S. Poljak and F. Rendl. Solving the max-cut problem using eigenvalues. Report 91735-OR, Institute fur Diskrete Mathematik, University of Bonn, 1991.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....has de ed any ecient solution so far, and indeed, it was proved to be NP hard [10] even in the case when all edge weights are equal to 1. In spite of that, many attempts have been made to tackle the problem with approximation and randomized algorithms. Delorme and Poljak [6] and Poljak and Rendl [20] solved a relaxation of the problem using eigenvalues. A similar approach based on semide nite programming was developed by Goemans and Williamson who presented a randomized algorithm in [12] with a performance guarantee of 0:878. Polynomial time methods to nd the exact solution of the Ising ....

S. Poljak and F. Rendl. Solving the max-cut problem using eigenvalues. Discr. Appl. Math., 62:249-278, 1995.

....The general Max Cut problem has defied efficient solution so far, and indeed, it was proved to be NP hard [GJ] even in the case when all edge weights are equal to 1 or Gamma1. In spite of that, many attempts have been made to tackle the problem with approximation and randomized algorithms ( DP] [PR], GW] Polynomial time methods for toroidal square lattices were suggested in the early 60 s by Kasteleyn [Kast, Kast1] and Kac and Ward [KW] Kac and Ward tried to calculate the partition function as a determinant of a 4n Theta 4n matrix over complex numbers and even though their original ....

S. Poljak and F. Rendl. Solving the max-cut problem using eigenvalues. Discr. Appl. Math., 62:249--278, 1995.

....has defied efficient solution so far, and indeed, it was proved to be NP hard [9] even in the case when all edge weights are equal to 1. In spite of that, many attempts have been made to tackle the problem with approximation and randomized algorithms. Delorme and Poljak [6] and Poljak and Rendl [18] solved a relaxation of the problem using eigenvalues. A similar approach based on semidefinite programming was developed by Goemans and Williamson who presented a randomized algorithm in [11] with a performance guarantee of 0:878. Polynomial time methods to find the exact solution of the Ising ....

S. Poljak and F. Rendl. Solving the max-cut problem using eigenvalues. Discr. Appl. Math., 62:249--278, 1995. 19

.... problem is to solve the graph partitioning problem with k = 2, m 1 = i, and m 2 = n Gamma i for all i = 1; Delta Delta Delta bn=2c (notice that in graph partitioning problem max and min characterizations are essentially equivalent by simply changing the weights w i with P w j Gamma w i ) In [DP90, PR91] the following SDP bound is proposed: minf n 4 1 (A Diag(x) 1 T x = ag MC(G) 76) where MC(G) is the size of maximum cut in G. 5.76) is equivalent to primal dual pair: min z (1=n)1 T x s.t. zI Gamma Diag(x) A max A ffl Y s.t. Y ii = 1=n Y 0 (77) 33 and may be solved by ....

S. Poljak and F. Rendl. Solving the max-cut problem using eigenvalues. Technical Report 91735-OR, Forschungsinstitut Fur Diskrete Mathematik, Institut Fur okonometrie und Operations Research, Rheinische Friedrich--Wilhelms-Universitat, Bonn, November 1991.

....means that in any recursive call, the upper bound should not be much larger than the largest solution permitted by the corresponding set of constraints. We use a method of efficiently deriving the upper bound in each recursive call from previously computed upper bounds, similar to the one used in [9]. The resulting algorithm is shown in Figure 1. The details are omitted due to space limitations. Although the PSD upper bound is fairly tight and eliminates a very large fraction of the search space, it still leads to unreasonably long running times for 5 Theta 5 Theta 5 Ising graphs. However, ....

S. Poljak and F. Rendl. Solving the max-cut problem using eigenvalues. Technical Report 91735-OR, Universitat Bonn, 1991.

....known for a long time that this basic optimization problem is NP hard [8] but has a (straightforward) 5approximation algorithm [18] The best approximation in the general case is a recent exciting . 87856 approximation algorithm due to Goemans and Williamson [9, 10] building upon previous work [4, 5, 15]. Unfortunately, there is not much room for improvement since the problem is Max SNP hard [14] and hence [17] has no ffl approximation scheme if P 6= NP. Thus one is led to consider restricted versions of MAX CUT. In [16, 2] polynomial time approximation schemes were presented for dense ....

S. Poljak and F. Rendl. Solving the max-cut problem using eigenvalues. Report 91735-OR, Institute fur Diskrete Mathematik, University of Bonn, 1991.

