arXiv subjects

Hauke Brinkop

Publications and source records attributed to Hauke Brinkop.

5 recordsLinked to original sources

arXiv2025

Approximation algorithms for integer programming with resource augmentation

The classic algorithm [Papadimitriou, J.ACM '81] for IPs has a running time $n^{O(m)}(m\cdot\max\{Δ,\|\textbf{b}\|_{\infty}\})^{O(m^2)}$, where $m$ is the number of constraints, $n$ is the number of variables, and $Δ$ and $\|\textbf{b}\|_{\infty}$ are, respectively, the largest absolute values among the entries in the constraint matrix and the right-hand side vector of the constraint. The running time is exponential in $m$, and becomes pseudo-polynomial if $m$ is a constant. In recent years, there has been extensive research on FPT (fixed parameter tractable) algorithms for the so-called $n$-fold IPs, which may possess a large number of constraints, but the constraint matrix satisfies a specific block structure. It is remarkable that these FPT algorithms take as parameters $Δ$ and the number of rows and columns of some small submatrices. If $Δ$ is not treated as a parameter, then the running time becomes pseudo-polynomial even if all the other parameters are taken as constants. This paper explores the trade-off between time and accuracy in solving an IP. We show that, for arbitrary small $\varepsilon>0$, there exists an algorithm for IPs with $m$ constraints that runs in ${f(m,\varepsilon)}\cdot\textnormal{poly}(|I|)$ time, and returns a near-feasible solution that violates the constraints by at most $\varepsilonΔ$. Furthermore, for $n$-fold IPs, we establish a similar result -- our algorithm runs in time that depends on the number of rows and columns of small submatrices together with $1/\varepsilon$, and returns a solution that slightly violates the constraints. Meanwhile, both solutions guarantee that their objective values are no worse than the corresponding optimal objective values satisfying the constraints. As applications, our results can be used to obtain additive approximation schemes for multidimensional knapsack as well as scheduling.

math.OC↗

arXiv2025

Robust Scheduling on Uniform Machines -- New Results Using a Relaxed Approximation Guarantee

We consider the problem of scheduling $n$ jobs on $m$ uniform machines while minimizing the makespan ($Q||C_{\max}$) and maximizing the minimum completion time ($Q||C_{\min}$) in an online setting with migration of jobs. In this online setting, the jobs are inserted or deleted over time, and at each step, the goal is to compute a near-optimal solution while reassigning some jobs, such that the overall processing time of reassigned jobs, called migration, is bounded by some factor $β$ times the processing time of the job added or removed. We propose Efficient Polynomial Time Approximation Schemes (EPTASs) with an additional load error of $\mathcal{O}(\varepsilon p_{\max})$ for both problems, with constant amortized migration factor $β$, where $p_{\max}$ is the maximum processing time in the instance over all steps. As an intermediate step, we obtain Efficient Parameterized Approximation Schemes (EPASs) for both problems, $(1+\varepsilon)$-competitive algorithms parameterized by $p_{\max}$ and the number of different processing times $d$ in an instance, with $β$ bounded in a function of $p_{\max}$, $d$ and $\varepsilon$. This is the first result in the direction of a polynomial time approximation scheme in the field of online scheduling with bounded reassignment on uniform machines; before, such results were known only for the considered problems on identical machines. Crucial to our result is a division of the machines into large and small machines depending on the current approximate objective value, allowing for different approaches on either machine set, as well as a new way of rounding the instance that does not depend on the current objective value.

