Frauke Liers

A branch-and-cut algorithm based on semidefinite programming for the minimum k-partition problem

The minimum k-partition (MkP) problem is the problem of partitioning the set of vertices of a graph into k disjoint subsets so as to minimize the total weight of the edges joining vertices in the same partition. The main contribution of this paper is the design and implementation of a branch-and-cut algorithm based on semidefinite programming (SBC) for the MkP problem. The two key ingredients for this algorithm are: the combination of semidefinite programming with polyhedral results; and a novel iterative clustering heuristic (ICH) that finds feasible solutions for the MkP problem. We compare ICH to the hyperplane rounding techniques of Goemans and Williamson and of Frieze and Jerrum, and the computational results support the conclusion that ICH consistently provides better feasible solutions for the MkP problem. ICH is used in our SBC algorithm to provide feasible solutions at each node of the branch-and-bound tree. The SBC algorithm computes globally optimal solutions for dense graphs with up to 60 vertices, for grid graphs with up to 100 vertices, and for different values of k, providing a fast exact approach for k≥3.

Global Approaches for Facility Layout and VLSI Floorplanning

This chapter provides an overview of conic optimization models for facility layout and VLSI floorplanning problems. We focus on two classes of problems to which conic optimization approaches have been successfully applied, namely the single-row facility layout problem, and fixed-outline floorplanning in VLSI circuit design. For the former, a close connection to the cut polytope has been exploited in positive semidefinite and integer programming approaches. In particular, the semidefinite optimization approaches can provide global optimal solutions for instances with up to 40 facilities, and tight global bounds for instances with up to 100 facilities. For the floorplanning problem, a conic optimization model provided the first non-trivial lower bounds in the literature.

Exact ground states of large two-dimensional planar Ising spin glasses.

Studying spin-glass physics through analyzing their ground-state properties has a long history. Although there exist polynomial-time algorithms for the two-dimensional planar case, where the problem of finding ground states is transformed to a minimum-weight perfect matching problem, the reachable system sizes have been limited both by the needed CPU time and by memory requirements. In this work, we present an algorithm for the calculation of exact ground states for two-dimensional Ising spin glasses with free boundary conditions in at least one direction. The algorithmic foundations of the method date back to the work of Kasteleyn from the 1960s for computing the complete partition function of the Ising model. Using Kasteleyn cities, we calculate exact ground states for huge two-dimensional planar Ising spin-glass lattices (up to 3000;{2} spins) within reasonable time. According to our knowledge, these are the largest sizes currently available. Kasteleyn cities were recently also used by Thomas and Middleton in the context of extended ground states on the torus. Moreover, they show that the method can also be used for computing ground states of planar graphs. Furthermore, we point out that the correctness of heuristically computed ground states can easily be verified. Finally, we evaluate the solution quality of heuristic variants of the L. Bieche approach.

Ground state of the Bethe lattice spin glass and running time of an exact optimization algorithm

We study the Ising spin glass on random graphs with fixed connectivity z and with a Gaussian distribution of the couplings, with mean

Speeding up IP-based algorithms for constrained quadratic 0–1 optimization

\ensuremath{\mu}

Solving k-Way Graph Partitioning Problems to Optimality: The Impact of Semidefinite Relaxations and the Bundle Method

and unit variance. We compute exact ground states by using a sophisticated branch-and-cut method for

Local cuts revisited

z=4,6

Universality-class dependence of energy distributions in spin glasses

and system sizes up to 1280 spins, for different values of

Solving network design problems via iterative aggregation

\ensuremath{\mu}.

An exact algorithm for robust network design

We locate the spin-glass/ferromagnet phase transition at