fractional stable set polytope FR(G) (defined in (14)). See =-=[22]-=- for more details. Lower bounds for the rank of the Lasserre procedure. It would be interesting to find lower bounds for the Lasserre rank of a graph G, which is defined as the smallest integer t for ... |

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Citation Context ...for constructing projection representations for 0 \Gamma 1 polytopes; in particular, by Balas, Ceria and Cornu'ejols [3], Sherali and Adams [29], Lov'asz and Schrijver [27] and, recently, by Lasserre =-=[19, 20]-=-. A common feature of these methods is the construction of a hierarchy K ' K 1 ' : : : ' K d ' P of relaxations of P obtained as projections of higher dimensional polyhedra, that finds P in d steps; t... |

47 | Semidefinite Programming and Integer Programming, Centrum voor Wiskunde en Informatica
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(Show Context)
Citation Context ...approximation algorithm for max-cut using a semidefinite relaxation of the cut polytope. Since then semidefinite relaxations have been widely used for approximating combinatorial problems (see, e.g., =-=[25]-=- for a survey). A comparison of the various lift and project methods can be found in [22]. In particular, if we denote the t-th iterate in the Lov'asz-Schrijver hierarchy by N t + (K) and the t-th ite... |

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Citation Context ...e matrices of order n with an all ones diagonal while Fn\Gamma2 (n) is the 2 n\Gamma1 -dimensional simplex with the cut matrices as vertices. We study adjacency properties of cuts on F t (n). Padberg =-=[28]-=- showed that any two cuts form an edge of the metric polytope and Laurent and Poljak [23] showed the analogous result for F 0 (n). We address here the question whether, more generally, any 2 t+1 cuts ... |

43 | On the optimality of the random hyperplane rounding technique for MAX CUT. Random Structures and Algorithms
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Citation Context ...weight of a cut with respect to the weights c by the maximum obtained by optimizing over the relaxation Q t (Kn ). Goemans and Williamson [16] showed that ae 0s0:878 when cs0 and Feige and Schechtman =-=[14]-=- have constructed graphs for which the integrality ratio ae 0 attains the worst case value 0:878. It is however known that, in practice, the integrality ratio ae 0 is larger than the worst case value.... |

29 | On the matrix-cut rank of polyhedra
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(Show Context)
Citation Context ... polytope 1 P = fx 2 R d j P d i=1 x is1g if we start from its relaxation K = fx 2 R d j P d i=1 x is1 2 g; then the same number d of iterations is needed for finding P using the N or the N+ operator =-=[8]-=-. Other examples are given in [8], [15]. Moreover, geometric conditions are studied in [15] under which the N+ operator yields a tighter relaxation than the N operator. If we apply the Lov'asz-Schrijv... |

29 | When does the positive semidefiniteness constraint help in lifting procedures
- Goemans, Tunçel
(Show Context)
Citation Context ...s1g if we start from its relaxation K = fx 2 R d j P d i=1 x is1 2 g; then the same number d of iterations is needed for finding P using the N or the N+ operator [8]. Other examples are given in [8], =-=[15]-=-. Moreover, geometric conditions are studied in [15] under which the N+ operator yields a tighter relaxation than the N operator. If we apply the Lov'asz-Schrijver construction to the pair P = CUT(G),... |

27 | Bounds on the Chvátal rank of polytopes in the 0/1 cube
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Citation Context ...k of K. Although the Chv'atal rank can be very large in general (as it depends on the dimension d and on the coefficients of A), it is bounded by O(d 2 log d) when K is contained in the cube [0; 1] d =-=[13]-=-. From an algorithmic point of view, the first Chv'atal closure does not yield an efficient relaxation in general, since optimizing a linear objective function over K 0 is a co-NP-hard problem [12]. A... |

25 | On a positive semidefinite relaxation of the cut polytope. Linear Algebra and its Applications
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Citation Context ...mensional simplex with the cut matrices as vertices. We study adjacency properties of cuts on F t (n). Padberg [28] showed that any two cuts form an edge of the metric polytope and Laurent and Poljak =-=[23]-=- showed the analogous result for F 0 (n). We address here the question whether, more generally, any 2 t+1 cuts form a face of F t (n); we show that this property holds for t = 1 and n \Gamma 2. The ma... |

