arXiv subjects

Kai Kahler

Publications and source records attributed to Kai Kahler.

4 recordsLinked to original sources

arXiv2025

Equivalent Instances for Scheduling and Packing Problems

Two instances $(I,k)$ and $(I',k')$ of a parameterized problem $P$ are equivalent if they have the same set of solutions (static equivalent) or if the set of solutions of $(I,k)$ can be constructed by the set of solutions for $(I',k')$ and some computable pre-solutions. If the algorithm constructing such a (static) equivalent instance whose size is polynomial bounded runs in fixed-parameter tractable (FPT) time, we say that there exists a (static) equivalent instance for this problem. In this paper we present (static) equivalent instances for Scheduling and Knapsack problems. We improve the bound for the $\ell_1$-norm of an equivalent vector given by Eisenbrand, Hunkenschröder, Klein, Koutecký, Levin, and Onn and show how this yields equivalent instances for integer linear programs (ILPs) and related problems. We obtain an $O(MN^2\log(NU))$ static equivalent instance for feasibility ILPs where $M$ is the number of constraints, $N$ is the number of variables and $U$ is an upper bound for the $\ell_\infty$-norm of the smallest feasible solution. With this, we get an $O(n^2\log(n))$ static equivalent instance for Knapsack where $n$ is the number of items. Moreover, we give an $O(M^2N\log(NMΔ))$ kernel for feasibility ILPs where $Δ$ is an upper bound for the $\ell_\infty$-norm of the given constraint matrix. Using balancing results by Knop et al., the ConfILP and a proximity result by Eisenbrand and Weismantel we give an $O(d^2\log(p_{\max}))$ equivalent instance for LoadBalancing, a generalization of scheduling problems. Here $d$ is the number of different processing times and $p_{\max}$ is the largest processing time.

cs.CC↗

arXiv2025

New Algorithm for Combinatorial $n$-folds and Applications

Block-structured integer linear programs (ILPs) play an important role in various application fields. We address $n$-fold ILPs where the matrix $\mathcal{A}$ has a specific structure, i.e., where the blocks in the lower part of $\mathcal{A}$ consist only of the row vectors $(1,\dots,1)$. In this paper, we propose an approach tailored to exactly these combinatorial $n$-folds. We utilize a divide and conquer approach to separate the original problem such that the right-hand side iteratively decreases in size. We show that this decrease in size can be calculated such that we only need to consider a bounded amount of possible right-hand sides. This, in turn, lets us efficiently combine solutions of the smaller right-hand sides to solve the original problem. We can decide the feasibility of, and also optimally solve, such problems in time $(n r Δ)^{O(r)} \log(\|b\|_\infty),$ where $n$ is the number of blocks, $r$ the number of rows in the upper blocks and $Δ=\|A\|_\infty$. We complement the algorithm by discussing applications of the $n$-fold ILPs with the specific structure we require. We consider the problems of (i) scheduling on uniform machines, (ii) closest string and (iii) (graph) imbalance. Regarding (i), our algorithm results in running times of $p_{\max}^{O(d)}|I|^{O(1)},$ matching a lower bound derived via ETH. For (ii) we achieve running times matching the current state-of-the-art in the general case. In contrast to the state-of-the-art, our result can leverage a bounded number of column-types to yield an improved running time. For (iii), we improve the parameter dependency on the size of the vertex cover.

cs.DS↗

arXiv2025

Exact and Approximate High-Multiplicity Scheduling on Identical Machines

Goemans and Rothvoss (SODA'14) gave a framework for solving problems which can be described as finding a point in int$.$cone$(P\cap\mathbb{Z}^N)\cap Q$, where $P,Q\subset\mathbb{R}^N$ are (bounded) polyhedra. The running time for solving such a problem is $enc(P)^{2^{O(N)}}enc(Q)^{O(1)}$. This framework can be used to solve various scheduling problems, but the encoding length $enc(P)$ usually involves large parameters like the makespan. We describe three tools to improve the framework: - Problem-specific preprocessing can be used to greatly reduce $enc(P)$. - By solving a certain LP relaxation and then using the classical result by Frank and Tardos (J. Comb. '87), we get a more compact encoding of $P$ in general. - A result by Jansen and Klein (SODA'17) makes the running time depend on the number of vertices of the integer hull of $P$. We provide a new bound for this number that is similar to the one by Berndt et al. (SOSA'21) but better for our setting. For example, applied to the scheduling problem $P||C_{\max}$, these tools improve the running time from $(\log(C_{\max}))^{2^{O(d)}}enc(I)^{O(1)}$ to the possibly much better $(\log(p_{\max}))^{2^{O(d)}}enc(I)^{O(1)}$. Here, $p_{\max}$ is the largest processing time, $d$ is the number of different processing times, $C_{\max}$ is the makespan and $enc(I)$ is the encoding length of the instance. On the complexity side, we use reductions from the literature to provide new parameterized lower bounds for $P||C_{\max}$. Finally, we show that the big open question asked by Mnich and van Bevern (Comput. Oper. Res. '18) whether $P||C_{\max}$ is FPT w.r.t. the number of job types $d$ has the same answer as the question whether $Q||C_{\max}$ is FPT w.r.t. the number of job and machine types $d+τ$ (all in high-multiplicity encoding). The same holds for objective $C_{\min}$.

cs.DS↗

arXiv2022

On the Complexity of Scheduling Problems With a Fixed Number of Parallel Identical Machines

In parallel machine scheduling, we are given a set of jobs, together with a number of machines and our goal is to decide for each job, when and on which machine(s) it should be scheduled in order to minimize some objective function. Different machine models, job characteristics and objective functions result in a multitude of scheduling problems and many of them are NP-hard, even for a fixed number of identical machines. In this work, we give conditional running time lower bounds for a large number of scheduling problems, indicating the optimality of some classical algorithms. Most notably, we show that the algorithm by Lawler and Moore for $1||\sum w_jU_j$ and $Pm||C_{max}$, as well as the algorithm by Lee and Uzsoy for $P2||\sum w_jC_j$ are probably optimal. There is still small room for improvement for the $1|Rej\leq Q|\sum w_jU_j$ algorithm by Zhang et al., the algorithm for $1||\sum T_j$ by Lawler and the FPTAS for $1||\sum w_jU_j$ by Gens and Levner. We also give a lower bound for $P2|any|C_{max}$ and improve the dynamic program by Du and Leung from $\mathcal{O}(nP^2)$ to $\mathcal{O}(nP)$, matching this new lower bound. Here, $P$ is the sum of all processing times. The same idea also improves the algorithm for $P3|any|C_{max}$ by Du and Leung from $\mathcal{O}(nP^5)$ to $\mathcal{O}(nP^2)$. While our results suggest the optimality of some classical algorithms, they also motivate future research in cases where the best known algorithms do not quite match the lower bounds.

cs.DS↗