arXiv subjects

Julian Golak

Publications and source records attributed to Julian Golak.

8 recordsLinked to original sources

arXiv2026

Optimal Service Commitments in Traveling Salesman Problems with Stochastic Demand

In ridepooling services, mobility providers must communicate an expected arrival time before customers decide whether to book. An ambitious commitment attracts higher demand than a conservative one, but is correspondingly harder to fulfill. Provider revenue is therefore governed by an endogenous interplay between the announced commitment and the demand it induces. We study this interplay in a traveling salesman setting in which all customers share a shuttle to a common destination, formalizing it as the Traveling Salesman Problem with Service Commitments (TSP-SC), a two-stage stochastic optimization problem. In the first stage, the provider announces an arrival commitment to all customers interested in the trip, each of whom independently accepts or rejects the offer with a probability decreasing in the committed time. In the second stage, an Orienteering Problem determines a revenue-maximizing subset of accepting customers to be served within the commitment; the remaining customers are rejected. We show that optimal commitments lie in a finite candidate set induced by the tour lengths of customer subsets, and develop two exact algorithms, a general and a faster variant for homogeneous customers, together with an Adaptive Binomial Sampling Heuristic (BSH) for larger instances. In a numerical study on instances with up to 100 customers, the heuristic deviates from the optimal expected revenue by at most 0.32~\% on average across all exactly solvable sizes. The results further reveal a striking operational regularity: as the customer base grows, the revenue-maximizing policy converges to a regime in which roughly one in three accepting customers is rejected, a rejection rate that remains stable across instance sizes.

math.OC↗

arXiv2026

On the Diameter of Arrangements of Topological Disks

Let $\mathcal{D}=\{D_0,\ldots,D_{n-1}\}$ be a set of $n$ topological disks in the plane and let $\mathcal{A} := \mathcal{A}(\mathcal{D})$ be the arrangement induced by $\mathcal{D}$. For two disks $D_i,D_j\in\mathcal{D}$, let $Δ_{ij}$ be the number of connected components of $D_i\cap D_j$, and let $Δ:= \max_{i,j} Δ_{ij}$. We show that the diameter of $\mathcal{G}^*$, the dual graph of $\mathcal{A}$, can be bounded as a function of $n$ and $Δ$. Thus, any two points in the plane can be connected by a Jordan curve that crosses the disk boundaries a number of times bounded by a function of $n$ and $Δ$. In particular, for the case of two disks, we prove that the diameter of $\mathcal{G}^*$ is at most $\max\{2,2Δ\}$ and this bound is tight. For the general case of $n>2$ disks, we show that the diameter of $\mathcal{G}^*$ is $O(n^3 2^n Δ)$. We achieve this by proving that the number of maximal faces in $\mathcal{A}$ -- faces whose ply is more than the ply of their neighboring faces -- is $O(n^2 2^n Δ)$. To this end, we first show that the number of maximum faces -- faces whose ply is $n$ -- is $O(n^2Δ)$; the latter bound, which is of independent interest, is tight in the worst case.

math.CO↗

arXiv2025

Order Retrieval in Compact Storage Systems

Growing demand for sustainable logistics and higher space utilization, driven by e-commerce and urbanization, increases the need for storage systems that are both energy- and space-efficient. Compact storage systems aim to maximize space utilization in limited storage areas and are therefore particularly suited in densely-populated urban areas where space is scarce. In this paper, we examine a recently introduced compact storage system in which uniformly shaped bins are stacked directly on top of each other, eliminating the need for aisles used to handle materials. Target bins are retrieved in a fully automated process by first lifting all other bins that block access and then accessing the target bin from the side of the system by a dedicated robot. Consequently, retrieving a bin can require substantial lifting effort, and thus energy. However, this energy can be reduced through smart retrieval strategies. From an operational perspective, we investigate how retrievals can be optimized with respect to energy consumption. We model the retrieval problem within a mathematical framework. We show that the problem is strongly NP-complete and derive structural insights. Building on these insights, we propose two exact methods: a mixed-integer programming (MIP) formulation and a dynamic programming algorithm, along with a simple, practitioner-oriented greedy algorithm that yields near-instant solutions. Numerical experiments reveal that dynamic programming consistently outperforms state-of-the-art MIP solvers in small to medium sized instances, while the greedy algorithm delivers satisfactory performance, especially when exact methods become computationally impractical.

cs.CC↗

arXiv2025

Optimizing Periodic Operations for Efficient Inland Waterway Lock Management

In inland waterways, the efficient management of water lock operations impacts the level of congestion and the resulting uncertainty in inland waterway transportation. To achieve reliable and efficient traffic, schedules should be easy to understand and implement, reducing the likelihood of errors. The simplest schedules follow periodic patterns, reducing complexity and facilitating predictable management. Since vessels do not arrive in perfectly regular intervals, periodic schedules may lead to more wait time. The aim of this research is to estimate this cost by evaluating how effective these periodic schedules manage vessel traffic at water locks. The first objective is to estimate a periodic arrival pattern that closely matches a dataset of irregular vessel arrivals at a specific lock. We develop an algorithm that, given a fixed number of vessel streams, solves the problem in polynomial time. The solution then serves as input for the subsequent part, where we consider algorithms that compute operational schedules by formulating an optimisation problem with periodic arrival patterns as input, and the goal is to determine a periodic schedule that minimises the long-run average waiting time of vessels. We present a polynomial-time algorithm for the two-stream case and a pseudo-polynomial-time algorithm for the general case, along with incremental polynomial-time approximation schemes. In our numerical experiments, use AIS data to construct a periodic arrival pattern closely matching the observed data. Our experiments demonstrate that when evaluated against actual data, intuitive and straightforward policies often outperform optimal policies specifically trained on the periodic arrival pattern.

