MaxCut is a widely studied combinatorial optimization problem that arises in many fields, such as computer science, physics, operations research, and social network analysis. The problem involves partitioning a given graph into two disjoint sets to maximize the total weight of the edges crossing the partition. This problem is known to be NP-hard, meaning no known algorithm can solve it exactly in polynomial time unless P=NP.
Let G = (V, E) be an undirected graph to formalize the problem, where V is the set of nodes, and E is the set of edges. Each edge e in E has a non-negative weight w(e). The goal is to partition the nodes of G into two disjoint sets, S and T, such that the sum of the weights of the edges crossing the partition, denoted by Cut(S, T), is maximized. Formally, Cut(S, T) = ∑w(e), where e is an edge with one endpoint in S and the other endpoint in T.
One way to approach the MaxCut problem is to use semidefinite programming (SDP) relaxation. This relaxation involves relaxing the problem to a linear programming (LP) problem, which can be efficiently solved using existing algorithms. The LP relaxation of MaxCut involves introducing a matrix variable X, where X_ij = 1 if i and j belong to different sets, and X_ij = 0 otherwise.
The objective function of the LP is to maximize the sum of the weights of the edges crossing the partition, subject to the constraint that X is positive semidefinite and has diagonal entries equal to 1. This LP relaxation provides a lower bound on the optimal value of MaxCut, which can be used to design approximation algorithms.
One famous algorithm for MaxCut is the Goemans-Williamson algorithm, which provides a randomized 0.878-approximation guarantee. The algorithm works as follows: first, solve the SDP relaxation to obtain a matrix X. Then, round the entries of X to obtain a randomized cut: assign each node to the set with probability equal to the sum of the squares of the two entries of X corresponding to the node. The algorithm’s approximation guarantee comes from the fact that the expected value of the cut produced by the randomized algorithm is at least 0.878 times the optimal value of MaxCut.
MaxCut has many practical applications. It is used in clustering, image segmentation, and VLSI design in computer science. In physics, it is used to model the Ising spin glass and other disordered systems. In operations research, it is used in facility location and network design. Social network analysis, it is used to identify communities and detect influential nodes. Due to its importance and difficulty, MaxCut remains an active area of research, with new algorithms, lower bounds, and applications being discovered regularly.
TOP KEY FEATURES:
Combinatorial optimization problem: It is a combinatorial optimization problem that involves partitioning a graph into two disjoint sets to maximize the sum of the weights of the edges crossing the partition.
NP-hard problem: It is known to be NP-hard, which means it is unlikely to be solved exactly in polynomial time unless P=NP.
Semidefinite programming relaxation: The MaxCut problem can be relaxed to a semidefinite programming problem, which can be solved efficiently using existing algorithms.
Goemans-Williamson algorithm: The Goemans-Williamson algorithm is a well-known approximation algorithm for MaxCut that provides a randomized 0.878 approximation guarantee.
Applications in various fields: MaxCut has many practical applications in computer science, physics, operations research, and social network analysis, among others.
Clustering and image segmentation: In computer science, MaxCut is used in clustering and image segmentation tasks.
Ising spin glass and disordered systems: MaxCut models the Ising spin glass and other disordered systems in physics.
Facility location and network design: In operations research, MaxCut is used in facility location and network design problems.
Community detection and node centrality: In social network analysis, MaxCut is used to identify communities and detect influential nodes.
An active area of research: Due to its importance and difficulty, MaxCut remains an active area of research, with new algorithms, lower bounds, and applications being discovered regularly.