59,290 research outputs found
The optimal layout of football players: A case study for AC Milan
This paper attempts to find the optimal formation of three midfielders and three forward football players on ground, using the classic Quadratic Assignment Problem or Facility Layout problem. Players are treated as âmachinesâ, their positions as locations, and the flow of materials between machines as âflow of passesâ and âflow of markingsâ. Based on detailed statistics from four matches of AC Milan, and formulated the problem as minimum (quick strategy), maximum (slow strategy), and mixed or balanced strategies, a number of various layouts emerged. Compared to the initial formation of players, the efficiency time gains in the unconditioned layouts are between 3 and 6.8%. Also, when the manager claims that his three forwards shouldnât shift positions with the midfielders, the efficiency gains in these restricted layouts is about 14´´ to 74´´, which is about 1 to 3% of the approximately 40´ effective time spent into passes and markings from both teams.sports; layout; assignment; football players; passes; markings; time;
A nonmonotone GRASP
A greedy randomized adaptive search procedure (GRASP) is an itera-
tive multistart metaheuristic for difficult combinatorial optimization problems. Each
GRASP iteration consists of two phases: a construction phase, in which a feasible
solution is produced, and a local search phase, in which a local optimum in the
neighborhood of the constructed solution is sought. Repeated applications of the con-
struction procedure yields different starting solutions for the local search and the
best overall solution is kept as the result. The GRASP local search applies iterative
improvement until a locally optimal solution is found. During this phase, starting from
the current solution an improving neighbor solution is accepted and considered as the
new current solution. In this paper, we propose a variant of the GRASP framework that
uses a new ânonmonotoneâ strategy to explore the neighborhood of the current solu-
tion. We formally state the convergence of the nonmonotone local search to a locally
optimal solution and illustrate the effectiveness of the resulting Nonmonotone GRASP
on three classical hard combinatorial optimization problems: the maximum cut prob-
lem (MAX-CUT), the weighted maximum satisfiability problem (MAX-SAT), and
the quadratic assignment problem (QAP)
Speeding up IP-based Algorithms for Constrained Quadratic 0-1 Optimization
In many practical applications, the task is to optimize a non-linear objective function over the vertices of a well-studied polytope as, e.g., the matching polytope or the travelling salesman polytope (TSP).Prominent examples are the quadratic assignment problem and the quadratic knapsack problem; further applications occur in various areas such as production planning or automatic graph drawing. In order to apply branch-and-cut methods for the exact solution of such problems, the objective function has to be linearized. However, the standard linearization usually leads to very weak relaxations. On the other hand, problem-specific polyhedral studies are often time-consuming.Our goal is the design of general separation routines that can replace detailed polyhedral studies of the resulting polytope and that can be used as a black box. As unconstrained binary quadratic optimization is equivalent to the maximum cut problem, knowledge about cut polytopes can be used in our setting. Other separation routines are inspired by the local cuts that have been developed by Applegate, Bixby, Chvatal and Cook for faster solution of large-scale traveling salesman instances. Finally, we apply quadratic reformulations of the linear constraints as proposed by Helmberg, Rendl and Weismantel for the quadratic knapsack problem. By extensive experiments, we show that a suitable combination of these methods leads to a drastical speedup in the solution of constrained quadratic 0-1 problems. We also discuss possible generalizations of these methods to arbitrary non-linear objective functions
Speeding up IP-based Algorithms for Constrained Quadratic 0-1 Optimization
In many practical applications, the task is to optimize a non-linear objective function over the vertices of a well-studied polytope as, e.g., the matching polytope or the travelling salesman polytope (TSP).Prominent examples are the quadratic assignment problem and the quadratic knapsack problem; further applications occur in various areas such as production planning or automatic graph drawing. In order to apply branch-and-cut methods for the exact solution of such problems, the objective function has to be linearized. However, the standard linearization usually leads to very weak relaxations. On the other hand, problem-specific polyhedral studies are often time-consuming.Our goal is the design of general separation routines that can replace detailed polyhedral studies of the resulting polytope and that can be used as a black box. As unconstrained binary quadratic optimization is equivalent to the maximum cut problem, knowledge about cut polytopes can be used in our setting. Other separation routines are inspired by the local cuts that have been developed by Applegate, Bixby, Chvatal and Cook for faster solution of large-scale traveling salesman instances. Finally, we apply quadratic reformulations of the linear constraints as proposed by Helmberg, Rendl and Weismantel for the quadratic knapsack problem. By extensive experiments, we show that a suitable combination of these methods leads to a drastical speedup in the solution of constrained quadratic 0-1 problems. We also discuss possible generalizations of these methods to arbitrary non-linear objective functions
The Random Quadratic Assignment Problem
Optimal assignment of classes to classrooms \cite{dickey}, design of DNA
microarrays \cite{carvalho}, cross species gene analysis \cite{kolar}, creation
of hospital layouts cite{elshafei}, and assignment of components to locations
on circuit boards \cite{steinberg} are a few of the many problems which have
been formulated as a quadratic assignment problem (QAP). Originally formulated
in 1957, the QAP is one of the most difficult of all combinatorial optimization
problems. Here, we use statistical mechanical methods to study the asymptotic
behavior of problems in which the entries of at least one of the two matrices
that specify the problem are chosen from a random distribution .
Surprisingly, this case has not been studied before using statistical methods
despite the fact that the QAP was first proposed over 50 years ago
\cite{Koopmans}. We find simple forms for and , the
costs of the minimal and maximum solutions respectively. Notable features of
our results are the symmetry of the results for and
and the dependence on only through its mean and standard deviation,
independent of the details of . After the asymptotic cost is determined for
a given QAP problem, one can straightforwardly calculate the asymptotic cost of
a QAP problem specified with a different random distribution
The assignment problem in distributed computing
This dissertation focuses on the problem of assigning the modules of a program to the processors in a distributed system with the goal of minimizing the overall cost of running the program. The cost depends on the execution times of the modules on the processors and on the cost of communication between modules. This module allocation problem arises in a variety of situations where one is interested in making optimum use of available computer resources. The general module allocation problem is intractable; however it becomes polynomially-solvable when the communication graph is restricted. In this dissertation, we restrict our attention to k-trees;As the first problem, we study parametric module allocation on partial k-trees. We allow the costs, both execution and communication, to vary linearly as functions of a real parameter t. We show that if the number of processors is fixed, the sequence of optimum assignments that are obtained, as t varies from zero to infinity, can be constructed in polynomial time. As an auxiliary result, we develop a linear-time algorithm to find a separator in a k-tree. We discuss the implications of our results for parametric versions of the weighted vertex cover, independent set, and 0-1 quadratic programming problems on partial k-trees;Next, we consider two variants of the assignment problem. The first problem is to find a minimum-cost assignment when one of the processors has a limited memory. The second is to find an assignment that minimizes the maximum processor load. We present exact dynamic programming algorithms for both problems, which lead to approximation schemes for the case where the communication graph is a partial k-tree. Faster algorithms are presented for trees with uniform costs. In contrast to these results, we show that, for arbitrary graphs, no fully polynomial time approximation schemes exist unless P = NP. Both dynamic programming algorithms have been implemented. The implementation details and our experimental results are presented
- âŚ