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In this paper we describe a cutting plane based algorithm for the multiple knapsack problem. We use our algorithm to solve some practical problem instances arising in the layout of electronic circuits and in the design of main frame computers, and we report on our computational experience. This includes a discussion and evaluation of separation algorithms, an LP-based primal heuristic and some implementation details. The paper is based on the polyhedral theory for the multiple knapsack polytope developed in our companion paper SC 93-04 and meant to turn this theory into an algorithmic tool for the solution of practical problems.
{\def\N{{\mbox{{\rm I\kern-0.22emN}}}} Given a set $N$ of items and a capacity $b \in \N$, and let $N_j$ be the set of items with weight $j$, $1 \leq j \leq b$. The 0/1 knapsack polytope is the convex hull of all 0/1 vectors that satisfy the inequality $$\sum_{j=1}^b \sum_{i \in N_j} jx_i \leq b.$$ In this paper we first present a complete linear description of the 0/1 knapsack polytope for two special cases: (a) $N_j = \emptyset$ for all $1 < j \leq \lfloor {b \over 2} \rfloor$ and (b) $N_j = \emptyset$ for all $1 < j \leq \lfloor {b \over 3} \rfloor$ and $N_j = \emptyset$ for all $j \geq \lfloor {b \over 2} \rfloor + 1$. It turns out that the inequalities that are needed for the complete description of these special polytopes are derived by means of some ``reduction principle''. This principle is then generalized to yield valid and in many cases facet defining inequalities for the general 0/1 knapsack polytope. The separation problem for this class of inequalities can be solved in pseudo polynomial time via dynamic programming techniques.}
{\def\N{{\mbox{{\rm I\kern-0.22emN}}}} Let a set $N$ of items, a capacity $F \in \N$ and weights $a_i \in \N$, $i \in N$ be given. The 0/1 knapsack polytope is the convex hull of all 0/1 vectors that satisfy the inequality $$\sum_{i \in N} a_i x_i \leq F.$$ In this paper we present a linear description of the 0/1 knapsack polytope for the special case where $a_i \in \{\mu,\lambda\}$ for all items $i \in N$ and $1 \leq \mu < \lambda \leq b$ are two natural numbers. The inequalities needed for this description involve elements of the Hilbert basis of a certain cone. The principle of generating inequalities based on elements of a Hilbert basis suggests further extensions.}
Die vorliegende Arbeit beschäftigt sich mit dem Plazierungsproblem, welches beim Entwurf elektronischer Schaltungen auftritt. Das Plazierungsproblem modellieren wir als ein quadratisches 0/1 Optimierungsproblem unter linearen Nebenbedingungen und untersuchen das Modell komplexitätstheoretisch. Der zweite Aspekt der Arbeit bezieht sich auf die Lösung praktischer Problembeispiele im sogenannten Sea of cells"-Entwurfsstil. Zur Lösung dieser Beispiele wurde ein Prototyp implementiert und mit state of the art"-Plazierungsverfahren verglichen. Schlie\ss lich werden wir uns mit dem Clusteringproblem, das eine Variante des Mehrfachschnitt-Problems darstellt, beschäftigen. Dabei steht einerseits im Vordergrund, wie diese Probleme heuristisch gelöst werden können und wie die Integration des Ansatzes in das Plazierungsprogramm erfolgt. Andererseits soll das Clusteringproblem polyedrisch untersucht werden.
In this paper we modify Buchberger's $S$-pair reduction algorithm for computing a Gröbner basis of a toric ideal so as to apply to an integer program in inequality form with fixed right hand sides and fixed upper bounds on the variables. We formulate the algorithm in the original space and interpret the reduction steps geometrically. In fact, three variants of this algorithm are presented and we give elementary proofs for their correctness. A relationship between these (exact) algorithms, iterative improvement heuristics and the Kernighan-Lin procedure is established.
{\def\N{{\mbox{{\rm I\kern-0.22emN}}}}In this paper we introduce a multivariate grading of the toric ideal associated with the integer program $min \{ cx : Ax = b, x \in \N^n \}$, and a truncated Buchberger algorithm to solve the program. In the case of $max \{ cx : Ax \leq b, x \leq u, x \in \N^n \}$ in which all data are non-negative, this algebraic method gives rise to a combinatorial algorithm presented in UWZ94}.
This paper presents some connections between test sets and valid inequalities of integer programs. The reason for establishing such relationships is the hope that information (even partial) on one of these objects can be used to get information on the other and vice versa. We approach this study from two directions: On the one hand we examine the geometric process by which the secondary polytope associated with a matrix $A$ transforms to the state polytope as we pass from linear programs that have $A$ as coefficient matrix to the associated integer programs. The second direction establishes the notion of classes of augmentation vectors parallel to the well known concept of classes of facet defining inequalities for integer programs. We show how certain inequalities for integer programs can be derived from test sets for these programs.
We investigate the generating sets (``Gröbner bases'') of integer lattices which correspond to the Gröbner bases of the associated binomial ideals. Extending results in Sturmfels and Thomas, preprint 1994, we obtain a geometric characterization of the universal Gröbner basis in terms of the vertices and edges of the associated corner polyhedra. We emphasize the special case where the lattice has finite index. In this case the corner polyhedra were studied by Gomory, and there is a close connection to the ``group problem in integer programming'' Schrijver, p.~363. We present exponential lower and upper bounds for the size of a reduced Gröbner basis. The initial complex of (the ideal of) a lattice is shown to be dual to the boundary of a certain simple polyhedron.
{\def\xnew{x^{\mbox{\tiny new}}}\def\Z{{{\rm Z}\!\! Z}}For every fixed set ${\cal F}\subseteq\{0,1\}^n$ the following problems are strongly polynomial time equivalent: given a feasible point $x\in\cal F$ and a linear objective function $c\in\Z^n$, \begin{itemize} \item find a feasible point $x^*\in\cal F$ that maximizes $cx$ (Optimization), \item find a feasible point $\xnew\in\cal F$ with $c\xnew>cx$ (Augmentation), and \item find a feasible point $\xnew\in\cal F$ with $c\xnew>cx$ such that $\xnew-x$ is ``irreducible''\\(Irreducible Augmentation). \end{itemize} This generalizes results and techniques that are well known for $0/1$--integer programming problems that arise from various classes of combinatorial optimization problems.}