arXiv subjects

Klaus Jansen

Publications and source records attributed to Klaus Jansen.

69 records · Page 4Linked to original sources

arXiv2018

Online Strip Packing with Polynomial Migration

We consider the relaxed online strip packing problem: Rectangular items arrive online and have to be packed without rotations into a strip of fixed width such that the packing height is minimized. Thereby, repacking of previously packed items is allowed. The amount of repacking is measured by the migration factor, defined as the total size of repacked items divided by the size of the arriving item. First, we show that no algorithm with constant migration factor can produce solutions with asymptotic ratio better than 4/3. Against this background, we allow amortized migration, i.e. to save migration for a later time step. As a main result, we present an AFPTAS with asymptotic ratio $1 + \mathcal{O}(ε)$ for any $ε> 0$ and amortized migration factor polynomial in $1 / ε$. To our best knowledge, this is the first algorithm for online strip packing considered in a repacking model.

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arXiv2018

Scheduling Monotone Moldable Jobs in Linear Time

A moldable job is a job that can be executed on an arbitrary number of processors, and whose processing time depends on the number of processors allotted to it. A moldable job is monotone if its work doesn't decrease for an increasing number of allotted processors. We consider the problem of scheduling monotone moldable jobs to minimize the makespan. We argue that for certain compact input encodings a polynomial algorithm has a running time polynomial in n and log(m), where n is the number of jobs and m is the number of machines. We describe how monotony of jobs can be used to counteract the increased problem complexity that arises from compact encodings, and give tight bounds on the approximability of the problem with compact encoding: it is NP-hard to solve optimally, but admits a PTAS. The main focus of this work are efficient approximation algorithms. We describe different techniques to exploit the monotony of the jobs for better running times, and present a (3/2+ε)-approximate algorithm whose running time is polynomial in log(m) and 1/ε, and only linear in the number n of jobs.

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arXiv2017

Improved approximation for two dimensional strip packing with polynomial bounded width

We study the well-known two-dimensional strip packing problem. Given is a set of rectangular axis-parallel items and a strip of width $W$ with infinite height. The objective is to find a packing of these items into the strip, which minimizes the packing height. Lately, it has been shown that the lower bound of $3/2$ of the absolute approximation ratio can be beaten when we allow a pseudo-polynomial running-time of type $(n W)^{f(1/\varepsilon)}$. If $W$ is polynomially bounded by the number of items, this is a polynomial running-time. We present a pseudo-polynomial algorithm with approximation ratio $4/3 +\varepsilon$ and running time $(n W)^{1/\varepsilon^{\mathcal{O}(2^{1/\varepsilon})}}$.

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arXiv2017

An EPTAS for Scheduling on Unrelated Machines of Few Different Types

In the classical problem of scheduling on unrelated parallel machines, a set of jobs has to be assigned to a set of machines. The jobs have a processing time depending on the machine and the goal is to minimize the makespan, that is the maximum machine load. It is well known that this problem is NP-hard and does not allow polynomial time approximation algorithms with approximation guarantees smaller than $1.5$ unless P$=$NP. We consider the case that there are only a constant number $K$ of machine types. Two machines have the same type if all jobs have the same processing time for them. This variant of the problem is strongly NP-hard already for $K=1$. We present an efficient polynomial time approximation scheme (EPTAS) for the problem, that is, for any $\varepsilon > 0$ an assignment with makespan of length at most $(1+\varepsilon)$ times the optimum can be found in polynomial time in the input length and the exponent is independent of $1/\varepsilon$. In particular we achieve a running time of $2^{\mathcal{O}(K\log(K) \frac{1}{\varepsilon}\log^4 \frac{1}{\varepsilon})}+\mathrm{poly}(|I|)$, where $|I|$ denotes the input length. Furthermore, we study three other problem variants and present an EPTAS for each of them: The Santa Claus problem, where the minimum machine load has to be maximized; the case of scheduling on unrelated parallel machines with a constant number of uniform types, where machines of the same type behave like uniformly related machines; and the multidimensional vector scheduling variant of the problem where both the dimension and the number of machine types are constant. For the Santa Claus problem we achieve the same running time. The results are achieved, using mixed integer linear programming and rounding techniques.