cs.DS↗

arXiv2025

The Line Traveling Salesman and Repairman Problem with Collaboration

In this work, we consider extensions of both the Line Traveling Salesman and Line Traveling Repairman Problem, in which a single server must service a set of clients located along a line segment under the assumption that not only the server, but also the clients can move along the line and seek to collaborate with the server to speed up service times. We analyze the structure of different problem versions and identify hard and easy subproblems by building up on prior results from the literature. Specifically, we investigate problem versions with zero or general processing times, clients that are either slower or faster than the server, as well as different time window restrictions. Collectively, these results map out the complexity landscape of the Line Traveling Salesman and Repairman Problem with collaboration.

cs.CC↗

arXiv2024

Exponentially faster fixed-parameter algorithms for high-multiplicity scheduling

We consider so-called $N$-fold integer programs (IPs) of the form $\max\{c^T x : Ax = b, \ell \leq x \leq u, x \in \mathbb Z^{nt}\}, where $A \in \mathbb Z^{(r+sn)\times nt} consists of $n$ arbitrary matrices $A^{(i)} \in \mathbb Z^{r\times t}$ on a horizontal, and $n$ arbitrary matrices $B^{(j)} \in \mathbb Z^{s\times t} on a diagonal line. Several recent works design fixed-parameter algorithms for $N$-fold IPs by taking as parameters the numbers of rows and columns of the $A$- and $B$-matrices, together with the largest absolute value $Δ$ over their entries. These advances provide fast algorithms for several well-studied combinatorial optimization problems on strings, on graphs, and in machine scheduling. In this work, we extend this research by proposing algorithms that additionally harness a partition structure of submatrices $A^{(i)}$ and $B^{(j)}$, where row indices of non-zero entries do not overlap between any two sets in the partition. Our main result is an algorithm for solving any $N$-fold IP in time $nt log(nt)L^2(S_A)^{O(r+s)}(p_Ap_BΔ)^{O(rp_Ap_B+sp_Ap_B)}$, where $p_A$ and $p_B$ are the size of the largest set in such a partition of $A^{(i)}$ and $B^{(j)}$, respectively, $S_A$ is the number of parts in the partition of $A = (A^{(1)},..., A^{(n)}), and $L = (log(||u - \ell||_\infty)\cdot (log(max_{x:\ell \leq x \leq u} |c^Tx|))$ is a measure of the input. We show that these new structural parameters are naturally small in high-multiplicity scheduling problems, such as makespan minimization on related and unrelated machines, with and without release times, the Santa Claus objective, and the weighted sum of completion times. In essence, we obtain algorithms that are exponentially faster than previous works by Knop et al. (ESA 2017) and Eisenbrand et al./Kouteck{ý} et al. (ICALP 2018) in terms of the number of job types.

cs.DS↗

arXiv2023

Envy-free dynamic pricing schemes

A combinatorial market consists of a set of indivisible items and a set of agents, where each agent has a valuation function that specifies for each subset of items its value for the given agent. From an optimization point of view, the goal is usually to determine a pair of pricing and allocation of the items that provides an efficient distribution of the resources, i.e., maximizes the social welfare, or is as profitable as possible for the seller, i.e., maximizes the revenue. To overcome the weaknesses of mechanisms operating with static prices, a recent line of research has concentrated on dynamic pricing schemes. In this model, agents arrive in an unspecified sequential order, and the prices can be updated between two agent-arrivals. Though the dynamic setting is capable of maximizing social welfare in various scenarios, the assumption that the agents arrive one after the other eliminates the standard concept of fairness. In this paper, we study the existence of optimal dynamic prices under fairness constraints in unit-demand markets. We propose four possible notions of envy-freeness of different strength depending on the time period over which agents compare themselves to others: the entire time horizon, only the past, only the future, or only the present. For social welfare maximization, while the first definition leads to Walrasian equilibria, we give polynomial-time algorithms that always find envy-free optimal dynamic prices in the remaining three cases. In contrast, for revenue maximization, we show that the corresponding problems are APX-hard if the ordering of the agents is fixed. On the positive side, we give polynomial-time algorithms for the setting when the seller can choose the order in which agents arrive.

cs.GT↗

arXiv2018

SciSports: Learning football kinematics through two-dimensional tracking data

SciSports is a Dutch startup company specializing in football analytics. This paper describes a joint research effort with SciSports, during the Study Group Mathematics with Industry 2018 at Eindhoven, the Netherlands. The main challenge that we addressed was to automatically process empirical football players' trajectories, in order to extract useful information from them. The data provided to us was two-dimensional positional data during entire matches. We developed methods based on Newtonian mechanics and the Kalman filter, Generative Adversarial Nets and Variational Autoencoders. In addition, we trained a discriminator network to recognize and discern different movement patterns of players. The Kalman-filter approach yields an interpretable model, in which a small number of player-dependent parameters can be fit; in theory this could be used to distinguish among players. The Generative-Adversarial-Nets approach appears promising in theory, and some initial tests showed an improvement with respect to the baseline, but the limits in time and computational power meant that we could not fully explore it. We also trained a Discriminator network to distinguish between two players based on their trajectories; after training, the network managed to distinguish between some pairs of players, but not between others. After training, the Variational Autoencoders generated trajectories that are difficult to distinguish, visually, from the data. These experiments provide an indication that deep generative models can learn the underlying structure and statistics of football players' trajectories. This can serve as a starting point for determining player qualities based on such trajectory data.

cs.LG↗