@misc{GroetschelMartinWeismantel, author = {Gr{\"o}tschel, Martin and Martin, Alexander and Weismantel, Robert}, title = {The Steiner Tree Packing Problem in VLSI-Design.}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-1328}, number = {SC-94-02}, abstract = {In this paper we describe several versions of the routing problem arising in VLSI design and indicate how the Steiner tree packing problem can be used to model these problems mathematically. We focus on switchbox routing problems and provide integer programming formulations for routing in the knock-knee and in the Manhattan model. We give a brief sketch of cutting plane algorithms that we developed and implemented for these two models. We report on computational experiments using standard test instances. Our codes are able to determine optimum solutions in most cases, and in particular, we can show that some of the instances have no feasible solution if Manhattan routing is used instead of knock-knee routing.}, language = {en} } @misc{FerreiraGroetschelKiefletal., author = {Ferreira, Carlos E. and Gr{\"o}tschel, Martin and Kiefl, Stefan and Krispenz, Ludwig and Martin, Alexander and Weismantel, Robert}, title = {Some Integer Programs Arising in the Design of Main Frame Computers.}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-957}, number = {SC-92-25}, abstract = {In this paper we describe and discuss a problem that arises in the (global) design of a main frame computer. The task is to assign certain functional units to a given number of so called multi chip modules or printed circuit boards taking into account many technical constraints and minimizing a complex objective function. We describe the real world problem. A thorough mathematical modelling of all aspects of this problem results in a rather complicated integer program that seems to be hopelessly difficult -- at least for the present state of integer programming technology. We introduce several relaxations of the general model, which are also \$NP\$-hard, but seem to be more easily accessible. The mathematical relations between the relaxations and the exact formulation of the problem are discussed as well.}, language = {en} } @misc{GroetschelMartinWeismantel, author = {Gr{\"o}tschel, Martin and Martin, Alexander and Weismantel, Robert}, title = {Routing in Grid Graphs by Cutting Planes.}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-967}, number = {SC-92-26}, abstract = {{\def\N{{\cal N}} \def\R{\hbox{\rm I\kern-2pt R}} \def\MN{{\rm I\kern-2pt N}} In this paper we study the following problem, which we call the weighted routing problem. Let be given a graph \$G=(V,E)\$ with non-negative edge weights \$w_e\in\R_+\$ and integer edge capacities \$c_e\in\MN\$ and let \$\N=\{T_1,\ldots,T_N\}\$, \$N\ge 1\$, be a list of node sets. The weighted routing problem consists in finding edge sets \$S_1,\ldots,S_N\$ such that, for each \$k\in\{1,\ldots,N\}\$, the subgraph \$(V(S_k),S_k)\$ contains an \$[s,t]\$-path for all \$s,t\in T_k\$, at most \$c_e\$ of these edge sets use edge \$e\$ for each \$e\in E\$, and such that the sum of the weights of the edge sets is minimal. Our motivation for studying this problem arises from the routing problem in VLSI-design, where given sets of points have to be connected by wires. We consider the weighted routing problem from a polyhedral point of view. We define an appropriate polyhedron and try to (partially) describe this polyhedron by means of inequalities. We briefly sketch our separation algorithms for some of the presented classes of inequalities. Based on these separation routines we have implemented a branch and cut algorithm. Our algorithm is applicable to an important subclass of routing problems arising in VLSI-design, namely to problems where the underlying graph is a grid graph and the list of node sets is located on the outer face of the grid. We report on our computational experience with this class of problem instances.}}, language = {en} } @misc{GroetschelMartinWeismantel, author = {Gr{\"o}tschel, Martin and Martin, Alexander and Weismantel, Robert}, title = {Packing Steiner Trees: Further Facets.}, doi = {10.1006/eujc.1996.0004}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-975}, number = {SC-93-01}, abstract = {In this paper we continue the investigations in [GMW92a] for the \def\sbppo{Steiner tree packing polyhedron} \sbppo. We present several new classes of valid inequalities and give sufficient (and necessary) conditions for these inequalities to be facet-defining. It is intended to incorporate these inequalities into an existing cutting plane algorithm that is applicable to practical problems arising in the design of electronic circuits.}, language = {en} } @misc{GroetschelMartinWeismantel, author = {Gr{\"o}tschel, Martin and Martin, Alexander and Weismantel, Robert}, title = {Packing Steiner Trees: Separation Algorithms.}, doi = {10.1137/S0895480193258716}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-984}, number = {SC-93-02}, abstract = {In this paper we investigate separation problems for classes of inequalities valid for the polytope associated with the Steiner tree packing problem, a problem that arises, e.~g., in VLSI routing. The separation problem for Steiner partition inequalities is \${\cal N}\hskip-2pt{\cal P}\$-hard in general. We show that it can be solved in polynomial time for those instances that come up in switchbox routing. Our algorithm uses dynamic programming techniques. These techniques are also applied to the much more complicated separation problem for alternating cycle inequalities. In this case we can compute in polynomial time, given some point \$y\$, a lower bound for the gap \$\alpha-a^Ty\$ over all alternating cycle inequalities \$a^Tx\ge\alpha\$. This gives rise to a very effective separation heuristic. A by-product of our algorithm is the solution of a combinatorial optimization problem that is interesting in its own right: Find a shortest path in a graph where the ``length'' of a path is its usual length minus the length of its longest edge.