22 |
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(Show Context)
Citation Context ...'atal rank of K. Although the Chv'atal rank can be very large in general (as it depends on the dimension d and on the coefficients of A), it is bounded by O(d 2 log d) when K is contained in the cube =-=[0; 1]-=- d [13]. From an algorithmic point of view, the first Chv'atal closure does not yield an efficient relaxation in general, since optimizing a linear objective function over K 0 is a co-NP-hard problem ... |

22 |
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(Show Context)
Citation Context ...tions are given in section 2. Let En := fij j 1si ! jsng denote the edge set of the complete graph Kn and let E denote the projection from R En onto R E . Obviously, CUT(G) = E(CUT(Kn )) and Barahona =-=[4]-=- shows that MET(G) = E(MET(Kn )). In the linear description of MET(G), it suffices to consider the circuit inequalities (4) for chordless circuits [6]; therefore, MET(Kn ) is defined by the 4 \Gamma n... |

21 |
On a representation of the matching polytope via semidefinite liftings
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- 1999
(Show Context)
Citation Context ...In the case of the stable set problem, it has been shown in [27] that the smallest t for which equality N t + (FR(G)) = ST(G) holds satisfies tsff(G), with equality when G is the line graph of K 2n+1 =-=[30]-=-. In the case of max-cut, the LS rank of Kn is conjectured to be equal to n \Gamma 4; equality has been shown for n = 4; 5; 6; 7 [21]. We saw that the Lasserre rank of Kn is equal to 1,2,2,3 for n = 4... |

20 |
On the membership problem for the elementary closure of a polyhedron
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(Show Context)
Citation Context ... d [13]. From an algorithmic point of view, the first Chv'atal closure does not yield an efficient relaxation in general, since optimizing a linear objective function over K 0 is a co-NP-hard problem =-=[12]-=-. Another idea has been investigated for constructing cutting planes in an implicit way, which consists of trying to represent P as the projection of another polytope Q lying in a higher dimensional s... |

16 | A strengthened sdp relaxation via a second lifting for the max-cut problem - ANJOS, WOLKOWICZ - 1999 |

14 | The cut cone III: On the role of triangle facets
- Deza, Laurent, et al.
- 1992
(Show Context)
Citation Context ...r any set ffi(S 1 ); : : : ; ffi(S k ) of cuts in general position (meaning that each cell in the Venn diagram of the sets S 1 ; : : : ; S k is non empty) form a face of MET(Kn ) and thus of CUT(Kn ) =-=[10]-=-. One may wonder whether some analogous result holds for the matrix set F t (n). We saw above that any two cut matrices form a face of F 0 (n); note that this does not extend to a set of three cut mat... |

12 |
All facets of the cut cone Cn for n = 7 are known
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- 1990
(Show Context)
Citation Context ...al inequalities: 2 6 X i=2 y 1i + X 2i!j6 y ijs\Gamma4 (54) and the inequalities obtained from (53) and (54) by permutation of the nodes and switching by cuts. We now treat the case n = 7. Grishukhin =-=[17]-=- has computed that all the facets of CUT(K 7 ) are, up to permutation and switching, induced by one of the following eleven inequalities: (1) the triangle inequality (5); (2) the pentagonal inequality... |

11 |
On the cone of positive semidefinite matrices. Linear Algebra and its Applications 90
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(Show Context)
Citation Context ...: : ; x k g) is a face of K. Consider a convex set K of the form K = fX 2 PSDn j hA i ; Xi = b i for i = 1; : : : ; mg where the A i 's are symmetric matrices and b i 2 R. It follows from a result in =-=[11]-=- that the smallest face F (A) of K containing A is given by F (A) = fX 2 K j ker X ' ker Ag: This description of the faces applies in particular to any set F t (n). Analogously to CUT(Kn ), the set F ... |

10 | Optimality conditions and LMI relaxations for 0-1 programs - Lasserre - 2000 |