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arXiv2017

Complexity and Inapproximability Results for Parallel Task Scheduling and Strip Packing

We study the Parallel Task Scheduling problem $Pm|size_j|C_{\max}$ with a constant number of machines. This problem is known to be strongly NP-complete for each $m \geq 5$, while it is solvable in pseudo-polynomial time for each $m \leq 3$. We give a positive answer to the long-standing open question whether this problem is strongly $NP$-complete for $m=4$. As a second result, we improve the lower bound of $\frac{12}{11}$ for approximating pseudo-polynomial Strip Packing to $\frac{5}{4}$. Since the best known approximation algorithm for this problem has a ratio of $\frac{4}{3} + \varepsilon$, this result narrows the gap between approximation ratio and inapproximability result by a significant step. Both results are proven by a reduction from the strongly $NP$-complete problem 3-Partition.

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arXiv2017

Structural Parameters for Scheduling with Assignment Restrictions

We consider scheduling on identical and unrelated parallel machines with job assignment restrictions. These problems are NP-hard and they do not admit polynomial time approximation algorithms with approximation ratios smaller than $1.5$ unless P$=$NP. However, if we impose limitations on the set of machines that can process a job, the problem sometimes becomes easier in the sense that algorithms with approximation ratios better than $1.5$ exist. We introduce three graphs, based on the assignment restrictions and study the computational complexity of the scheduling problem with respect to structural properties of these graphs, in particular their tree- and rankwidth. We identify cases that admit polynomial time approximation schemes or FPT algorithms, generalizing and extending previous results in this area.

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arXiv2017

On the Configuration-LP of the Restricted Assignment Problem

We consider the classical problem of Scheduling on Unrelated Machines. In this problem a set of jobs is to be distributed among a set of machines and the maximum load (makespan) is to be minimized. The processing time $p_{ij}$ of a job $j$ depends on the machine $i$ it is assigned to. Lenstra, Shmoys and Tardos gave a polynomial time $2$-approximation for this problem. In this paper we focus on a prominent special case, the Restricted Assignment problem, in which $p_{ij}\in\{p_j,\infty\}$. The configuration-LP is a linear programming relaxation for the Restricted Assignment problem. It was shown by Svensson that the multiplicative gap between integral and fractional solution, the integrality gap, is at most $2 - 1/17 \approx 1.9412$. In this paper we significantly simplify his proof and achieve a bound of $2 - 1/6 \approx 1.8333$. As a direct consequence this provides a polynomial $(2 - 1/6 + ε)$-estimation algorithm for the Restricted Assignment problem by approximating the configuration-LP. The best lower bound known for the integrality gap is $1.5$ and no estimation algorithm with a guarantee better than $1.5$ exists unless $\mathrm{P} = \mathrm{NP}$.

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arXiv2016

About the Structure of the Integer Cone and its Application to Bin Packing

We consider the bin packing problem with $d$ different item sizes and revisit the structure theorem given by Goemans and Rothvoß[6] about solutions of the integer cone. We present new techniques on how solutions can be modified and give a new structure theorem that relies on the set of vertices of the underlying integer polytope. As a result of our new structure theorem, we obtain an algorithm for the bin packing problem with running time $|V|^{2^{O(d)}} \cdot enc(I)^{O(1)}$, where $V$ is the set of vertices of the integer knapsack polytope and $enc(I)$ is the encoding length of the bin packing instance. The algorithm is fixed parameter tractable, parameterized by the number of vertices of the integer knapsack polytope $|V|$. This shows that the bin packing problem can be solved efficiently when the underlying integer knapsack polytope has an easy structure, i.e. has a small number of vertices. Furthermore, we show that the presented bounds of the structure theorem are asymptotically tight. We give a construction of bin packing instances using new structural insights and classical number theoretical theorems which yield the desired lower bound.

