arXiv subjects

Andreas S. Schulz

Publications and source records attributed to Andreas S. Schulz.

11 recordsLinked to original sources

arXiv2026

Scheduling with Testing: Competitive Algorithms for Minimizing the Total Weighted Completion Time in the Adversarial Model

We study scheduling with testing on a single machine and on identical parallel machines to minimize the total \emph{weighted} completion time in the adversarial model. In this setting, each job is equipped with a weight, an upper bound on its processing time, and a testing time. An algorithm can either execute a job for an amount of time equal to the upper bound or test it first to reveal a potentially lower processing time used to schedule the job later. We establish the first constant-competitive algorithms for this problem with job-dependent weights that reflect each job's relative importance. For single-machine scheduling, we present a deterministic algorithm with a competitive ratio of 2.3166 and show that a randomized variant has a competitive ratio of 2.1523. These guarantees match the best-known upper bounds in the unweighted setting. Combining these algorithms with list scheduling yields competitive ratios of 2.7763 and 2.5110 for identical-parallel-machine scheduling, improving the previously best-known bounds even in the unweighted case.

cs.DS↗

arXiv2026

Block Stacking, Airplane Refueling, and Robust Appointment Scheduling

How can a stack of identical blocks be arranged to extend beyond the edge of a table as far as possible? We consider a generalization of this classic puzzle to blocks that differ in width and mass. Despite the seemingly simple premise, we demonstrate that it is unlikely that one can efficiently determine a stack configuration of maximum overhang. Formally, we prove that the Block-Stacking Problem is NP-hard, partially answering an open question from the literature. Furthermore, we demonstrate that the restriction to stacks without counterweights has a surprising connection to the Airplane Refueling Problem, another famous puzzle, and to Robust Appointment Scheduling, a problem of practical relevance. In addition to revealing a remarkable relation to the real-world challenge of devising schedules under uncertainty, their equivalence unveils a polynomial-time approximation scheme, that is, a $(1+ε)$-approximation algorithm, for Block Stacking without counterbalancing and a $(2+ε)$-approximation algorithm for the general case.

math.CO↗

arXiv2026

Robust Appointment Scheduling with Waiting Time Guarantees

Appointment scheduling problems under uncertainty encounter a fundamental trade-off between cost minimization and customer waiting times. Most existing studies address this trade-off using a weighted sum approach, which puts little emphasis on individual waiting times and, thus, customer satisfaction. In contrast, we study how to minimize total cost while providing waiting time guarantees to all customers. Given box uncertainty sets for service times and no-shows, we introduce the Robust Appointment Scheduling Problem with Waiting Time Guarantees. We show that the problem is NP-hard in general and introduce a mixed-integer linear program that can be solved in reasonable computation time. For special cases, we prove that polynomial-time variants of the well-known Smallest-Variance-First sequencing rule and the Bailey-Welch scheduling rule are optimal. Furthermore, a case study with data from the radiology department of a large university hospital demonstrates that the approach not only guarantees acceptable waiting times but, compared to existing robust approaches, may simultaneously reduce costs incurred by idle time and overtime. This work suggests that limiting instead of minimizing customer waiting times is a win-win solution in the trade-off between customer satisfaction and cost minimization. Additionally, it provides an easy-to-implement and customizable appointment scheduling framework with waiting time guarantees.

econ.GN↗

arXiv2023

On the complexity of a maintenance problem for hierarchical systems

We prove that a maintenance problem on frequency-constrained maintenance jobs with a hierarchical structure is integer-factorization hard. This result holds even on simple systems with just two components to maintain. As a corollary, we provide a first hardness result for Levi et al.'s modular maintenance scheduling problem (Naval Research Logistics 61, 472-488, 2014).

