## A New Trust Region Technique for the Maximum Weight Clique Problem (2002)

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Venue: | Discrete Applied Mathematics |

Citations: | 19 - 2 self |

### BibTeX

@ARTICLE{Busygin02anew,

author = {Stanislav Busygin},

title = {A New Trust Region Technique for the Maximum Weight Clique Problem},

journal = {Discrete Applied Mathematics},

year = {2002},

volume = {154},

pages = {2006}

}

### OpenURL

### Abstract

A new simple generalization of the Motzkin-Straus theorem for the maximum weight clique problem is formulated and directly proved. Within this framework a new trust region heuristic is developed. In contrast to usual trust region methods, it regards not only the global optimum of a quadratic objective over a sphere, but also a set of other stationary points of the program. We formulate and prove a condition when a Motzkin-Straus optimum coincides with such a point. The developed method has complexity O(n ), where n is the number of graph vertices. It was implemented in a publicly available software package QUALEX-MS.

### Citations

10888 |
Computers and Intractability: A guide to the theory of NP-completeness
- Garey, Johnson
- 1979
(Show Context)
Citation Context ...i : The maximum weight clique problem asks for a clique Q of the maximum W (Q) value. We denote this value by !(w; G). Both the maximum cardinality and the maximum weight clique problems are NP -hard =-=[2-=-], so it is considered unlikely that an exact polynomial time algorithm for them exists. Approximation Copyright csStanislav Busygin, 2001, 2002. All rights reserved. 1 of large cliques is also hard.... |

208 |
Computing a trust region step
- Mor6, Sorensen
(Show Context)
Citation Context ...er a sphere is polynomially solvable in contrast to general nonconvex programming [17]. There is a vast range of other sources describing theoretical and practical results on the trust region problem =-=[5, 7, 8, 10]-=-. Here we outline the complete diagonalization method deriving not only the global optimum at a given sphere radius, but all stationary points corresponding to particular radii we want to consider. Th... |

121 |
Maxima for graphs and a new proof of a theorem of TurĂ¡n
- Motzkin, Straus
- 1965
(Show Context)
Citation Context ...esearch work. 2 The Motzkin{Straus Theorem for Maximum Clique and Its Generalization In 1965 Motzkin and Straus formulated the maximum clique problem as a certain quadratic programming over a simplex =-=[-=-11]. Theorem 1 (Motzkin{Straus) The global optimum value of the quadratic program max f(x) = 1 2 x T AGx (1) subject to X i2V x i = 1; x 0 (2) is 1 2 1 1 !(G) : (3) See [3] for a recent direct proo... |

68 |
Clique is hard to approximate within n 1
- Hastad
- 1999
(Show Context)
Citation Context ...ered unlikely that an exact polynomial time algorithm for them exists. Approximation Copyright csStanislav Busygin, 2001, 2002. All rights reserved. 1 of large cliques is also hard. It was shown in [=-=-=-14] that unless NP = ZPP no polynomial time algorithm can approximate the clique number within a factor of n 1 for any > 0. Recently this margin was tightened in [15] to n=2 (log n) 1 . Hence, any ... |

59 | Better inapproximability results for MaxClique, Chromatic Number and Min-3Lin-Deletion
- Khot, Ponnuswami
- 2006
(Show Context)
Citation Context ...es is also hard. It was shown in [14] that unless NP = ZPP no polynomial time algorithm can approximate the clique number within a factor of n 1 for any > 0. Recently this margin was tightened in [1=-=-=-5] to n=2 (log n) 1 . Hence, any heuristic algorithm nicely approximating maximum clique in a short time in practice is a signicant achievement. In this paper we present such an algorithm named QUALE... |

37 | Continuous characterizations of the maximum clique problem
- Gibbons, Hearn, et al.
- 1997
(Show Context)
Citation Context ...s 1 2 1 1 !(G) : (3) See [3] for a recent direct proof. We formulate a simple generalization of this result for the maximum weight clique problem. In contrast to the generalization established in [6=-=]-=-, this one does not require any reformulation of the maximum clique quadratic program to another minimization problem. It maximally preserves the form of the original Motzkin{Straus result. Let w min ... |

37 |
SPARTAN: A modelbased semantic compression system for massive data tables
- Babu, Garofalakis, et al.
- 2001
(Show Context)
Citation Context ... our method strictly outperforms the algorithm PBH and the weight dierence between heaviest cliques and those found by QUALEX-MS is rather negligible. This success consists with the fact reported in [=-=12]-=- regarding the ability of the QUALEX 2.0 routine to solve always exactly certain maximum weight clique instances arisen in practice in a short time. Table 2: Performance of QUALEX-MS vs. PBH on random... |

33 | Camouflaging independent sets in quasi-random graphs
- Brockington, Culberson
(Show Context)
Citation Context ...hose denitely hard for existing combinatorial algorithms and possibly even intractable for any combinatorial approach. Such an extreme case is Brockington-Culberson graphs from the DIMACS test suite [=-=13]-=- (brock*). Apart from the fact that no combinatorial algorithm succeeded on them so far, they were claimed hard for algorithms based on the Motzkin{Straus continuous formulation in [9]. However, the Q... |

29 |
On the stationary values of a second-degree polynomial on the unit sphere
- Forsythe, Golub
- 1965
(Show Context)
Citation Context ...er a sphere is polynomially solvable in contrast to general nonconvex programming [17]. There is a vast range of other sources describing theoretical and practical results on the trust region problem =-=[5, 7, 8, 10]-=-. Here we outline the complete diagonalization method deriving not only the global optimum at a given sphere radius, but all stationary points corresponding to particular radii we want to consider. Th... |