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arXiv2016

Closing the Gap for Makespan Scheduling via Sparsification Techniques

Makespan scheduling on identical machines is one of the most basic and fundamental packing problems studied in the discrete optimization literature. It asks for an assignment of $n$ jobs to a set of $m$ identical machines that minimizes the makespan. The problem is strongly NP-hard, and thus we do not expect a $(1+ε)$-approximation algorithm with a running time that depends polynomially on $1/ε$. Furthermore, Chen et al. [3] recently showed that a running time of $2^{(1/ε)^{1-δ}}+\text{poly}(n)$ for any $δ>0$ would imply that the Exponential Time Hypothesis (ETH) fails. A long sequence of algorithms have been developed that try to obtain low dependencies on $1/ε$, the better of which achieves a running time of $2^{\tilde{O}(1/ε^2)}+O(n\log n)$ [11]. In this paper we obtain an algorithm with a running time of $2^{\tilde{O}(1/ε)}+O(n\log n)$, which is tight under ETH up to logarithmic factors on the exponent. Our main technical contribution is a new structural result on the configuration-IP. More precisely, we show the existence of a highly symmetric and sparse optimal solution, in which all but a constant number of machines are assigned a configuration with small support. This structure can then be exploited by integer programming techniques and enumeration. We believe that our structural result is of independent interest and should find applications to other settings. In particular, we show how the structure can be applied to the minimum makespan problem on related machines and to a larger class of objective functions on parallel machines. For all these cases we obtain an efficient PTAS with running time $2^{\tilde{O}(1/ε)} + \text{poly}(n)$.

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arXiv2015

A Faster FPTAS for the Unbounded Knapsack Problem

The Unbounded Knapsack Problem (UKP) is a well-known variant of the famous 0-1 Knapsack Problem (0-1 KP). In contrast to 0-1 KP, an arbitrary number of copies of every item can be taken in UKP. Since UKP is NP-hard, fully polynomial time approximation schemes (FPTAS) are of great interest. Such algorithms find a solution arbitrarily close to the optimum $\mathrm{OPT}(I)$, i.e. of value at least $(1-\varepsilon) \mathrm{OPT}(I)$ for $\varepsilon > 0$, and have a running time polynomial in the input length and $\frac{1}{\varepsilon}$. For over thirty years, the best FPTAS was due to Lawler with a running time in $O(n + \frac{1}{\varepsilon^3})$ and a space complexity in $O(n + \frac{1}{\varepsilon^2})$, where $n$ is the number of knapsack items. We present an improved FPTAS with a running time in $O(n + \frac{1}{\varepsilon^2} \log^3 \frac{1}{\varepsilon})$ and a space bound in $O(n + \frac{1}{\varepsilon} \log^2 \frac{1}{\varepsilon})$. This directly improves the running time of the fastest known approximation schemes for Bin Packing and Strip Packing, which have to approximately solve UKP instances as subproblems.

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arXiv2015

Algorithmic Aspects of Upper Domination

In this paper we study combinatorial and algorithmic resp. complexity questions of upper domination, i.e., the maximum cardinality of a minimal dominating set in a graph. We give a full classification of the related maximisation and minimisation problems, as well as the related parameterised problems, on general graphs and on graphs of bounded degree, and we also study planar graphs.

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arXiv2015

Fully Dynamic Bin Packing Revisited

We consider the fully dynamic bin packing problem, where items arrive and depart in an online fashion and repacking of previously packed items is allowed. The goal is, of course, to minimize both the number of bins used as well as the amount of repacking. A recently introduced way of measuring the repacking costs at each timestep is the migration factor, defined as the total size of repacked items divided by the size of an arriving or departing item. Concerning the trade-off between number of bins and migration factor, if we wish to achieve an asymptotic competitive ration of $1 + ε$ for the number of bins, a relatively simple argument proves a lower bound of $Ω(\frac{1}ε)$ for the migration factor. We establish a nearly matching upper bound of $O(\frac{1}ε^4 \log \frac{1}ε)$ using a new dynamic rounding technique and new ideas to handle small items in a dynamic setting such that no amortization is needed. The running time of our algorithm is polynomial in the number of items $n$ and in $\frac{1}ε$. The previous best trade-off was for an asymptotic competitive ratio of $\frac{5}{4}$ for the bins (rather than $1+ε$) and needed an amortized number of $O(\log n)$ repackings (while in our scheme the number of repackings is independent of $n$ and non-amortized).

