134 research outputs found
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
Combinatorial optimization with one quadratic term
Diese Arbeit befasst sich mit einer neuen Herangehensweise für binäre kombinatorische Optimierungsprobleme. Die wesentliche Idee hierbei ist, die Anzahl der quadratischen Terme in der Zielfunktion auf einen einzigen zu beschränken, und das durch eine Linearisierung entstehende Polyeder zu analysieren. Für diesen Ansatz gibt es mehrere Motivationsgründe. Im Allgemeinen ist das ursprüngliche Problem mit beliebig vielen quadratischen Termen NP-schwer. Doch obwohl eine gute polyedrische Beschreibung mit schnellen Separierungsroutinen die Optimierung in einem Branch-and-Cut-Verfahren signifikant beschleunigen könnte, gibt es bislang nur wenige Erkenntnisse zur polyedrischen Struktur des binären quadratischen Optimierungsproblems. Betrachtet man das reduzierte Problem mit einem quadratischen Term, dann ist eine effiziente Optimierung möglich, falls die zugrundeliegende lineare Version effizient lösbar ist. Somit können hier auch die facettendefinierenden Ungleichungen effizient separiert werden. Darüberhinaus bleiben alle zulässigen Ungleichungen des reduzierten Problems zulässig für das ursprüngliche Problem. In Kombination bedeutet dies, dass Erkenntnisse zur Facettenstruktur des Problems mit einem quadratischen Term direkt zu einer verbesserten polyedrische Beschreibung des Ursprungsproblems führen. Für eine praktische Anwendung dieses theoretischen Ansatzes betrachten wir verschiedene konkrete Optimierungsprobleme mit einem
quadratischen Term und analysieren deren jeweilige polyedrische
Struktur, die sich nach der Linearisierung ergibt. Konkret betrachten
wir das Minimale Spannwald- und das Minimale Spannbaumproblem, das
Minimale Branching- und das Minimale Arboreszenzproblem, das Minimale
Assignmentproblem und das Maximale Matchingproblem. Für jedes dieser
Optimierungsprobleme leiten wir neue Klassen von facettendefinierenden
Ungleichungen her. Außerdem präsentieren wir für das Minimale
Spannwald- und das Minimale Spannbaumproblem eine vollständige
Beschreibung der jeweiligen Polytope. Für die verwandten gerichteten
Probleme, das Minimale Branching- und das Minimale Arboreszenzproblem,
zeigen wir zwar einerseits einige Gemeinsamkeiten mit den
ungerichteten Problemen, andererseits aber auch, dass sich die
polyedrischen Strukturen im gerichteten Fall durch die
zusätzlichen Gradbedingungen deutlich verkomplizieren. Bei der
Untersuchung des Minimalen Assignmentproblems mit einem quadratischen
Term stellen wir nicht nur die Vermutung über die vollständige
polyedrische Beschreibung auf, sondern kommen insbesondere zu der
überraschenden Erkenntnis, dass bereits ein einziger quadratischer
Term genügen kann, um die Anzahl der Facetten von polynomiell auf
exponentiell zu erhöhen. Die größte Vielfalt an Facettenklassen
leiten wir für das Polyeder des Maximalen Matchingproblems mit einem
quadratischen Term her. Wir zeigen jedoch auch, dass diese noch nicht genügen, um die vollständige Beschreibung des Polyeders zu erhalten. Da die meisten der hergeleiteten Facettenklassen von exponentieller Größe sind, leiten wir verschiedene Routinen für eine polynomielle Separierung her. Unsere exemplarischen Rechenergebnisse für das quadratische Minimale Spannwald- und das quadratische Minimale Spannbaumproblem zeigen die praktische Relevanz unseres Ansatzes
Convex Algebraic Geometry Approaches to Graph Coloring and Stable Set Problems
The objective of a combinatorial optimization problem is to find an element that maximizes a given function defined over a large and possibly high-dimensional finite set. It is often the case that the set is so large that solving the problem by inspecting all the elements is intractable. One approach to circumvent this issue is by exploiting the combinatorial structure of the set (and possibly the function) and reformulate the problem into a familiar set-up where known techniques can be used to attack the problem.
Some common solution methods for combinatorial optimization problems involve formulations that make use of Systems of Linear Equations, Linear Programs (LPs), Semidefinite Programs (SDPs), and more generally, Conic and Semi-algebraic Programs. Although, generality often implies flexibility and power in the formulations, in practice, an increase in sophistication usually implies a higher running time of the algorithms used to solve the problem. Despite this, for some combinatorial problems, it is hard to rule out the applicability of one formulation over the other.
One example of this is the Stable Set Problem. A celebrated result of Lovász's states that it is possible to solve (to arbitrary accuracy) in polynomial time the Stable Set Problem for perfect graphs. This is achieved by showing that the Stable Set Polytope of a perfect graph is the projection of a slice of a Positive Semidefinite Cone of not too large dimension. Thus, the Stable Set Problem can be solved with the use of a reasonably sized SDP. However, it is unknown whether one can solve the same problem using a reasonably sized LP. In fact, even for simple classes of perfect graphs, such as Bipartite Graphs, we do not know the right order of magnitude of the minimum size LP formulation of the problem.
Another example is Graph Coloring. In 2008 Jesús De Loera, Jon Lee, Susan Margulies and Peter Malkin proposed a technique to solve several combinatorial problems, including Graph Coloring Problems, using Systems of Linear Equations. These systems are obtained by reformulating the decision version of the combinatorial problem with a system of polynomial equations. By a theorem of Hilbert, known as Hilbert's Nullstellensatz, the infeasibility of this polynomial system can be determined by solving a (usually large) system of linear equations. The size of this system is an exponential function of a parameter that we call the degree of the Nullstellensatz Certificate.