.... [HL93b] simulated annealing (SA) introduced in [KGV83] and applied to graph partitioning in [JAMS89] and genetic algorithms [Hol75] Different approaches to the problem include network flow based method ( SM86] LR88] LR93] spectral and polyhedral approaches ( DH73] Bop87] RW90] [PR91], PT93] and approximation algorithms ( GWed] GW94] HP95] 3 Path Optimization Path Optimization can be viewed as a variation of the hill climbing local optimization partitioning procedure. Given an initial partitioning = S; S) PO performs a variation of simple neighborhood search. ....

S. Poljak and F. Rendl. Solving the max-cut problem using eigenvalues. Technical Report 199, Technische Universitat Graz, Institute fur Mathematik, 1991.

....eigenvalue minimization problems can in general be formulated as semidefinite programs. This is potentially quite useful, since there is an abundant literature on eigenvalue bounds for combinatorial optimization problems; see the survey paper by Mohar and Poljak [49] As shown by Poljak and Rendl [60, 59] and Delorme and Poljak [14] the eigenvalue bound provides a very good bound on the maximum cut in practice. Delorme and Poljak [13, 12] study the worst case ratio between the maximum cut and their eigenvalue bound. The worst instance they are aware of is the 5 cycle for which the ratio is 32 ....

....is less than :8796 in the unweighted case. The convex hull of the optimum vectors is depicted on the right; the circle represents the center of the sphere. 5 Computational Results In practice, we expect that the algorithm will perform much better than the worst case bound of ff. Poljak and Rendl [60, 59] (see also Delorme and Poljak [14] report computational results showing that the bound Z EIG is typically less than 2 5 and, in the instances they tried, never worse than 8 away from Z MC . We also performed our own computational experiments, in which the cuts computed by the algorithm ....

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S. Poljak and F. Rendl. Solving the max-cut problem using eigenvalues. In M. Lucertini, G. Rinaldi, A. Sassano, and B. Simeone, editors, Partitioning and Decomposition in Combinatorial Optimization, Discrete Applied Mathematics. 1995. To appear.

.... is to solve the graph partitioning problem with k = 2, m 1 = i, and m 2 = n Gamma i for all i = 1; Delta Delta Delta bn=2c (notice that in graph partitioning problem max and min characterizations are essentially equivalent by simply changing the weights w i with P w j Gamma w i ) In [16, 56] the following SDP bound is proposed: minf n 4 1 (A Diag(x) 1 T x = ag MC(G) 5.16) where MC(G) is the size of maximum cut in G. 5.16) is equivalent to: min z (1=n)1 T x s.t. zI Gamma Diag(x) A max A ffl Y s.t. Y ii = 1=n Y 0 (5.17) and may be solved by interior point ....

S. Poljak and F. Rendl, Solving the max-cut problem using eigenvalues, Tech. Report 91735-OR, Forschungsinstitut Fur Diskrete Mathematik, Institut Fur okonometrie und Operations Research, Rheinische Friedrich--Wilhelms-Universitat, Bonn, November 1991.

.... of KL employing graph contraction ( GB83] Bui86] JAMS89] GG84] GLR86] HL93b] simulated annealing (SA) KGV83] JAMS89] and genetic algorithms [Hol75] Different approaches to the problem include network flow based methods [SM86, LR88, LR93] spectral and polyhedral approaches [DH73, Bop87, RW90, PR91, PT93], and approximation algorithms [GWed, GW94, HP95] 4 3 Path Optimization Path Optimization can be viewed as a variation of the hill climbing local optimization partitioning procedure. Given an initial partitioning = S; S) PO performs a variation of simple neighborhood search. The ....

S. Poljak and F. Rendl. Solving the max-cut problem using eigenvalues. Technical Report 199, Technische Universitat Graz, Institute fur Mathematik, 1991.

....can be formulated as semidefinite programs. For MAX CUT, the nonlinear relaxation we consider is equivalent to a spectral bound proposed by Delorme and Poljak [8, 7] This equivalence to the semidefinite program we consider was established by Poljak and Rendl [45] As shown by Poljak and Rendl [44, 46] and Delorme and Poljak [9] the spectral bound provides a very good bound on the maximum cut in practice. Delorme and Poljak [8, 7] study the worst case ratio between the maximum cut and their spectral bound. The worst instance they are aware of is the 5 cycle for which the ratio is 32 25 5 p ....

....moreover, weights can be selected such that the ratio drops to :8788. A more complicated weighted instance on 103 vertices can be shown to lead to a bound E[W ] Z P 0:8786. In practice, however, we expect that the algorithm will perform much better than this worstcase bound. Poljak and Rendl [44, 46] (see also Delorme and Poljak [9] report computational results showing that the bound Z EIG is typically less than 2 5 and never worse than 8 away from Z MC . We also performed our own computational experiments, in which the cuts computed by the algorithm were usually within 4 of the ....