math.OC↗

arXiv2023

Primal Separation and Approximation for the $\{0, 1/2\}$-closure

We advance the theoretical study of $\{0, 1/2\}$-cuts for integer programming problems $\max\{c^T x \colon A x \leq b, x \text{ integer}\}$. Such cuts are Gomory-Chvátal cuts that only need multipliers of value $0$ or $1/2$ in their derivation. The intersection of all $\{0, 1/2\}$-cuts derived from $Ax \le b$ is denoted by $P_{1/2}$ and called the $\{0,1/2\}$-closure of $P = \{x : Ax \le b\}$. The primal separation problem for $\{0, 1/2\}$-cuts is: Given a vertex $\hat x$ of the integer hull of $P$ and some fractional point $x^* \in P$, does there exist a $\{0,1/2\}$-cut that is tight at $\hat x$ and violated by $x^*$? Primal separation is the key ingredient of primal cutting-plane approaches to integer programming. In general, primal separation for $\{0,1/2\}$-cuts is NP-hard. We present two cases for which primal separation is solvable in polynomial time. As an interesting side product, we obtain a(nother) simple proof that matching can be solved in polynomial time. Furthermore, since optimization over the Gomory-Chvátal closure is also NP-hard, there has been recent research on solving the optimization problem over the Gomory-Chvátal closure approximately. In a similar spirit, we show that the optimization problem over the $\{0,1/2\}$-closure can be solved in polynomial time up to a factor $(1 + \varepsilon)$, for any fixed $\varepsilon > 0$.

cs.DM↗

arXiv2021

On the Complexity of Recognizing Integrality and Total Dual Integrality of the $\{0,1/2\}$-Closure

The $\{0,\frac{1}{2}\}$-closure of a rational polyhedron $\{ x \colon Ax \le b \}$ is obtained by adding all Gomory-Chvátal cuts that can be derived from the linear system $Ax \le b$ using multipliers in $\{0,\frac{1}{2}\}$. We show that deciding whether the $\{0,\frac{1}{2}\}$-closure coincides with the integer hull is strongly NP-hard. A direct consequence of our proof is that, testing whether the linear description of the $\{0,\frac{1}{2}\}$-closure derived from $Ax \le b$ is totally dual integral, is strongly NP-hard.

cs.DM↗

arXiv2021

Genetic column generation: Fast computation of high-dimensional multi-marginal optimal transport problems

We introduce a simple, accurate, and extremely efficient method for numerically solving the multi-marginal optimal transport (MMOT) problems arising in density functional theory. The method relies on (i) the sparsity of optimal plans [for $N$ marginals discretized by $\ell$ gridpoints each, general Kantorovich plans require $\ell^N$ gridpoints but the support of optimizers is of size $O(\ell\cdot N)$ [FV18]], (ii) the method of column generation (CG) from discrete optimization which to our knowledge has not hitherto been used in MMOT, and (iii) ideas from machine learning. The well-known bottleneck in CG consists in generating new candidate columns efficiently; we prove that in our context, finding the best new column is an NP-complete problem. To overcome this bottleneck we use a genetic learning method tailormade for MMOT in which the dual state within CG plays the role of an "adversary", in loose similarity to Wasserstein GANs. On a sequence of benchmark problems with up to 120 gridpoints and up to 30 marginals, our method always found the exact optimizers. Moreover, empirically the number of computational steps needed to find them appears to scale only polynomially when both $N$ and $\ell$ are simultaneously increased (while keeping their ratio fixed to mimic a thermodynamic limit of the particle system).

