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FPTAS for mixed-integer polynomial optimization with a fixed number of variables [article]

Jesús A. De Loera , Robert Weismantel
2005 arXiv   pre-print
We show the existence of an FPTAS for the problem of maximizing a non-negative polynomial over mixed-integer sets in convex polytopes, when the number of variables is fixed.  ...  The first author gratefully acknowledges support from NSF grant DMS-0309694, a 2003 UC-Davis Chancellor's fellow award, the Alexander von Humboldt foundation, and IMO-Magdeburg.  ...  Introduction A well-known result by H.W. Lenstra Jr. states that linear mixed integer programming problems with fixed number of variables can be solved in polynomial time on the input size [10] .  ... 
arXiv:math/0505677v1 fatcat:g6onwutvqjgptln3dsbpkozs3y

FPTAS for optimizing polynomials over the mixed-integer points of polytopes in fixed dimension

Jesús A. De Loera, Raymond Hemmecke, Matthias Köppe, Robert Weismantel
2007 Mathematical programming  
We show the existence of a fully polynomial-time approximation scheme (FPTAS) for the problem of maximizing a non-negative polynomial over mixed-integer sets in convex polytopes, when the number of variables  ...  Moreover, using a weaker notion of approximation, we show the existence of a fully polynomial-time approximation scheme for the problem of maximizing or minimizing an arbitrary polynomial over mixed-integer  ...  Likewise, mixed integer linear programming problems with a fixed number of integer variables can be solved in polynomial time.  ... 
doi:10.1007/s10107-007-0175-8 fatcat:ivvels5xrbhpzkmmxwoqdy7bcy

Integer Polynomial Optimization in Fixed Dimension

Jesús A. De Loera, Raymond Hemmecke, Matthias Köppe, Robert Weismantel
2006 Mathematics of Operations Research  
We classify, according to their computational complexity, integer optimization problems whose constraints and objective functions are polynomials with integer coefficients and the number of variables is  ...  For the optimization of an integer polynomial over the lattice points of a convex polytope, we show an algorithm to compute lower and upper bounds for the optimal value.  ...  Alexander Barvinok, who communicated to us that Lemma 3.1 was true for variable D, and thus they had indeed obtained an FPTAS from the construction of the upper and lower bounds.  ... 
doi:10.1287/moor.1050.0169 fatcat:meybqyoyd5ainor6rwt4uqiv7u

On the complexity of nonlinear mixed-integer optimization [article]

Matthias Köppe
2010 arXiv   pre-print
This is a survey on the computational complexity of nonlinear mixed-integer optimization.  ...  It highlights a selection of important topics, ranging from incomputability results that arise from number theory and logic, to recently obtained fully polynomial time approximation schemes in fixed dimension  ...  The author wishes to thank the referees, in particular for their comments on the presentation of the Lenstra-type algorithm, and his student Robert Hildebrand for a subsequent discussion about this topic  ... 
arXiv:1006.4895v1 fatcat:2ixomlaycnhldo3q3peetlreiq

Preemptive scheduling with rejection

Han Hoogeveen, Martin Skutella, Gerhard J. Woeginger
2003 Mathematical programming  
We provide a complete classification of these scheduling problems with respect to complexity and approximability. Our main results are on the variant with an arbitrary number of unrelated machines.  ...  The scheduler may reject a subset of the jobs and thereby incur jobdependent penalties for each rejected job, and he must construct a schedule for the remaining jobs so as to optimize the preemptive makespan  ...  A family of polynomial time approximation algorithms with performance guarantee 1 + ε for all fixed ε > 0 is called a polynomial time approximation scheme (PTAS).  ... 
doi:10.1007/s10107-002-0324-z fatcat:yxit66xtvba4ld452su6z3duy4