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arXiv2013

On the optimality of approximation schemes for the classical scheduling problem

We consider the classical scheduling problem on parallel identical machines to minimize the makespan, and achieve the following results under the Exponential Time Hypothesis (ETH) 1. The scheduling problem on a constant number $m$ of identical machines, which is denoted as $Pm||C_{max}$, is known to admit a fully polynomial time approximation scheme (FPTAS) of running time $O(n) + (1/ε)^{O(m)}$ (indeed, the algorithm works for an even more general problem where machines are unrelated). We prove this algorithm is essentially the best possible in the sense that a $(1/ε)^{O(m^{1-δ})}+n^{O(1)}$ time FPTAS for any $δ>0$ implies that ETH fails. 2. The scheduling problem on an arbitrary number of identical machines, which is denoted as $P||C_{max}$, is known to admit a polynomial time approximation scheme (PTAS) of running time $2^{O(1/ε^2\log^3(1/ε))}+n^{O(1)}$. We prove this algorithm is nearly optimal in the sense that a $2^{O((1/ε)^{1-δ})}+n^{O(1)}$ time PTAS for any $δ>0$ implies that ETH fails, leaving a small room for improvement. To obtain these results we will provide two new reductions from 3SAT, one for $Pm||C_{max}$ and another for $P||C_{max}$. Indeed, the new reductions explore the structure of scheduling problems and can also lead to other interesting results. For example, using the framework of our reduction for $P||C_{max}$, Chen et al. (arXiv:1306.3727) is able to prove the APX-hardness of the scheduling problem in which the matrix of job processing times $P=(p_{ij})_{m\times n}$ is of rank 3, solving the open problem mentioned by Bhaskara et al. (SODA 2013).

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arXiv2013

A Robust AFPTAS for Online Bin Packing with Polynomial Migration

In this paper we develop general LP and ILP techniques to find an approximate solution with improved objective value close to an existing solution. The task of improving an approximate solution is closely related to a classical theorem of Cook et al. in the sensitivity analysis for LPs and ILPs. This result is often applied in designing robust algorithms for online problems. We apply our new techniques to the online bin packing problem, where it is allowed to reassign a certain number of items, measured by the migration factor. The migration factor is defined by the total size of reassigned items divided by the size of the arriving item. We obtain a robust asymptotic fully polynomial time approximation scheme (AFPTAS) for the online bin packing problem with migration factor bounded by a polynomial in $\frac{1}ε$. This answers an open question stated by Epstein and Levin in the affirmative. As a byproduct we prove an approximate variant of the sensitivity theorem by Cook at el. for linear programs.

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arXiv2008

Trimming of Graphs, with Application to Point Labeling

For $t,g>0$, a vertex-weighted graph of total weight $W$ is $(t,g)$-trimmable if it contains a vertex-induced subgraph of total weight at least $(1-1/t)W$ and with no simple path of more than $g$ edges. A family of graphs is trimmable if for each constant $t>0$, there is a constant $g=g(t)$ such that every vertex-weighted graph in the family is $(t,g)$-trimmable. We show that every family of graphs of bounded domino treewidth is trimmable. This implies that every family of graphs of bounded degree is trimmable if the graphs in the family have bounded treewidth or are planar. Based on this result, we derive a polynomial-time approximation scheme for the problem of labeling weighted points with nonoverlapping sliding labels of unit height and given lengths so as to maximize the total weight of the labeled points. This settles one of the last major open questions in the theory of map labeling.

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