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S. Poljak and F. Rendl. Solving the max-cut problem using eigenvalues. Report 91735-OR, Institute fur Diskrete Mathematik, Universitat Bonn, 1991.

....jh j , or equivalently, maximize f 0 (P k ) min 1hk ff 2 jh (H Gamma j ) The corresponding partitioning objective is to maximize g(S k ) min 1hk jjY n h jj 2 . Max Cut: Another problem which has received much attention in the recent literature is the max cut problem [13] 14] [35]: Maximize: f(P k ) k X h=1 jE(C h )j: The same mapping of y n i to row i of Delta 1 2 U that was used to reduce min cut graph partitioning to min sum vector partitioning can be used to reduce the max cut problem to max sum vector partitioning. 4 Linear Ordering Algorithm We now ....

S. Poljak and F. Rendl, Solving the max-cut problem using eigenvalues, Discrete Applied Mathematics 62 (1995) 249-278.

....of cut polyhedra 35 2.4) the max cut problem and the unconstrained quadratic programming problem are equivalent. Other approaches, beside the polyhedral approach, have been proposed for attacking the max cut problem. In particular, an approach based on eigenvalue methods is investigated in [45] [138]. We mention briefly some facts, permitting to connect it with polyhedral aspects. The Laplacian matrix L of the graph G is the n Theta n matrix defined by L ii = deg G (i) for i 2 V n and L ij = Gammaa ij for i 6= j 2 V n , where A = a ij ) 1i;jn is the adjacency matrix of G. Set (G) n 4 ....

S. Poljak and F. Rendl. Solving the max-cut problem using eigenvalues. Research report No. 91735-OR, Institut fur Diskrete Mathematik, Universitat Bonn, 1991.

....Therefore one can compute the upper bound OE(G; c) n 4 inf n max (L(G; c) diag(z) j z 2 IR V ; e T z = 0 o : 1) This bound can be computed quite efficiently if G is sparse, since the sparsity of the matrix L(G; c) can be exploited. The bound has been computed by Poljak and Rendl [12] on graphs of sizes up to a few hundred nodes. The same authors also report on optimal solutions, obtained by inserting the computation of the bound in a branch and bound framework, for graphs of sizes up to 80 nodes. 3.2 Edge formulation: the semidefinite relaxation Let x be an n dimensional ....

S. Poljak and F. Rendl. Solving the max-cut problem using eigenvalues. Discrete Applied Mathematics, 62:249--278, 1995.

....techniques of positive semidefinite programming and randomized rounding. However, solving semidefinite programs is computationally expensive. Previous implementations of the algorithm on serial machines could only handle relatively small inputs (200 to 500 vertices) See Poljak and Rendl [31] for related work. Our goals are twofold. Firstly, we want to establish the practical possibility of using the GW algorithm for much larger input graphs of thousands of vertices within realistic amounts of time. In order to achieve this goal, we make use of parallel computation techniques and ....

....of the algorithm on graphs with negative edge weights a case which is not covered by the 0:878 performance guarantee of the algorithm. By way of comparison, we have also implemented simulated annealing and the randomized greedy algorithm. Previous experimental work with the GW algorithm [16, 7, 31, 12] has been limited to much smaller inputs. Our parallel implementation makes it possible to compare the trends reported previously with results obtained on much larger graphs. 4.1 The Graphs and the Algorithms We have tested the algorithm on the following graphs: 1. Graphs derived from circuit ....

S. Poljak and F. Rendl. Solving the max-cut problem using eigenvalues. Technical Report 91735-OR, Universitat Bonn, 1991.

....the eigenvalue approach are reported in [9] However, the bound considered there was not the best possible for the approach. The computational experiments with the bound (G; c; 0) graph bisection into equal sizes) are reported in [10] and the bound (G; c) on the max cut problem is computed in [14]. Lower bounds are obtained by rounding a suitable eigenvector to a 610vector, and a consecutive local improvement of the cuts. The approach provides solutions with about 5 relative error between the upper bound and a cut found. The eigenvalue bound (G; c) has several interesting properties ....

S. Poljak and F. Rendl, Solving the max-cut problem using eigenvalues, Research Report No. 91735-OR, Institut fur Diskrete Mathematik, Univ. Bonn, 1991.