Parametric Integer Programming Algorithm for Bilevel Mixed Integer Programs

M. Köppe, M. Queyranne, C. T. Ryan
2010 Journal of Optimization Theory and Applications  
We consider discrete bilevel optimization problems where the follower solves an integer program with a fixed number of variables.  ...  For the pure integer case where the leader's variables are integer, and hence optimal solutions are guaranteed to exist, we present two algorithms which run in polynomial time when the total number of  ...  Open Access This article is distributed under the terms of the Creative Commons Attribution Noncommercial License which permits any noncommercial use, distribution, and reproduction in any medium, provided  ... 
doi:10.1007/s10957-010-9668-3 fatcat:zwas47utrbgchfbt5lldmdbjoy

Nonlinear Integer Programming [chapter]

Raymond Hemmecke, Matthias Köppe, Jon Lee, Robert Weismantel
2009 50 Years of Integer Programming 1958-2008  
Research efforts of the past fifty years have led to a development of linear integer programming as a mature discipline of mathematical optimization.  ...  Such a level of maturity has not been reached when one considers nonlinear systems subject to integrality requirements for the variables. This chapter is dedicated to this topic.  ...  Acknowledgments We would like to thank our coauthors, Jesús De Loera and Shmuel Onn, for their permission to base the presentation of some of the material in this chapter on our joint papers [43, 86,  ... 
doi:10.1007/978-3-540-68279-0_15 fatcat:hpo6uuxsorg6pcvh7jtwpa6w6e

Preemptive Scheduling with Rejection [chapter]

Han Hoogeveen, Martin Skutella, Gerhard J. Woeginger
2000 Lecture Notes in Computer Science  
We provide a complete classification of these scheduling problems with respect to complexity and approximability. Our main results are on the variant with an arbitrary number of unrelated machines.  ...  The scheduler may reject a subset of the jobs and thereby incur job-dependent penalties for each rejected job, and he must construct a schedule for the remaining jobs so as to optimize the preemptive makespan  ...  Consider an optimal solution $x_{ij}^{*},$ $y_{j}^{*}$ , and $\tau*$ of the mixed integer linear program (1) .  ... 
doi:10.1007/3-540-45253-2_25 fatcat:6nyrgt7rbngm5hinpqodhg2nmi

An FPTAS for Minimizing Indefinite Quadratic Forms over Integers in Polyhedra [article]

Robert Hildebrand, Robert Weismantel, Kevin Zemmer
2015 arXiv   pre-print
We apply it, for instance, to the Motzkin polynomial and to indefinite quadratic forms x^T Q x in a fixed number of variables, where Q has at most one positive, or at most one negative eigenvalue.  ...  We present a generic approach that allows us to develop a fully polynomial-time approximation scheme (FTPAS) for minimizing nonlinear functions over the integer points in a rational polyhedron in fixed  ...  The computational complexity of mixed-integer polynomial optimization in fixed dimension was surveyed in [8, 7] , and they develop an FPTAS for maximizing non-negative polynomials over integer and mixedinteger  ... 
arXiv:1507.00969v2 fatcat:sv6sxetbuzdfbkppwnhvbwoqda

A Fully Polynomial Time Approximation Scheme for Packing While Traveling [article]

Frank Neumann, Sergey Polyakovskiy, Martin Skutella, Leen Stougie, Junhua Wu
2017 arXiv   pre-print
We give an exact dynamic programming approach for this problem and a fully polynomial time approximation scheme (FPTAS) when maximising the benefit that can be gained over the baseline travel cost.  ...  We investigate the underlying non-linear packing while traveling (PWT) problem of the TTP where items have to be selected along a fixed route.  ...  A FPTAS for a given maximisation problem is an algorithm A that obtains for any valid input I and ǫ, 0 < ǫ ≤ 1, a solution of objective value A(I) ≥ (1 − ǫ)OP T (I) in time polynomial in the input size  ... 
arXiv:1702.05217v1 fatcat:3vktt45ujrc4be4s4mgi66if74