8 | On the facial structure of the set of correlation matrices
- Laurent, Poljak
- 1995
(Show Context)
Citation Context ...ij ) ij2En belongs to CUT(Kn ). We now examine some properties of the faces of the convex set F t (n). All the 2 n\Gamma1 cut matrices are vertices of F t (n) (since they have rank 1). It is shown in =-=[24]-=- that the cut matrices are the only vertices of F 0 (n). Moreover, it is shown in [23] that any two distinct cut matrices form a face (edge) of F 0 (n). This adjacency property extends to each of the ... |

7 | Geometry of semidefinite max-cut relaxations via matrix ranks - Anjos, Wolkowicz |

7 | Tighter linear and semidefinite relaxations for max-cut based on the Lovász-Schrijver lift-andproject procedure
- Laurent
(Show Context)
Citation Context ...ly obtain a semidefinite relaxation of CUT(G) by first applying the N+ operator to MET(Kn ) and then projecting onto R E ; namely define N t + (G) := E (N t + (MET(Kn ))): (6) It can be verified (see =-=[21]-=-) that N t + (G) ` N t + (MET(G)); (7) it is not known whether equality holds, i.e., whether the two operators N+ and E commute. The node model. A second possibility is to apply the lift and project c... |

7 |
On the Shannon Capacity of a Graph
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(Show Context)
Citation Context ... linear or, in the case of Lov'asz-Schrijver and of Lasserre, semidefinite. This idea of using semidefinite relaxations for a combinatorial 0 \Gamma 1 problem goes back to the seminal work of Lov'asz =-=[26]-=- who introduced the theta function #(G) as bound for the stability number of a graph G, obtained by optimizing over the semidefinite relaxation TH(G) of the stable set polytope. This idea was again us... |

6 | All facets of the cut cone for n = 7 are known - Grishukhin - 1990 |

4 | When does the positive semide constraint help in lifting procedures - Goemans, Tuncel |

3 | Tighter linear and semide relaxations for max-cut based on the Lovasz-Schrijver lift-and-project procedure - Laurent - 2001 |

3 | On a representation of the matching polytope via semide liftings - Stephen, Tuncel - 1999 |

2 |
Cones of matrices and set-functions and 0 \Gamma 1 optimization
- asz, Schrijver
- 1991
(Show Context)
Citation Context ...pose methods have been proposed for constructing projection representations for 0 \Gamma 1 polytopes; in particular, by Balas, Ceria and Cornu'ejols [3], Sherali and Adams [29], Lov'asz and Schrijver =-=[27]-=- and, recently, by Lasserre [19, 20]. A common feature of these methods is the construction of a hierarchy K ' K 1 ' : : : ' K d ' P of relaxations of P obtained as projections of higher dimensional p... |

1 |
Edmonds polytopes and a hierarchy of combinatorial problems
- atal
- 1973
(Show Context)
Citation Context ...0 := fx 2 R d j u T Axsbu T bc for all us0 such that u T A integerg; which satisfies P ` K 0 ` K. Define iteratively K (1) := K 0 and K (t+1) := (K (t) ) 0 for ts1. Then, K (t) = P for some integer t =-=[7]-=-; the smallest such t is the Chv'atal rank of K. Although the Chv'atal rank can be very large in general (as it depends on the dimension d and on the coefficients of A), it is bounded by O(d 2 log d) ... |

1 |
Optimality conditions and LMI relaxations for 0 \Gamma 1 programs
- Lasserre
- 2000
(Show Context)
Citation Context ... ` Q t (K) ` K; the first inclusion follows from Lemma 2 and the second one from the fact that g `sy ( ; )s0 for all ` and y 2 Q t (K). The hierarchy of relaxations Q t (K) was introduced by Lasserre =-=[18, 20]-=- who showed that P is found after d steps; that is, P = Q d (K). His construction is motivated by results about representations of polynomials as sums of squares and his original presentation involves... |

1 | Geometry of semide Max-Cut relaxations via ranks - Anjos, Wolkowicz - 2001 |

1 | The cut cone III: On the role of triangle facets. Graphs and Combinatorics - Deza, Laurent, et al. - 1992 |

1 | On the cone of positive semide matrices. Linear Algebra and its Applications - Hill, Waters - 1987 |

1 | On a positive semide relaxation of the cut polytope - Laurent, Poljak - 1995 |