}, language = {en} } @misc{FerreiraMartinWeismantel, author = {Ferreira, Carlos E. and Martin, Alexander and Weismantel, Robert}, title = {Facets for the Multiple Knapsack Problem.}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-1007}, number = {SC-93-04}, abstract = {In this paper we consider the multiple knapsack problem which is defined as follows: given a set \$N\$ of items with weights \$f_i\$, \$i \in N\$, a set \$M\$ of knapsacks with capacities \$F_k\$, \$k \in M\$, and a profit function \$c_{ik}, i \in N, k \in M\$; find an assignment of a subset of the set of items to the set of knapsacks that yields maximum profit (or minimum cost). With every instance of this problem we associate a polyhedron whose vertices are in one to one correspondence to the feasible solutions of the instance. This polytope is the subject of our investigations. In particular, we present several new classes of inequalities and work out necessary and sufficient conditions under which the corresponding inequality defines a facet. Some of these conditions involve only properties of certain knapsack constraints, and hence, apply to the generalized assignment polytope as well. The results presented here serve as the theoretical basis for solving practical problems. The algorithmic side of our study, i.e., separation algorithms, implementation details and computational experience with a branch and cut algorithm are discussed in the companion paper SC 93-07.}, language = {en} } @misc{FerreiraMartinSouzaetal., author = {Ferreira, Carlos E. and Martin, Alexander and Souza, Cid C. de and Weismantel, Robert and Wolsey, Laurence}, title = {Formulations and Valid Inequalities for the Node Capacitated Graph Partitioning Problem.}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-1450}, number = {SC-94-16}, abstract = {We investigate the problem of partitioning the nodes of a graph under capacity restriction on the sum of the node weights in each subset of the partition. The objective is to minimize the sum of the costs of the edges between the subsets of the partition. This problem has a variety of applications, for instance in the design of electronic circuits and devices. We present alternative integer programming formulations for this problem and discuss the links between these formulations. Having chosen to work in the space of edges of the multicut, we investigate the convex hull of incidence vectors of feasible multicuts. In particular, several classes of inequalities are introduced, and their strength and robustness are analyzed as various problem parameters change.}, language = {en} } @misc{FerreiraMartinSouzaetal., author = {Ferreira, Carlos E. and Martin, Alexander and Souza, Cid C. de and Weismantel, Robert and Wolsey, Laurence}, title = {The Node Capacitated Graph Partitioning Problem: A Computational Study.}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-1469}, number = {SC-94-17}, abstract = {In this paper we consider the problem of \$k\$-partitioning the nodes of a graph with capacity restrictions on the sum of the node weights in each subset of the partition, and the objective of minimizing the sum of the costs of the edges between the subsets of the partition. Based on a study of valid inequalities, we present a variety of separation heuristics for so-called cycle, cycle with ears, knapsack tree and path-block-cycle inequalities. The separation heuristics, plus primal heuristics, have been implemented in a branch-and-cut routine using a formulation including the edges with nonzero costs and node variables. Results are presented for three classes of problems: equipartitioning problems arising in finite element methods and partitioning problems associated with electronic circuit layout and compiler design.}, language = {en} } @misc{GroetschelMartinWeismantel, author = {Gr{\"o}tschel, Martin and Martin, Alexander and Weismantel, Robert}, title = {Optimum Path Packing on Wheels: The Noncrossing Case.}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-1225}, number = {SC-93-26}, abstract = {We show that, given a wheel with nonnegative edge lengths and pairs of terminals located on the wheel's outer cycle such that no two terminal pairs cross, then a path packing, i.~e.,a collection of edge disjoint paths connecting the given terminal pairs, of minimum length can be found in strongly polynomial time. Moreover, we exhibit for this case a system of linear inequalities that provides a complete and nonredundant description of the path packing polytope, which is the convex hull of all incidence vectors of path packings and their supersets.}, language = {en} } @misc{MartinWeismantel, author = {Martin, Alexander and Weismantel, Robert}, title = {Packing Paths and Steiner Trees: Routing of Electronic Circuits.}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-1113}, number = {SC-93-15}, abstract = {One of the challenging problems in the design of electronic circuits is the so-called routing problem. Roughly speaking, the task is to connect so-called terminal sets via wires on a predefined area. In addition, certain design rules are to be taken into account and an objective function such as the wiring length must be minimized. The routing problem in general is too complex to be solved in one step. Depending on the user's choice of decomposing the chip design problem into a hierarchy of stages, on the underlying technology, and on the given design rules, various subproblems arise. We discuss several variants of practically relevant routing problems and give a short overview on the underlying technologies and design rules. Many of the routing problems that come up this way can be formulated as the problem of packing so-called Steiner trees in certain graphs. We consider the Steiner tree packing problem from a polyhedral point of view and present three possibilities to define an appropriate polyhedron. Weighing their pros and cons we decide for one of these polytopes and sketch some of our investigations.}, language = {en} }