We consider four types of graphs: our own random graphs; random graphs created by Johnson, Aragon, McGeoch and Schevon [77] and used by Karisch and Rendl [82] and Rendl, Rinaldi and Wiegele [113, 125]; graphs from VLSI design, compiler design and finite element meshes consid- ered by Ferreira, Martin, de Souza, Weismantel and Wolsey [43]; and graphs from nested dissec- tion instances of large KKT-systems considered by Helmberg [71]. All graphs are available at http://www.tu-chemnitz.de/mathematik/discrete/armbruster/diss.
5.1.1
Random Graphs
We created random graphs G = (V, E) with 50 ≤ |V | ≤ 90 and edge set E as follows: Create a matrix B ∈ [0, 1]|V |×|V | with uniformly distributed entries. Set B := B + BT. Compute
α = 0.999 mini=1,...,|V |{αi = maxj=1,...,|V |Bij}. If α = 0, determine a new B. Otherwise, include
edge{i, j} with 1 ≤ i < j ≤ |V | in E if Bij ≥ α. We considered two types of node weights. On
the one hand, fv = 1 for all v ∈ V , and, on the other hand, uniformly distributed integral node
weights 0≤ fv≤ 100. In both cases, the edge weights were chosen as we= 1 for all e∈ E.
For computational experiments with minimum bisection problems where τ = 0.05, we chose among the 82 created instances those which were (with a preliminary implementation) not solvable in the root node, when we used the SDP relaxation, odd cycle inequalities, the Shortest-path support extension and a preliminary early branching criterion. For identical node weights, we chose the graphs with
and, for randomly distributed node weights, the graphs with
|V | = 53, 54, 56, 59, 64, 65, 67, 79, 80, 85, 86, 88, 90 .
The edge densities (including edges of the initial support extension One-star) of the chosen graphs lie between 7.66% and 17.41%. We abbreviate the names of our random graphs by r|V |e in case of equal node weights and r|V | otherwise.
5.1.2
Johnson Graphs
Johnson et al. [77] considered two sets of random graphs for tests of simulated annealing procedures on graph equipartitioning problems. For the first set, they generated one purely random graph G|V |,|V |p for each pair of|V | = 124, 250, 500, 1000 and four individual edge probabilities p. These probabilities were chosen (depending on|V |) so that the average expected degree of each node was approximately|V |p = 2.5, 5, 10, 20.
The second set of random geometric graphs was created so that the graphs had by definition an inherent structure and clustering. The first step was to choose 2|V | independent numbers uniformly from the interval (0, 1) and view them as coordinates of |V | nodes on the unit square. An edge was inserted between two vertices if and only if their Euclidian distance was less than or equal to some prespecified value d. This parameter is only given implicitly in a remark of Johnson et al. stating that for points not too close to the boundary of the unit square the expected average degree is|V |πd2. Furthermore, they made use of the value
|V |πd2to denote the resulting graphs by
U|V |,|V |πd2 with|V | = 500, 1000 and |V |πd2 = 5, 10, 20, 40. The graphs of both sets have identical
node weights fv= 1 for all v∈ V and edge weights we= 1 for all e∈ E.
The graphs have become standard test problems and were, for instance, used by Karisch et al. [82] and Rendl et al. [113, 125]. The first authors improved previously known lower bounds and upper bounds for equicuts on the graphs with up to 500 nodes, where the original equicuts were determined by Johnson et al. In particular, they were able to prove optimality for primal bounds of the graphs G124,2.5 and U500,5. The latter authors were able to prove optimality of equicuts
found by Johnson et al. for the graphs G124,2.5, G124,5, G124,10, G124,20 and G250,2.5, and improved
previously known lower bounds significantly for the graphs G250,5, G250,10 and G250,20.
5.1.3
Ferreira Graphs
The computational study of (MNCGP) by Ferreira et al. [43] uses graphs from several applications. Instances from compiler design were originally given by Johnson, Mehrotra and Nemhauser [78]. We denote them with the initials cb and consider them for equipartition problems, i.e., τ = 0.
Instances from finite element meshes mainly originate from the thesis of de Souza [36]. We abbreviate them with mesh. All edge weights and node weights are equal to one and we will solve them as equipartition problems, i.e., τ = 0. An example is displayed in Figure 5.1.
Finally, there are instances from VLSI design, which were created by Ferreira et al. using a placement and decomposition code developed by J¨unger, Martin, Reinelt and Weismantel [79]. We use the initials alue, alut, diw, dmxa, gap, and taq that denote the chips, for which the design problem had to be solved. Note that the study of Ferreira et al. only considered instances on less
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Figure 5.1: An optimal equipartition with 7 cut edges for graph mesh.274.469
than 300 nodes. All larger instances, and those that do not appear in [43], were given to us by Alexander Martin. On all VLSI instances, we will consider the minimum bisection problem with τ = 0.05.
Ferreira et al. [43] mainly consider problem (MNCGP). Explicit solutions for the equiparti- tioning problem are only given for four of the finite element mesh instances. Therefore, a direct comparison will be rather limited.
5.1.4
KKT Graphs
The last set of test graphs was considered by Helmberg [71]. They originate from nested bisec- tion approaches for solving sparse symmetric linear systems, like KKT-systems, communicated to Helmberg by Sharon Filipowski from Boeing, with a note that standard bisection heuristics seem not to work well on the problems and no bounding method was available to judge the quality of the produced solutions. We denote them with the initials kkt. More details on the adjacency matrices can be found in [62]. They have the following structure.
A = H B
T
B 0
! ,
where the block H represents an approximate Hessian of the Lagrangian for a nonlinear optimisa- tion problem and the block B corresponds to constraints. On the KKT instances we will always consider (MB) with τ = 0.05.
5.1.5
Computational Environment and Relevance of Computation Times
We have performed all test runs on identical HP Compaq DC7100 Pentium 4 540 (3.2 GHz) HT with 800 MHz FSB, 1 MByte level 2 cache, 1 GByte RAM and SuSE Linux 9.3. Computation times
between individual machines did not differ significantly, we observed differences of at most 1%. Furthermore, multiple users also did not effect the accuracy of our time measurements. However, not all times that we give in this section are directly comparable with each other, because we used the GCC compiler in debug mode during the implementation phase. Some of our tests had to be executed already at this time. When we finally switched to the optimisation mode of the compiler, we observed that run times decreased by about 60%.
A direct comparison of our run times to those of other computational studies is difficult, since the computational environments may differ significantly. Whenever possible, we adjusted our time limits so that they reflected the time limits of the other studies by looking at the computational parameters given by these studies, e.g., processor speed. Of course, developments in the software used by older studies, for instance CPLEX as the LP, solver cannot be fully accounted for.