arXiv subjects

Lars Rohwedder

Publications and source records attributed to Lars Rohwedder.

44 records · Page 3Linked to original sources

arXiv2020

Robust Algorithms under Adversarial Injections

In this paper, we study streaming and online algorithms in the context of randomness in the input. For several problems, a random order of the input sequence---as opposed to the worst-case order---appears to be a necessary evil in order to prove satisfying guarantees. However, algorithmic techniques that work under this assumption tend to be vulnerable to even small changes in the distribution. For this reason, we propose a new \emph{adversarial injections} model, in which the input is ordered randomly, but an adversary may inject misleading elements at arbitrary positions. We believe that studying algorithms under this much weaker assumption can lead to new insights and, in particular, more robust algorithms. We investigate two classical combinatorial-optimization problems in this model: Maximum matching and cardinality constrained monotone submodular function maximization. Our main technical contribution is a novel streaming algorithm for the latter that computes a $0.55$-approximation. While the algorithm itself is clean and simple, an involved analysis shows that it emulates a subdivision of the input stream which can be used to greatly limit the power of the adversary.

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arXiv2019

A Quasi-Polynomial Approximation for the Restricted Assignment Problem

The Restricted Assignment Problem is a prominent special case of Scheduling on Parallel Unrelated Machines. For the strongest known linear programming relaxation, the configuration LP, we improve the non-constructive bound on its integrality gap from 1.9142 to 1.8334 and significantly simplify the proof. Then we give a constructive variant, yielding a 1.8334-approximation in quasi-polynomial time. This is the first quasi-polynomial algorithm for this problem improving on the long-standing approximation rate of 2.

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arXiv2019

Approximation results for makespan minimization with budgeted uncertainty

We study approximation algorithms for the problem of minimizing the makespan on a set of machines with uncertainty on the processing times of jobs. In the model we consider, which goes back to~\cite{BertsimasS03}, once the schedule is defined an adversary can pick a scenario where deviation is added to some of the jobs' processing times. Given only the maximal cardinality of these jobs, and the magnitude of potential deviation for each job, the goal is to optimize the worst-case scenario. We consider both the cases of identical and unrelated machines. Our main result is an EPTAS for the case of identical machines. We also provide a $3$-approximation algorithm and an inapproximability ratio of $2-ε$ for the case of unrelated machines

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arXiv2019

Online Bin Covering with Limited Migration

Semi-online models where decisions may be revoked in a limited way have been studied extensively in the last years. This is motivated by the fact that the pure online model is often too restrictive to model real-world applications, where some changes might be allowed. A well-studied measure of the amount of decisions that can be revoked is the migration factor $β$: When an object $o$ of size $s(o)$ arrives, the decisions for objects of total size at most $β\cdot s(o)$ may be revoked. Usually $β$ should be a constant. This means that a small object only leads to small changes. This measure has been successfully investigated for different, classic problems such as bin packing or makespan minimization. The dual of makespan minimization - the Santa Claus or machine covering problem - has also been studied, whereas the dual of bin packing - the bin covering problem - has not been looked at from such a perspective. In this work, we extensively study the bin covering problem with migration in different scenarios. We develop algorithms both for the static case - where only insertions are allowed - and for the dynamic case, where items may also depart. We also develop lower bounds for these scenarios both for amortized migration and for worst-case migration showing that our algorithms have nearly optimal migration factor and asymptotic competitive ratio (up to an arbitrary small $\eps$). We therefore resolve the competitiveness of the bin covering problem with migration.

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arXiv2018

Near-Linear Time Algorithm for n-fold ILPs via Color Coding

We study an important case of ILPs $\max\{c^Tx \ \vert\ \mathcal Ax = b, l \leq x \leq u,\, x \in \mathbb{Z}^{n t} \} $ with $n\cdot t$ variables and lower and upper bounds $\ell, u\in\mathbb Z^{nt}$. In $n$-fold ILPs non-zero entries only appear in the first $r$ rows of the matrix $\mathcal A$ and in small blocks of size $s\times t$ along the diagonal underneath. Despite this restriction many optimization problems can be expressed in this form. It is known that $n$-fold ILPs can be solved in FPT time regarding the parameters $s, r,$ and $Δ$, where $Δ$ is the greatest absolute value of an entry in $\mathcal A$. The state-of-the-art technique is a local search algorithm that subsequently moves in an improving direction. Both, the number of iterations and the search for such an improving direction take time $Ω(n)$, leading to a quadratic running time in $n$. We introduce a technique based on Color Coding, which allows us to compute these improving directions in logarithmic time after a single initialization step. This leads to the first algorithm for $n$-fold ILPs with a running time that is near-linear in the number $nt$ of variables, namely $(rsΔ)^{O(r^2s + s^2)} L^2 \cdot nt \log^{O(1)}(nt)$, where $L$ is the encoding length of the largest integer in the input. In contrast to the algorithms in recent literature, we do not need to solve the LP relaxation in order to handle unbounded variables. Instead, we give a structural lemma to introduce appropriate bounds. If, on the other hand, we are given such an LP solution, the running time can be decreased by a factor of $L$.

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arXiv2018

Local search breaks 1.75 for Graph Balancing

Graph Balancing is the problem of orienting the edges of a weighted multigraph so as to minimize the maximum weighted in-degree. Since the introduction of the problem the best algorithm known achieves an approximation ratio of $1.75$ and it is based on rounding a linear program with this exact integrality gap. It is also known that there is no $(1.5 - ε)$-approximation algorithm, unless $\mathrm{P}=\mathrm{NP}$. Can we do better than $1.75$? We prove that a different LP formulation, the configuration LP, has a strictly smaller integrality gap. Graph Balancing was the last one in a group of related problems from literature, for which it was open whether the configuration LP is stronger than previous, simple LP relaxations. We base our proof on a local search approach that has been applied successfully to the more general Restricted Assignment problem, which in turn is a prominent special case of makespan minimization on unrelated machines. With a number of technical novelties we are able to obtain a bound of $1.749$ for the case of Graph Balancing. It is not clear whether the local search algorithm we present terminates in polynomial time, which means that the bound is non-constructive. However, it is a strong evidence that a better approximation algorithm is possible using the configuration LP and it allows the optimum to be estimated within a factor better than $1.75$. A particularly interesting aspect of our techniques is the way we handle small edges in the local search. We manage to exploit the configuration constraints enforced on small edges in the LP. This may be of interest to other problems such as Restricted Assignment as well.

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arXiv2018

A note on the integrality gap of the configuration LP for restricted Santa Claus

In the restricted Santa Claus problem we are given resources $\mathcal R$ and players $\mathcal P$. Every resource $j\in\mathcal R$ has a value $v_j$ and every player $i$ desires a set $\mathcal R(i)$ of resources. We are interested in distributing the resources to players that desire them. The quality of a solution is measured by the least happy player, i.e., the lowest sum of resource values. This value should be maximized. The local search algorithm by Asadpour et al. and its connection to the configuration LP has proved itself to be a very influential technique for this and related problems. In the original proof, a local search was used to obtain a bound of $4$ for the ratio of the fractional to the integral optimum of the configuration LP (integrality gap). This bound is non-constructive since the local search has not been shown to terminate in polynomial time. On the negative side, the worst instance known has an integrality gap of $2$. Although much progress was made in this area, neither bound has been improved since. We present a better analysis that shows the integrality gap is not worse than $3 + 5/6 \approx 3.8333$.

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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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