10,072 research outputs found
Aspects Of Combinatorial Geometry
This thesis presents solutions to various problems in the expanding field of combinatorial geometry. Chapter 1 gives an introduction to the theory of the solution of an integer programming problem, that is maximising a linear form with integer variables subject to a number of constraints. Since the maximum value of the linear form occurs at a vertex of the convex hull of integer points defined by the constraints, it is of interest to estimate the number of these vertices. Chapter 2 describes the application of certain geometrical interpretations of number theory to the solution of integer programming problems in the plane. By using, in part, the well-known Klein interpretation of continued fractions, a method of constructing the vertices of the convex hull of integer points defined by particular constraints is developed. Bounds for the number of these vertices and properties of certain special cases are given. Chapter 3 considers the general d-dimensional integer programming problem. Upper and lower bounds are presented for the number of vertices of the convex hull of integer points defined by particular constraints. Chapter 4 is concerned with the approximation of convex sets by convex polytopes. First, a detailed description of recent work on minimal circumscribing triangles for convex polygons and the extension to minimal circumscribing equilateral triangles is given. This leads to a new approach to constructing a Borsuk Division and finding a regular hexagon circumscribing a convex polygon. Then, a method of approximating general convex sets by convex polytopes is presented, leading to consideration of the problem of a d-simplex approximating a d-ball. Chapter 5 develops algorithms for finding points with particular combinatorial properties, using containment objects such as balls, closed half-spaces and ellipsoids. Chapter 6 gives a new approach to the problem of inscribing a square in a convex polygon, leading to possible ideas for an algorithm
Determining All Integer Vertices of the PESP Polytope by Flipping Arcs
We investigate polyhedral aspects of the Periodic Event Scheduling Problem (PESP), the mathematical basis for periodic timetabling problems in public transport. Flipping the orientation of arcs, we obtain a new class of valid inequalities, the flip inequalities, comprising both the known cycle and change-cycle inequalities. For a point of the LP relaxation, a violated flip inequality can be found in pseudo-polynomial time, and even in linear time for a spanning tree solution. Our main result is that the integer vertices of the polytope described by the flip inequalities are exactly the vertices of the PESP polytope, i.e., the convex hull of all feasible periodic slacks with corresponding modulo parameters. Moreover, we show that this flip polytope equals the PESP polytope in some special cases. On the computational side, we devise several heuristic approaches concerning the separation of cutting planes from flip inequalities. We finally present better dual bounds for the smallest and largest instance of the benchmarking library PESPlib
Note on the Complexity of the Mixed-Integer Hull of a Polyhedron
We study the complexity of computing the mixed-integer hull
of a polyhedron .
Given an inequality description, with one integer variable, the mixed-integer
hull can have exponentially many vertices and facets in . For fixed,
we give an algorithm to find the mixed integer hull in polynomial time. Given
and fixed, we compute a vertex description of
the mixed-integer hull in polynomial time and give bounds on the number of
vertices of the mixed integer hull
Lifting Linear Extension Complexity Bounds to the Mixed-Integer Setting
Mixed-integer mathematical programs are among the most commonly used models
for a wide set of problems in Operations Research and related fields. However,
there is still very little known about what can be expressed by small
mixed-integer programs. In particular, prior to this work, it was open whether
some classical problems, like the minimum odd-cut problem, can be expressed by
a compact mixed-integer program with few (even constantly many) integer
variables. This is in stark contrast to linear formulations, where recent
breakthroughs in the field of extended formulations have shown that many
polytopes associated to classical combinatorial optimization problems do not
even admit approximate extended formulations of sub-exponential size.
We provide a general framework for lifting inapproximability results of
extended formulations to the setting of mixed-integer extended formulations,
and obtain almost tight lower bounds on the number of integer variables needed
to describe a variety of classical combinatorial optimization problems. Among
the implications we obtain, we show that any mixed-integer extended formulation
of sub-exponential size for the matching polytope, cut polytope, traveling
salesman polytope or dominant of the odd-cut polytope, needs many integer variables, where is the number of vertices of the
underlying graph. Conversely, the above-mentioned polyhedra admit
polynomial-size mixed-integer formulations with only or (for the traveling salesman polytope) many integer variables.
Our results build upon a new decomposition technique that, for any convex set
, allows for approximating any mixed-integer description of by the
intersection of with the union of a small number of affine subspaces.Comment: A conference version of this paper will be presented at SODA 201
Quantitative Tverberg, Helly, & Carath\'eodory theorems
This paper presents sixteen quantitative versions of the classic Tverberg,
Helly, & Caratheodory theorems in combinatorial convexity. Our results include
measurable or enumerable information in the hypothesis and the conclusion.
Typical measurements include the volume, the diameter, or the number of points
in a lattice.Comment: 33 page
Convex Integer Optimization by Constantly Many Linear Counterparts
In this article we study convex integer maximization problems with composite
objective functions of the form , where is a convex function on
and is a matrix with small or binary entries, over
finite sets of integer points presented by an oracle or by
linear inequalities.
Continuing the line of research advanced by Uri Rothblum and his colleagues
on edge-directions, we introduce here the notion of {\em edge complexity} of
, and use it to establish polynomial and constant upper bounds on the number
of vertices of the projection \conv(WS) and on the number of linear
optimization counterparts needed to solve the above convex problem.
Two typical consequences are the following. First, for any , there is a
constant such that the maximum number of vertices of the projection of
any matroid by any binary matrix is
regardless of and ; and the convex matroid problem reduces to
greedily solvable linear counterparts. In particular, . Second, for any
, there is a constant such that the maximum number of
vertices of the projection of any three-index
transportation polytope for any by any binary
matrix is ; and the convex three-index transportation problem
reduces to linear counterparts solvable in polynomial time
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