arXiv subjects

Ahmed Ghazy

Publications and source records attributed to Ahmed Ghazy.

6 recordsLinked to original sources

arXiv2026

Faster Exponential Algorithms for Multi-Machine Scheduling Problems

Minimizing the weighted completion times ($P \mid \mid \Sigma w_j C_j$) and weighted number of tardy jobs ($P \mid \mid \Sigma w_j U_j$) on multiple identical machines are two classical NP-hard scheduling problems. As shown by Lent\'e et al. (2014), both problems can be solved in time ${O}^{\star}(3^n)$. In this paper, we improve these bounds to ${O}(2.755^n)$ and ${O}^{\star}(2^n)$, respectively. Our algorithm for $P \mid \mid \Sigma w_j C_j$ exploits the meet-in-the-middle paradigm and an efficient data structure answering linear programming queries. Additionally, when the number of machines is at most $6$, we show that the running time for $P \mid \mid \Sigma w_j C_j$ can further be improved. Both scheduling problems are generalizations of the classical Bin Packing problem, which can be solved in ${O}^{\star}(2^n)$ time. Improving this running time is an important open question. We show that, when assuming the Asymptotic Rank Conjecture (ARC), Bin Packing can be solved in time ${O}((2-\varepsilon)^n)$ for some $\varepsilon >0$. Our algorithm makes use of two main ingredients: the recent ${O}((2-\varepsilon)^n)$-time algorithm of Nederlof et al. [SICOMP'23] for Bin Packing when the number of bins is a fixed constant, and the ${O}((2-\varepsilon)^n)$-time algorithm of Bj\"orklund et al. [SODA'25] for special instances of the $3$-way Partitioning problem when assuming ARC.

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arXiv2026

Where Treewidth and Pathwidth Diverge: Towards a Uniform Kernel for Pathwidth-$\eta$ Deletion

For a constant $\eta \geq 0$, Pathwidth-$\eta$ Deletion is the problem of deciding whether, for a given graph $G$ and integer $k$, there is a set $S \subseteq V(G)$ of size at most $k$ such that the pathwidth of $G - S$ is at most $\eta$. The problems Treewidth-$\eta$ Deletion and Treedepth-$\eta$ Deletion are defined similarly for the parameters treewidth and treedepth, respectively. A landmark result of Fomin et al. [FOCS, 2012] shows that, for any constant $\eta$, all three problems admit a kernel on $O(k^{c(\eta)})$ vertices, where $c(\eta)$ is a constant depending on $\eta$. Giannopoulou et al. [ACM TALG, 2017] show that, in some sense, this result is optimal for Treewidth-$\eta$ Deletion: for $\eta \geq 2$ and even when parameterizing by the size of a vertex cover $M$ of the input graph, there is no kernel of size $O(|M|^{\frac{\eta+1}{2}-\varepsilon})$, for any $\varepsilon > 0$. Contrasting this result, they prove that Treedepth-$\eta$ Deletion admits a uniform polynomial kernel, that is, a kernel of size $O(k^c)$ for a constant $c$ that is independent of $\eta$. In comparison, the question whether Pathwidth-$\eta$ Deletion admits a uniform polynomial kernel has been neglected in the literature. As treewidth and pathwidth tend to behave similarly, it is natural to expect that no uniform kernel exists when parameterizing by the size of a vertex cover. Surprisingly, we show this not to be the case. More concretely, we prove the existence of a uniform polynomial kernel for Pathwidth-$\eta$ Deletion when parameterizing by (1) the solution size $k$ plus the size of a set $M$ such that $G - M$ has bounded treedepth; (2) the (vertex-deletion) distance to pathwidth-$1$ graphs; (3) the distance to the class of graphs with treedepth at most $\eta + 1$. This leads us to conjecture that Pathwidth-$\eta$ Deletion admits a uniform kernel when parameterizing by the solution size $k$.

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arXiv2025

From Chinese Postman to Salesman and Beyond II: Inapproximability and Parameterized Complexity

