F. Barahona, M. Junger, and G. Reinelt [1989]: Experiments in quadratic 0-1 programming. Mathematical Programming 44, 127-137.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....problems to optimality. We conclude with a discussion of our findings and show strong points and weaknesses of our approach, as compared to previous work. 2 Equivalent formulations of (MC) Problem (1) can be recast in several essentially equivalent forms. To start, it is well known, see e.g. [3], that the adjunction of a linear term c t x in the cost function of (1) does not change the problem significantly, because max x t Lx 2c t x such that x 2 f Gamma1; 1g n (2) is equivalent to max y t Ly such that y 2 f Gamma1; 1g n 1 ; y n 1 = 1; where L : L c c t 0 : ....

....inclusion of a linear term leads to a problem of the form (1) of size increased by one. It is also straightforward to see that (2) is equivalent to Quadratic (0; 1) programming (QP) QP ) min x t Qx q t x such that x 2 f0; 1g n : This has been observed by many researchers, see e.g. [3, 13]. Finally, we recall how (1) relates to the Max Cut Problem. This connection was established for instance by Mohar and Poljak [29] Let G be an undirected graph on vertex set V = f1; ng with edge weights fc e : e 2 E(G)g, given by its adjacency matrix A = a ij ) where a ij = a ji = c e ....

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F. BARAHONA, M. J UNGER, and G. REINELT. Experiments in quadratic 0-1 programming. Mathematical Programming, 44:127--137, 1989.

....the code on the same classes of problems as in [14] G :5 , G Gamma1=0=1 , Q 100 , and Q 100; 2 . G :5 consists of unweighted graphs with edge probability 1 2, G Gamma1=0=1 of weighted (complete) graphs with edge weights chosen uniformly from f Gamma1; 0; 1g. Q 100 and Q 100; 2 were used in [29, 4]. Formulating Q 100 with respect to (QP) the lower triangle of B is set to zero, the upper triangle (including the diagonal) is chosen uniformly from f Gamma100; 100g. The diagonal takes the role of the linear term. Q 100; 2 represents instances with a density of 20 . It was observed in ....

F. Barahona, M. Junger, and G. Reinelt. Experiments in quadratic 0-1 programming. Mathematical Programming, 44:127--137, 1989.

....graphs not contractible to K 5 (see [1] The polynomial solvability on these special graphs is related to a complete description of their cut polytopes and does not seem to allow a generalization to wider classes of graphs. Computational experiments based on polyhedral approaches are given in [2, 3, 14]. It turns out that almost planar graphs can be handled efficiently for quite large jV j: Graphs on up to 2501 nodes and 7500 edges are dealt with in [4] On the other hand it becomes clear from probabilistic arguments [15, 16] that the polyhedral approach is unlikely to be successful on ....

....for quite large jV j: Graphs on up to 2501 nodes and 7500 edges are dealt with in [4] On the other hand it becomes clear from probabilistic arguments [15, 16] that the polyhedral approach is unlikely to be successful on moderately dense graphs of even modest size. This can also be seen in [3, 6]. From these papers it becomes clear that random problems with 50 nodes and edge probability 1 2, or problems with 100 nodes and edge probability 3 10 are not manageable by the polyhedral approach under the present state of the art. This claim is also substantiated in [14] where a different ....

F. BARAHONA, M. J UNGER, and G. REINELT. Experiments in quadratic 0-1 programming. Mathematical Programming, 44:127--137, 1989.

.... 1) matrix D : Q Gammac Gammac T d : Then (P ) is equivalent to (P 0 ) max z T Dz such that z 2 f Gamma1; 1g n 1 : Note that without loss of generality we can require z n 1 = 1 because the function is symmetric, f(z) f( Gammaz) This transformation was already used in in [5] to solve quadratic (0,1) minimization using the max cut problem. A semidefinite relaxation for (P 0 ) is (R 0 ) max trDZ such that diag(Z) e; Z 0: It was pointed out by [4] that the two relaxations (R) R 0 ) are actually equivalent as well. The following proof and other important ....

F. BARAHONA, M. J UNGER and G. REINELT. Experiments in quadratic 0-1 programming, Mathematical Programming 44:127-137, 1989.

....max w T x subject to x 2 CutP (G) This is the polyhedral approach, classical in combinatorial optimization, which leads to the study of the facets of CutP (G) This approach has been used in practice for solving large instances of the max cut problem (see 34 M. Deza and M. Laurent e.g. 23] [24]) Its success depends, of course, on the degree of knowledge about the facets needed for the problem at hand and of their tractability, i.e. whether they can be separated in polynomial time or, at least, whether a good separation heuristic is available. For instance, CutP (G) Met(G) i.e. the ....

F. Barahona, M. Junger, and G. Reinelt. Experiments in quadratic 0-1 programming. Mathematical Programming, 44:127--137, 1989.

.... regard it as the maximisation of the sum of a linear term and a quadratic term, as in equation (3) above (e.g. 30] c) others regard it as a minimisation of a symmetric quadratic term (e.g. 35] d) others regard it as the minimisation of the sum of a linear term and a quadratic term (e.g. [4]) Although all of the above are (trivially) mathematical equivalent these different definitions can pose problems for the unwary reader. Since the computational focus of this paper is a comparison with the recent work of Glover, Kochenberger and Alidaee [15] we have adopted here the same ....

....and Rosen [43] discuss how a molecular conformation problem can be formulated as an UBQP. Chardaire and Sutter [12] mention an application of the problem in cellular radio channel assignment. It has long been known that UBQP is equivalent to the problem of finding a maximum cut in a graph (see [4,18]) 4 Pardalos and Rodgers [39] and Pardalos and Xue [40] have shown that a number of problems in graphs (maximum clique; maximum vertex packing; minimum vertex cover; maximum independent set; maximum weight independent set) can all be formulated as UBQP s. 5 2. LITERATURE SURVEY There have ....

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F. Barahona, M. Jünger and G. Reinelt, Experiments in quadratic 0-1 programming. Mathematical Programming 44 (1989) 127-137.

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F. Barahona, M. Junger, and G. Reinelt [1989]: Experiments in quadratic 0-1 programming. Mathematical Programming 44, 127-137.

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F. Barahona, M. Junger, and G. Reinelt [1989]: Experiments in quadratic 0-1 programming. Mathematical Programming 44, 127-137.

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