cs.LG↗

arXiv2020

Computing Approximate Equilibria in Weighted Congestion Games via Best-Responses

We present a deterministic polynomial-time algorithm for computing $d^{d+o(d)}$-approximate (pure) Nash equilibria in (proportional sharing) weighted congestion games with polynomial cost functions of degree at most $d$. This is an exponential improvement of the approximation factor with respect to the previously best deterministic algorithm. An appealing additional feature of the algorithm is that it only uses best-improvement steps in the actual game, as opposed to the previously best algorithms, that first had to transform the game itself. Our algorithm is an adaptation of the seminal algorithm by Caragiannis et al. [FOCS'11, TEAC 2015], but we utilize an approximate potential function directly on the original game instead of an exact one on a modified game. A critical component of our analysis, which is of independent interest, is the derivation of a novel bound of $[d/\mathcal{W}(d/ρ)]^{d+1}$ for the Price of Anarchy (PoA) of $ρ$-approximate equilibria in weighted congestion games, where $\mathcal{W}$ is the Lambert-W function. More specifically, we show that this PoA is exactly equal to $Φ_{d,ρ}^{d+1}$, where $Φ_{d,ρ}$ is the unique positive solution of the equation $ρ(x+1)^d=x^{d+1}$. Our upper bound is derived via a smoothness-like argument, and thus holds even for mixed Nash and correlated equilibria, while our lower bound is simple enough to apply even to singleton congestion games.

cs.GT↗

arXiv2020

Integer factorization and Riemann's hypothesis: Why two-item joint replenishment is hard

Distribution networks with periodically repeating events often hold great promise to exploit economies of scale. Joint replenishment problems are a fundamental model in inventory management, manufacturing, and logistics that capture these effects. However, finding an efficient algorithm that optimally solves these models, or showing that none may exist, has long been open, regardless of whether empty joint orders are possible or not. In either case, we show that finding optimal solutions to joint replenishment instances with just two products is at least as difficult as integer factorization. To the best of the authors' knowledge, this is the first time that integer factorization is used to explain the computational hardness of any kind of optimization problem. Under the assumption that Riemann's Hypothesis is correct, we can actually prove that the two-item joint replenishment problem with possibly empty joint ordering points is NP-complete under randomized reductions, which implies that not even quantum computers may be able to solve it efficiently. By relating the computational complexity of joint replenishment to cryptography, prime decomposition, and other aspects of prime numbers, a similar approach may help to establish (integer factorization) hardness of additional open periodic problems in supply chain management and beyond, whose solution has eluded standard methods.

cs.CC↗

arXiv2020

Approximation Algorithms and LP Relaxations for Scheduling Problems Related to Min-Sum Set Cover

We consider single-machine scheduling problems that are natural generalizations or variations of the min-sum set cover problem and the min-sum vertex cover problem. For each of these problems, we give new approximation algorithms. Some of these algorithms rely on time-indexed LP relaxations. We show how a variant of alpha-point scheduling leads to the best-known approximation ratios, including a guarantee of 4 for an interesting special case of the so-called generalized min-sum set cover problem. We also make explicit the connection between the greedy algorithm for min-sum set cover and the concept of Sidney decomposition for precedence-constrained single-machine scheduling, and show how this leads to a 4-approximation algorithm for single-machine scheduling with so-called bipartite OR-precedence constraints.

cs.DS↗

arXiv2017

Robust Monotone Submodular Function Maximization

We consider a robust formulation, introduced by Krause et al. (2008), of the classical cardinality constrained monotone submodular function maximization problem, and give the first constant factor approximation results. The robustness considered is w.r.t. adversarial removal of up to $τ$ elements from the chosen set. For the fundamental case of $τ=1$, we give a deterministic $(1-1/e)-1/Θ(m)$ approximation algorithm, where $m$ is an input parameter and number of queries scale as $O(n^{m+1})$. In the process, we develop a deterministic $(1-1/e)-1/Θ(m)$ approximate greedy algorithm for bi-objective maximization of (two) monotone submodular functions. Generalizing the ideas and using a result from Chekuri et al. (2010), we show a randomized $(1-1/e)-ε$ approximation for constant $τ$ and $ε\leq \frac{1}{\tildeΩ(τ)}$, making $O(n^{1/ε^3})$ queries. Further, for $τ\ll \sqrt{k}$, we give a fast and practical 0.387 algorithm. Finally, we also give a black box result result for the much more general setting of robust maximization subject to an Independence System.

cs.DS↗