A well-studied continuous model of graphs considers each edge as a continuous unit-length interval of points. In the problem $δ$-Tour defined within this model, the objective to find a shortest tour that comes within a distance of $δ$ of every point on every edge. This parameterized problem was introduced in the predecessor to this article and shown to be essentially equivalent to the Chinese Postman problem for $δ= 0$, to the graphic Travel Salesman Problem (TSP) for $δ= 1/2$, and close to first Vertex Cover and then Dominating Set for even larger $δ$. Moreover, approximation algorithms for multiple parameter ranges were provided. In this article, we provide complementing inapproximability bounds and examine the fixed-parameter tractability of the problem. On the one hand, we show the following: (1) For every fixed $0 < δ< 3/2$, the problem $δ$-Tour is APX-hard, while for every fixed $δ\geq 3/2$, the problem has no polynomial-time $o(\log{n})$-approximation unless P = NP. Our techniques also yield the new result that TSP remains APX-hard on cubic (and even cubic bipartite) graphs. (2) For every fixed $0 < δ< 3/2$, the problem $δ$-Tour is fixed-parameter tractable (FPT) when parameterized by the length of a shortest tour, while it is W[2]-hard for every fixed $δ\geq 3/2$ and para-NP-hard for $δ$ being part of the input. On the other hand, if $δ$ is considered to be part of the input, then an interesting nontrivial phenomenon occurs when $δ$ is a constant fraction of the number of vertices: (3) If $δ$ is part of the input, then the problem can be solved in time $f(k)n^{O(k)}$, where $k = \lceil n/δ\rceil$; however, assuming the Exponential-Time Hypothesis (ETH), there is no algorithm that solves the problem and runs in time $f(k)n^{o(k/\log k)}$.

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arXiv2025

From Chinese Postman to Salesman and Beyond I: Approximating Shortest Tours $δ$-Covering All Points on All Edges

A well-studied continuous model of graphs, introduced by Dearing and Francis [Transportation Science, 1974], considers each edge as a continuous unit-length interval of points. For $δ\geq 0$, we introduce the problem $δ$-Tour, where the objective is to find the shortest tour that comes within a distance of $δ$ of every point on every edge. It can be observed that 0-Tour is essentially equivalent to the Chinese Postman Problem, which is solvable in polynomial time. In contrast, 1/2-Tour is essentially equivalent to the Graphic Traveling Salesman Problem (TSP), which is NP-hard but admits a constant-factor approximation in polynomial time. We investigate $δ$-Tour for other values of $δ$, noting that the problem's behavior and the insights required to understand it differ significantly across various $δ$ regimes. We design polynomial-time approximation algorithms summarized as follows: (1) For every fixed $0 < δ< 3/2$, the problem $δ$-Tour admits a constant-factor approximation. (2) For every fixed $δ\geq 3/2$, the problem admits an $O(\log{n})$-approximation. (3) If $δ$ is considered to be part of the input, then the problem admits an $O(\log^3{n})$-approximation. This is the first of two articles on the $δ$-Tour problem. In the second one we complement the approximation algorithms presented here with inapproximability results and related to parameterized complexity.

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arXiv2024

Content-Oblivious Leader Election on Rings

In content-oblivious computation, n nodes wish to compute a given task over an asynchronous network that suffers from an extremely harsh type of noise, which corrupts the content of all messages across all channels. In a recent work, Censor-Hillel, Cohen, Gelles, and Sela (Distributed Computing, 2023) showed how to perform arbitrary computations in a content-oblivious way in 2-edge connected networks but only if the network has a distinguished node (called root) to initiate the computation. Our goal is to remove this assumption, which was conjectured to be necessary. Achieving this goal essentially reduces to performing a content-oblivious leader election since an elected leader can then serve as the root required to perform arbitrary content-oblivious computations. We focus on ring networks, which are the simplest 2-edge connected graphs. On oriented rings, we obtain a leader election algorithm with message complexity O(n*ID_max), where ID_max is the maximal assigned ID. As it turns out, this dependency on $ID_max$ is inherent: we show a lower bound of Omega(n*log(ID_max/n)) messages for content-oblivious leader election algorithms. We also extend our results to non-oriented rings, where nodes cannot tell which channel leads to which neighbor. In this case, however, the algorithm does not terminate but only reaches quiescence.

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arXiv2020

Leveraging Spatial Uncertainty for Online Error Compensation in EMT

Purpose: Electromagnetic Tracking (EMT) can potentially complement fluoroscopic navigation, reducing radiation exposure in a hybrid setting. Due to the susceptibility to external distortions, systematic error in EMT needs to be compensated algorithmically. Compensation algorithms for EMT in guidewire procedures are only practical in an online setting. Methods: We collect positional data and train a symmetric Artificial Neural Network (ANN) architecture for compensating navigation error. The results are evaluated in both online and offline scenarios and are compared to polynomial fits. We assess spatial uncertainty of the compensation proposed by the ANN. Simulations based on real data show how this uncertainty measure can be utilized to improve accuracy and limit radiation exposure in hybrid navigation. Results: ANNs compensate unseen distortions by more than 70%, outperforming polynomial regression. Working on known distortions, ANNs outperform polynomials as well. We empirically demonstrate a linear relationship between tracking accuracy and model uncertainty. The effectiveness of hybrid tracking is shown in a simulation experiment. Conclusion: ANNs are suitable for EMT error compensation and can generalize across unseen distortions. Model uncertainty needs to be assessed when spatial error compensation algorithms are developed, so that training data collection can be optimized. Finally, we find that error compensation in EMT reduces the need for x-ray images in hybrid navigation.

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