Graver basis and proximity techniques for block-structured separable convex integer minimization problems

Hemmecke, Raymond; Köppe, Matthias; Weismantel, Robert

doi:10.1007/s10107-013-0638-z

Cited by 19 publications

(35 citation statements)

References 21 publications

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“…However, the algorithm of Hemmecke et al [16] only works for linear and certain restricted separable convex objectives. Thus, using the ideas of Hemmecke, Köppe and Weismantel [15], we extend the previous result to optimizing any separable convex function.…”

Section: Our Contributionsupporting

confidence: 60%

Scheduling meets n-fold integer programming

Knop

Koutecký

2017

J Sched

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Scheduling problems are fundamental in combinatorial optimization. Much work has been done on approximation algorithms for NP-hard cases, but relatively little is known about exact solutions when some part of the input is a fixed parameter. In 2014, Mnich and Wiese initiated a systematic study in this direction. In this paper we continue this study and show that several additional cases of fundamental scheduling problems are fixed parameter tractable for some natural parameters. Our main tool is n-fold integer programming, a recent variable dimension technique which we believe to be highly relevant for the parameterized complexity community. This paper serves to showcase and highlight this technique. Specifically, we show the following four scheduling problems to be fixed-parameter tractable, where pmax is the maximum processing time of a job and wmax is the maximum weight of a job:-Makespan minimization on uniformly related machines (Q||Cmax) parameterized by pmax, -Makespan minimization on unrelated machines (R||Cmax) parameterized by pmax and the number of kinds of machines (defined later), -Sum of weighted completion times minimization on unrelated machines (R|| wjCj) parameterized by pmax + wmax and the number of kinds of machines, -The same problem, R|| wjCj, parameterized by the number of distinct job times and the number of machines.

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Section: Our Contributionsupporting

confidence: 60%

Scheduling meets n-fold integer programming

Knop

Koutecký

2017

J Sched

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“…In the following, we use the proximity technique pioneered by Hochbaum and Shantikumar [HS90] in the case of totally unimodular matrices and extended to the setting of Graver bases by Hemmecke, Köppe and Weismantel [HKW14]. This technique allows to show that, provided some structure of the constraints (e.g., total unimodularity or bounded ℓ ∞ -norm of its Graver elements), the continuous optimum is not too far from the integer optimum of the problem.…”

Section: Step Lengthsmentioning

confidence: 99%

“…This problem (1) is known as n-fold integer programming (IP ) E (n) ,b,l,u,f . Building on a dynamic program of Hemmecke, Onn and Romanchuk [HOR13] and a so-called proximity technique of Hemmecke, Köppe and Weismantel [HKW14], Knop and Koutecký [KK17] prove that:…”

Section: Introductionmentioning

confidence: 99%

Combinatorial n-fold integer programming and applications

2019

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Many fundamental NP-hard problems can be formulated as integer linear programs (ILPs). A famous algorithm by Lenstra solves ILPs in time that is exponential only in the dimension of the program, and polynomial in the size of the ILP. That algorithm became a ubiquitous tool in the design of fixedparameter algorithms for NP-hard problems, where one wishes to isolate the hardness of a problem by some parameter. However, in many cases using Lenstra's algorithm has two drawbacks: First, the run time of the resulting algorithms is often doubly-exponential in the parameter, and second, an ILP formulation in small dimension cannot easily express problems involving many different costs.Inspired by the work of Hemmecke, Onn and Romanchuk [Math. Prog. 2013], we develop a singleexponential algorithm for so-called combinatorial n-fold integer programs, which are remarkably similar to prior ILP formulations for various problems, but unlike them, also allow variable dimension. We then apply our algorithm to a few representative problems like Closest String, Swap Bribery, Weighted Set Multicover, and obtain exponential speedups in the dependence on the respective parameters, the input size, or both.Unlike Lenstra's algorithm, which is essentially a bounded search tree algorithm, our result uses the technique of augmenting steps. At its heart is a deep result stating that in combinatorial n-fold IPs, existence of an augmenting step implies existence of a "local" augmenting step, which can be found using dynamic programming. Our results provide an important insight into many problems by showing that they exhibit this phenomenon, and highlights the importance of augmentation techniques.1 Kannan's algorithm in parameterized complexity: they modeled Closest String with k input strings as an ILP of dimension k O(k) , and thereby concluded with the first fixed-parameter algorithm for Closest String. This success led Niedermeier [Nie04] to propose in his book:[...] It remains to investigate further examples besides Closest String where the described ILP approach turns out to be applicable. More generally, it would be interesting to discover more connections between fixed-parameter algorithms and (integer) linear programming.Since then, many more applications of Lenstra's and Kannan's algorithm for parameterized problems have been proposed. However, essentially all of them [BFN + 15, DS12, FLM + 08, HR15, Lam12, MW15] share a common trait with the algorithm for Closest String: they have a doubly-exponential run time dependence on the parameter. Moreover, it is difficult to find ILP formulations with small dimension for problems whose input contains many objects with varying cost functions, such as in Swap Bribery [BCF + 14, Challenge #2].

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“…However, there still remain powerful tools from the theory of integer programming which, surprisingly, have not found applications in the design of parameterized algorithms. For one example take a result of Hemmecke, Köppe and Weismantel [16] about 2-stage stochastic integer programming with n scenarios. They describe an FPT algorithm (which builds on a deep structural insight) for solving integer programs with a certain block structure and bounded coefficients.…”

Section: Discussion and Open Problemsmentioning

confidence: 99%