cs.DS↗

arXiv2024

Structural Results for High-Multiplicity Scheduling on Uniform Machines

Parameterizing by the largest processing time $p_{max}$ and the number of different job processing times $d$, we propose a proximity technique for High-Multiplicity Scheduling on Uniform Machines for the objectives Makespan Minimization ($C_{max}$) and Santa Claus ($C_{min}$) to obtain new structural results for these problems. The novelty in our approach is that we deal with a fractional solution for only a sub-instance, where the sub-instance itself is not known a priori. While the construction and computation of the fractional solution -- in contrast to usual proximity techniques -- is not done in polynomial time, this also allows us to formulate a comparably strong and general proximity statement. Eventually, this allows us to reduce the number of jobs that need to be distributed to a polynomial in $p_{max}$ for each machine and job type, by preassigning jobs according to the fractional solution, essentially returning a bounded number (at most $O(p_{max}^{O(d^2)})$) of kernels, one for each (guessed) sub-instance. We can use our structural results to obtain an algorithm with running time is $p_{max}^{O(d^2)}poly|I|$, matching the best-known so far by Knop et al. (Oper. Res. Lett. '21). Moreover, we propose an $p_{max}^{O(d^2)} poly |I|$ time algorithm for Envy Minimization $C_{envy}$ in the High-Multiplicity Setting on Uniform Machines, showing that this problem is \textsc{fpt} in $p_{max}$. Eventually, we also propose a general mechanism to bound the largest coefficient in the Configuration ILP for so called \emph{Load Balancing Problems} by $(dp_{max})^{O(d)}$, which we hope to be of interest for the development of algorithms.

cs.DS↗

arXiv2023

New Support Size Bounds for Integer Programming, Applied to Makespan Minimization on Uniformly Related Machines

Mixed-integer linear programming (MILP) is at the core of many advanced algorithms for solving fundamental problems in combinatorial optimization. The complexity of solving MILPs directly correlates with their support size, which is the minimum number of non-zero integer variables in an optimal solution. A hallmark result by Eisenbrand and Shmonin (Oper. Res. Lett., 2006) shows that any feasible integer linear program (ILP) has a solution with support size $s\leq 2m\cdot\log(4mΔ)$, where $m$ is the number of constraints, and $Δ$ is the largest coefficient in any constraint. Our main combinatorial result are improved support size bounds for ILPs. To improve granularity, we analyze for the largest $1$-norm $A_{\max}$ of any column of the constraint matrix, instead of $Δ$. We show a support size upper bound of $s\leq m\cdot(\log(3A_{\max})+\sqrt{\log(A_{\max})})$, by deriving a new bound on the -1 branch of the Lambert $\mathcal{W}$ function. Additionally, we provide a lower bound of $m\log(A_{\max})$, proving our result asymptotically optimal. Furthermore, we give support bounds of the form $s\leq 2m\cdot\log(1.46A_{\max})$. These improve upon the previously best constants by Aliev. et. al. (SIAM J. Optim., 2018), because all our upper bounds hold equally with $A_{\max}$ replaced by $\sqrt{m}Δ$. Using our combinatorial result, we obtain the fastest known approximation schemes (EPTAS) for the fundamental scheduling problem of makespan minimization of uniformly related machines ($Q\mid\mid C_{\max}$).

cs.DS↗

arXiv2022

Solving Cut-Problems in Quadratic Time for Graphs With Bounded Treewidth

In the problem (Unweighted) Max-Cut we are given a graph $G = (V,E)$ and asked for a set $S \subseteq V$ such that the number of edges from $S$ to $V \setminus S$ is maximal. In this paper we consider an even harder problem: (Weighted) Max-Bisection. Here we are given an undirected graph $G = (V,E)$ and a weight function $w \colon E \to \mathbb Q_{>0}$ and the task is to find a set $S \subseteq V$ such that (i) the sum of the weights of edges from $S$ is maximal; and (ii) $S$ contains $\left\lceil{\frac{n}{2}}\right\rceil$ vertices (where $n = \lvert V\rvert$). We design a framework that allows to solve this problem in time $\mathcal O(2^t n^2)$ if a tree decomposition of width $t$ is given as part of the input. This improves the previously best running time for Max-Bisection of [DBLP:journals/tcs/HanakaKS21] by a factor $t^2$. Under common hardness assumptions, neither the dependence on $t$ in the exponent nor the dependence on $n$ can be reduced [DBLP:journals/tcs/HanakaKS21,DBLP:journals/jcss/EibenLM21,DBLP:journals/talg/LokshtanovMS18]. Our framework can be applied to other cut problems like Min-Edge-Expansion, Sparsest-Cut, Densest-Cut, $β$-Balanced-Min-Cut, and Min-Bisection. It also works in the setting with arbitrary weights and directed edges.

cs.DS↗