TY - JOUR

T1 - Clustering improves the Goemans–Williamson approximation for the max-cut problem

AU - Rodriguez-Fernandez, Angel E.

AU - Gonzalez-Torres, Bernardo

AU - Menchaca-Mendez, Ricardo

AU - Stadler, Peter F.

N1 - Publisher Copyright:
© 2020 by the authors. Licensee MDPI, Basel, Switzerland.

PY - 2020/9

Y1 - 2020/9

N2 - MAX-CUT is one of the well-studied NP-hard combinatorial optimization problems. It can be formulated as an Integer Quadratic Programming problem and admits a simple relaxation obtained by replacing the integer “spin” variables xi by unitary vectors ⃗vi . The Goemans–Williamson rounding algorithm assigns the solution vectors of the relaxed quadratic program to a corresponding integer spin depending on the sign of the scalar product ⃗vi ·⃗r with a random vector⃗r. Here, we investigate whether better graph cuts can be obtained by instead using a more sophisticated clustering algorithm. We answer this question affirmatively. Different initializations of k-means and k-medoids clustering produce better cuts for the graph instances of the most well known benchmark for MAX-CUT. In particular, we found a strong correlation of cluster quality and cut weights during the evolution of the clustering algorithms. Finally, since in general the maximal cut weight of a graph is not known beforehand, we derived instance-specific lower bounds for the approximation ratio, which give information of how close a solution is to the global optima for a particular instance. For the graphs in our benchmark, the instance specific lower bounds significantly exceed the Goemans–Williamson guarantee.

AB - MAX-CUT is one of the well-studied NP-hard combinatorial optimization problems. It can be formulated as an Integer Quadratic Programming problem and admits a simple relaxation obtained by replacing the integer “spin” variables xi by unitary vectors ⃗vi . The Goemans–Williamson rounding algorithm assigns the solution vectors of the relaxed quadratic program to a corresponding integer spin depending on the sign of the scalar product ⃗vi ·⃗r with a random vector⃗r. Here, we investigate whether better graph cuts can be obtained by instead using a more sophisticated clustering algorithm. We answer this question affirmatively. Different initializations of k-means and k-medoids clustering produce better cuts for the graph instances of the most well known benchmark for MAX-CUT. In particular, we found a strong correlation of cluster quality and cut weights during the evolution of the clustering algorithms. Finally, since in general the maximal cut weight of a graph is not known beforehand, we derived instance-specific lower bounds for the approximation ratio, which give information of how close a solution is to the global optima for a particular instance. For the graphs in our benchmark, the instance specific lower bounds significantly exceed the Goemans–Williamson guarantee.

KW - Algorithms

KW - Approximation

KW - Clustering

KW - Max-Cut

KW - Semidefinite programming

UR - http://www.scopus.com/inward/record.url?scp=85096046847&partnerID=8YFLogxK

U2 - 10.3390/computation8030075

DO - 10.3390/computation8030075

M3 - Artículo

AN - SCOPUS:85096046847

VL - 8

SP - 1

EP - 12

JO - Computation

JF - Computation

SN - 2079-3197

IS - 3

M1 - 75

ER -