29 | A new O(n ) algorithm for the symmetric tridiagonal eigenvalue/eigenvector problem
- Dhillon
- 1997
(Show Context)
Citation Context ... QUALEX-MS package makes use of a new eigendecomposition routine DSYEVR from LAPACK involving Relatively Robust Representations to compute eigenpairs after the matrix is reduced to a tridiagonal form =-=[18-=-]. This explains improvement of the average running time versus the two other programs. As a BLAS implementation, the platform-specic prebuilt of ATLAS library 2 was used. Exact or best known solution... |

24 | Minimizing a quadratic over a sphere
- Hager
(Show Context)
Citation Context ...er a sphere is polynomially solvable in contrast to general nonconvex programming [17]. There is a vast range of other sources describing theoretical and practical results on the trust region problem =-=[5, 7, 8, 10]-=-. Here we outline the complete diagonalization method deriving not only the global optimum at a given sphere radius, but all stationary points corresponding to particular radii we want to consider. Th... |

20 |
Algorithms for Minimization without Derivatives, Englewood Cli s
- Brent
- 1973
(Show Context)
Citation Context ...em is bracketed between the minimizing point and one of the section bounds. Both univariate minimization and univariate rootsnding when a root is bracketed may be eciently performed by Brent's method =-=[1]-=-. Next, each of the obtained stationary points is passed to a greedy heuristic as a new vertex weight vector. For the result we take the clique heaviest among those obtained this way due to the statio... |

17 |
A new complexity result on minimization of a quadratic function with a sphere constraint
- Ye
- 1992
(Show Context)
Citation Context ...d feasible solution. This technique is very attractive in many cases since the optimization of a quadratic function over a sphere is polynomially solvable in contrast to general nonconvex programming =-=[17]-=-. There is a vast range of other sources describing theoretical and practical results on the trust region problem [5, 7, 8, 10]. Here we outline the complete diagonalization method deriving not only t... |

16 | A heuristic for the maximum independent set problem based on optimization of a quadratic over a sphere
- Busygin, Butenko, et al.
(Show Context)
Citation Context ...ue the experiment the same way, so it has not been completed. Next, we tested QUALEX-MS on all 80 DIMACS maximum clique instances 1 . At that we compared QUALEX-MS with the algorithm QSH presented in =-=[4]-=- and our previous maximum clique package QUALEX 2.0 (also available at [19].) These two algorithms previously showed considerable advances in handling DIMACS benchmark graphs comparing to the results ... |

16 |
The complexity of the matrix eigenproblem
- Pan, Chen
- 1999
(Show Context)
Citation Context ...above is O(n 3 ) if we derive O(n) stationary points and it takes not more than O(n 2 ) time to get one value. Indeed, the eigendecomposition may be computed up to anysxed precision in O(n 3 ) time [=-=16]-=-, and each basis conversion in (17) takes quadratic time, so generally we have one O(n 3 ) computation at the beginning of the procedure, and O(n) computations of O(n 2 ) complexity each afterwards. 7... |

12 |
The nauty
- McKay
(Show Context)
Citation Context ...tional Experiment Results The goal of thesrst computational experiment was tosnd a smallest maximum clique instance, where QUALEX-MS cannotsnd an exact solution. We used the program geng available at =-=[20]-=- to generate all non-isomorphic to each other graphs up to 10 vertices inclusive. QUALEX-MS successfully found exact solutions to all those instances. Though it may not be excepted for sure that with ... |

11 |
Clique is hard to approximate within n 1
- astad
- 1996
(Show Context)
Citation Context ... !(w; G). Both maximum cardinality and maximum weight clique problems are NP -hard [4], so it is considered unlikely that an exact polynomial time algorithm for them exists. Moreover, it was shown in =-=[-=-6] that unless NP = ZPP no polynomial time algorithm can approximate maximum clique within a factor of n 1 for any > 0. The paper is organized as follows. In Section 2 we revise the Motzkin-Straus t... |

10 | Finding independent sets in a graph using continuous multivariable polynomial formulations
- Abello, Butenko, et al.
- 2001
(Show Context)
Citation Context ...gramming over a simplex [11]. Theorem 1 (Motzkin{Straus) The global optimum value of the quadratic program max f(x) = 1 2 x T AGx (1) subject to X i2V x i = 1; x 0 (2) is 1 2 1 1 !(G) : (3) See [3]=-=-=- for a recent direct proof. We formulate a simple generalization of this result for the maximum weight clique problem. In contrast to the generalization established in [6], this one does not require a... |

8 | I.M.: A complementary pivoting approach to the maximum weight clique problem
- Massaro, Pelillo, et al.
- 2002
(Show Context)
Citation Context ...The last computational experiment performed with QUALEX-MS wassnding maximum weight cliques. Since there are no widely accepted maximum weight clique test suites, we followed the approach accepted in =-=[9]-=- and tested the algorithm against normal and irregular random graphs with various edge densities. To generate the irregular random graphs Algorithm 14 4.1 from [9] was used. Vertex weights were evenly... |

5 |
Local minima of the trust region problem
- Lyle, Szularz
- 1994
(Show Context)
Citation Context |

3 |
Stas Busygin's NP-Completeness
- Busygin
(Show Context)
Citation Context ...actice is a signicant achievement. In this paper we present such an algorithm named QUALEX-MS (QUick ALmost EXact Motzkin{Straus-based.) A software package implementing the algorithm is available at [=-=19]-=-. The paper is organized as follows. In Section 2 we revise the Motzkin{Straus theorem to use the quadratic programming formulation for the maximum weight clique problem. Section 3 reviews the trust r... |