....by dropping the rank one condition, and optimizing over the remaining sets. P sdp ) z sdp = maxftr LX : diag(X) e; X 0g: 3) This formulation and its dual form have been investigated quite intensively from a theoretical point of view, see e.g. 11, 22, 37, 51] as well as computationally, see [24, 29, 49, 50]. We come back to this problem in more detail in subsequent sections. 2.2 Problems in (0,1) variables Problem (P) can equivalently be expressed in (0; 1) variables z, by substituting x = 2z Gamma e: After homogenization and linearization, this results in the following problem. maxftr LZ : ....

S. POLJAK and F. RENDL. Solving the Max-Cut Problem using Eigenvalues. Discrete Applied Mathematics, 62: 249--278, 1995.

....factor 4 comes from (3) Therefore the largest eigenvalue of the Laplacian provides an upper bound on the weight of a maximum cut. A further improvement is proposed in [12] using (4) to obtain 4mc(G) G) min u t e=0 n max (L diag(u) 5) Computational experiments are contained in [37] and in [35] where this bound is applied to huge graphs (with up to 50,000 vertices and several million edges) Using some of the combinatorial properties of (G) investigated in [11] the paper [37] provides the first results of this bound in a Branch and Bound setting. In [36] it is shown, that ....

.... (G) min u t e=0 n max (L diag(u) 5) Computational experiments are contained in [37] and in [35] where this bound is applied to huge graphs (with up to 50,000 vertices and several million edges) Using some of the combinatorial properties of (G) investigated in [11] the paper [37] provides the first results of this bound in a Branch and Bound setting. In [36] it is shown, that the eigenvalue relaxation can alternatively be formulated as a semidefinite program, leading to computationally more stable algorithms. Computational results using this semidefinite setting are given ....

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S. POLJAK and F. RENDL. Solving the Max-Cut Problem using Eigenvalues. Discrete Applied Mathematics, 62: 249--278, 1995.

....the node relaxation of mc(G) and refer to [5] for further properties of this relaxation. Note that f(u) n 4 max (L diag(u) thus f(u) can be computed efficiently for all u. The fact that f attains its minimum is established e.g. in [5] Computational experiments using are reported in [18]. In this paper we want to further emphasize that provides a very tight upper bound on the max cut mc, which is manageable even for extremely large graphs. The edge model. In this model we describe cuts by introducing variables on the edge set. Let y 2 ( n 2 ) be the characteristic ....

....and also reasonably large complete graphs (up to 1000 nodes) can be handled quite efficiently using the dual approach. 3 The node relaxation on large graphs We first point out that computational results for graphs with a small number of nodes (n 500) and edges (jEj 13000) are contained in [18] on a wide variety of different types of graphs. The present paper is an outgrowth of [18, 17] We want n jEj upper bd. good cut gap ( BT time 100 2466 1464 1430 2.5 31 250 15594 8729 8535 2.3 141 500 62300 33804 33242 1.7 851 750 140607 75245 74117 1.5 2276 1000 250080 132679 130871 1.4 4046 ....

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F. RENDL and S. POLJAK. Solving the max-cut problem using eigenvalues. Technical Report, University of Technology Graz, 1991.

....the eigenvalue approach are reported in [9] However, the bound considered there was not the best possible for the approach. The computational experiments with the bound (G; c; 0) graph bisection into equal sizes) are reported in [10] and the bound (G; c) on the max cut problem is computed in [14]. Lower bounds are obtained by rounding a suitable eigenvector to a Sigma1 Gammavector, and a consecutive local improvement of the cuts. The approach provides solutions with about 5 relative error between the upper bound and a cut found. The eigenvalue bound (G; c) has several interesting ....

S. Poljak and F. Rendl, Solving the max-cut problem using eigenvalues, (Research Report No. 91735-OR, Institut fur Diskrete Mathematik, Univ. Bonn, 1991) Discrete Applied Mathematics, to appear.

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S. Poljak and F. Rendl. Solving the max-cut problem using eigenvalues. Report 91735-OR, Institute fur Diskrete Mathematik, University of Bonn, 1991.

No context found.

S. Poljak and F. Rendl. Solving the max-cut problem using eigenvalues. In M. Lucertini, G. Rinaldi, A. Sassano, and B. Simeone, editors, Partitioning and Decomposition in Combinatorial Optimization, Discrete Applied Mathematics. 1995. To appear.

No context found.

S. Poljak and F. Rendl. Solving the max-cut problem using eigenvalues. In M. Lucertini, G. Rinaldi, A. Sassano, and B. Simeone, editors, Partitioning and Decomposition in Combinatorial Optimization, Discrete Applied Mathematics. 1995. To appear.

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