Computational experiments of De Loera et al. showed that the Nullstellensatz method had potential applications for detecting non--colorability of graphs. Even for known hard instances of graph coloring with up to two thousand vertices and tens of thousands of edges the method was useful. Moreover, all of these graphs had very small Nullstellensatz Certificates. Although, the existence of hard non--colorable graph examples for the Nullstellensatz approach are known, determining what combinatorial properties makes the Nullstellensatz approach effective (or ineffective) is wide open.
The objective of this thesis is to amplify our understanding on the power and limitations of these methods, all of these falling into the umbrella of Convex Algebraic Geometry approaches, for combinatorial problems. We do this by studying the behavior of these approaches for Graph Coloring and Stable Set Problems.
First, we study the Nullstellensatz approach for graphs having large girth and chromatic number. We show that that every non--colorable graph with girth needs a Nullstellensatz Certificate of degree to detect its non--colorability. It is our general belief that the power of the Nullstellensatz method is tied with the interplay between local and global features of the encoding polynomial system. If a graph is locally -colorable, but globally non--colorable, we suspect that it will be hard for the Nullstellensatz to detect the non--colorability of the graph. Our results point towards that direction.
Finally, we study the Stable Set Problem for -regular Bipartite Graphs having no , i.e., having no cycle of length four. In 2017 Manuel Aprile \textit{et al.} showed that the Stable Set Polytope of the incidence graph of a Finite Projective Plane of order (hence, -regular) does not admit an LP formulation with fewer than facets. Although, we did not manage to improve this lower bound for general -regular graphs, we show that any -regular bipartite graph having no does not admit an LP formulation with fewer than facets. In addition, we obtain computational results showing the lower bound also holds for the Finite Projective Plane , a -regular graph. It is our belief that Aprile et al. bounds can be improved considerably
Exact methods for nonlinear combinatorial optimization
We consider combinatorial optimization problems with nonlinear objective functions.
Solution approaches for this class of problems proposed so far are either
highly problem-specific or they apply generic algorithms for constrained nonlinear
optimization, which often does not yield satisfactory results in practice.
Our aim is to develop, implement and experimentally evaluate exact algorithms
that address the nonlinearity of the objective function and at the same time exploit
the underlying combinatorial structure of the problem. To this end we follow
two approaches. The first combines good polyhedral descriptions of the objective
function and the feasible set in a branch and cut-algorithm. The second approach
is based on Lagrangean decomposition. By decomposing the original problem into
an unconstrained nonlinear problem and a linear combinatorial problem, we are
able to compute strong dual bounds for the optimal value. The computation of
lower bounds is then embedded into a branch and bound-algorithm. For many
applications there already exist efficient algorithms for the combinatorial subproblem,
thus an important aspect of this thesis is the study of the corresponding
unconstrained nonlinear subproblems.
Both approaches have the advantage that they can easily be adapted to a wide
range of nonlinear combinatorial problems.We devise both polyhedral and decomposition-
based algorithms for submodular applications from wireless network design
and portfolio optimization and evaluate their performance experimentally.
Exploiting the equivalence between unconstrained binary quadratic optimization
and the maximum cut problem gives rise to a branch and cut-algorithm for
quadratic combinatorial problems which we use to compute optimal layouts of
tanglegrams, an application from computational biology. Additionally we study
the effect of quadratic reformulation of linear constraints, both theoretically and
experimentally. The last class of nonlinear combinatorial problems we consider
are two-scenario problems. Here we propose a new technique to compute lower
bounds in the unconstrained subproblem of the decomposition. Our computational
study of the two-scenario minimum spanning tree problem shows that
the new Lagrangean decomposition-based algorithm is able to solve significantly
larger instances than the standard linearization approach
Polyhedral Approaches to Hypergraph Partitioning and Cell Formation
Ankara : Department of Industrial Engineering and Institute of Engineering and Science, Bilkent University, 1994.Thesis (Ph.D.) -- -Bilkent University, 1994.Includes bibliographical references leaves 152-161Hypergraphs are generalizations of graphs in the sense that each hyperedge
can connect more than two vertices. Hypergraphs are used to describe manufacturing
environments and electrical circuits. Hypergraph partitioning in manufacturing
models cell formation in Cellular Manufacturing systems. Moreover,
hypergraph partitioning in VTSI design case is necessary to simplify the layout
problem. There are various heuristic techniques for obtaining non-optimal hypergraph
partitionings reported in the literature. In this dissertation research,
optimal seeking hypergraph partitioning approaches are attacked from polyhedral
combinatorics viewpoint.
There are two polytopes defined on r-uniform hypergraphs in which every
hyperedge has exactly r end points, in order to analyze partitioning related problems.
Their dimensions, valid inequality families, facet defining inequalities are
investigated, and experimented via random test problems.
Cell formation is the first stage in designing Cellular Manufacturing systems.
There are two new cell formation techniques based on combinatorial optimization
principles. One uses graph approximation, creation of a flow equivalent tree by
successively solving maximum flow problems and a search routine. The other
uses the polynomially solvable special case of the one of the previously discussed
polytopes. These new techniques are compared to six well-known cell formation
algorithms in terms of different efficiency measures according to randomly generated
problems. The results are analyzed statistically.Kandiller, LeventPh.D
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