Toward breaking the curse of dimensionality: an FPTAS for stochastic dynamic programs with multidimensional actions and scalar states [article]

Nir Halman, Giacomo Nannicini
2018 arXiv   pre-print
We propose a Fully Polynomial-Time Approximation Scheme (FPTAS) for stochastic dynamic programs with multidimensional action, scalar state, convex costs and linear state transition function.  ...  Our paper enlarges the class of dynamic programs that admit an FPTAS by showing, under suitable conditions, how to deal with multidimensional action spaces and with vectors of continuous random variables  ...  Combining these tools, we obtain an FPTAS for a general class of DP models. These models can be seen as multistage stochastic LPs with one variable linking the stages.  ... 
arXiv:1811.11680v1 fatcat:eb3gnzzdnjfkneqtcyh6geqy3y

An FPTAS for optimizing a class of low-rank functions over a polytope

Shashi Mittal, Andreas S. Schulz
2012 Mathematical programming  
Our technique can also be used to obtain an FPTAS for combinatorial optimization problems with non-linear objective functions, for example when the objective is a product of a fixed number of linear functions  ...  We present a fully polynomial time approximation scheme (FPTAS) for optimizing a very general class of non-linear functions of low rank over a polytope.  ...  Ravi for useful discussions, and for sharing an unpublished draft [9] on a similar topic. The authors also thank Ilias Diakonikolas for sharing a detailed version of his paper [5] .  ... 
doi:10.1007/s10107-011-0511-x fatcat:kreupiw7zjeqlp4z4jchw24gnq

Exploiting Concavity in Bimatrix Games: New Polynomially Tractable Subclasses [chapter]

Spyros Kontogiannis, Paul Spirakis
2010 Lecture Notes in Computer Science  
We then proceed by proposing new subclasses of bimatrix games for which either an exact polynomial-time construction, or at least a FPTAS, is possible.  ...  For the mutually quasi-concave games, we provide (to our knowledge) the first FPTAS for the construction of a Nash equilibrium.  ...  Nevertheless, it should be noted that for the intersection of mutually concave games of fixed rank, rather than giving a FPTAS, we find in polynomial time an exact Nash equilibrium point via the optimal  ... 
doi:10.1007/978-3-642-15369-3_24 fatcat:ukvdyvu6qffhrhkm3tfyeuvrtm

An FPTAS for Computing Nash Equilibrium in Resource Graph Games

Hau Chan, Albert Xin Jiang
2018 Proceedings of the Twenty-Seventh International Joint Conference on Artificial Intelligence  
Our FPTAS can be generalized to compute optimal MSNE, and to games with a constant number of player types.  ...  We consider the problem of computing a mixed-strategy Nash equilibrium (MSNE) in resource graph games (RGGs), a compact representation for games with an exponential number of strategies.  ...  i , we can efficiently generate a mixed strategy σ i , represented as a sparse vector with polynomial number of nonzero entries, such that π i = si∈Si σ i (s i )s i .  ... 
doi:10.24963/ijcai.2018/21 dblp:conf/ijcai/ChanJ18 fatcat:5hs7oh44n5aolnztwrbe6mz4ea

Scheduling under Unavailability Constraints to Minimize Flow-time Criteria [chapter]

Imed Kacem
2007 Multiprocessor Scheduling, Theory and Applications  
Consequently, every node is represented by the following elements: • the number of scheduled jobs denoted by k, • a partial assignment vector: PA = {a 1 , a 2 , ..., a k } with a i {0, 1} i k and a i =  ...  Kellerer and Strusevich proposed also an FPTAS (Fully Polynomial Time Approximation Scheme) with O(n 4 / 2 ) time complexity [14] .  ...  This volume contains four major parts that cover the following directions: the state of the art in theory and algorithms for classical and non-standard scheduling problems; new exact optimization algorithms  ... 
doi:10.5772/5214 fatcat:nzwfuk6uwfezze355ih5inqktm
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