arXiv subjects

Seyed-Mohammad Seyed-Javadi

Publications and source records attributed to Seyed-Mohammad Seyed-Javadi.

3 recordsLinked to original sources

arXiv2026

On Hardness and Approximation of Broadcasting in Structured Graphs

We study the Telephone Broadcasting problem in graphs with restricted structure. Given a designated source in an undirected graph, the goal is to disseminate a message to all vertices in the minimum number of rounds, where in each round every informed vertex may inform at most one neighbor. For general graphs, the problem is NP-hard. Recent work shows that the problem remains NP-hard even on restricted graph classes such as graphs of treewidth 2 [Tale 2025], cactus graphs of pathwidth 2 [Aminian et~al. 2025] and graphs at distance 1 to a path forest [Egami et~al. 2025]. In this work, we investigate the problem in several graph families. We first prove NP-hardness for cycle-star graphs, graphs formed by k cycles sharing a single vertex, as well as melon graphs, graphs formed by k paths with shared endpoints. Despite multiple efforts to understand the problem in these simple graph families, the computational complexity of the problem remained unsettled. Our hardness results answer open questions by Bhabak and Harutyunyan [2015] and Harutyunyan and Hovhannisyan [2023] concerning the problem's complexity in cycle-star and melon graphs, respectively. On the positive side, we present EPTASs for cycle-star and melon graphs, improving over the best existing approximation factors of 2 for both graph families. Moreover, we identify a structural frontier for tractability by showing that the problem is solvable in polynomial time on graphs of bounded cutwidth, a class that generalizes other families such as graphs of bounded bandwidth. This result subsumes existing tractability results for graph families such as necklace graphs. Finally, for split graphs, a fundamental class of highly structured graphs, we obtain a polynomial-time algorithm with approximation factor 1.76. This improves on the previously known factor 2 bound; the same approach also applies to the multi-source setting.

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arXiv2025

On the Complexity of Telephone Broadcasting: From Cacti to Bounded Pathwidth Graphs

In the Telephone Broadcasting problem, the goal is to disseminate a message from a given source vertex of an input graph to all other vertices in the minimum number of rounds, where at each round, an informed vertex can send the message to at most one of its uninformed neighbors. For general graphs of n vertices, the problem is NP-complete, and the best existing algorithm has an approximation factor of O(log n/ log log n). The existence of a constant factor approximation for the general graphs is still unknown. In this paper, we study the problem in two simple families of sparse graphs, namely, cacti and graphs of bounded pathwidth. There have been several efforts to understand the complexity of the problem in cactus graphs, mostly establishing the presence of polynomial-time solutions for restricted families of cactus graphs. Despite these efforts, the complexity of the problem in arbitrary cactus graphs remained open. We settle this question by establishing the NP-completeness of telephone broadcasting in cactus graphs. For that, we show the problem is NP-complete in a simple subfamily of cactus graphs, which we call snowflake graphs. These graphs not only are cacti but also have pathwidth 2. These results establish that, despite being polynomial-time solvable in trees, the problem becomes NP-complete in very simple extensions of trees. On the positive side, we present constant-factor approximation algorithms for the studied families of graphs, namely, an algorithm with an approximation factor of 2 for cactus graphs and an approximation factor of O(1) for graphs of bounded pathwidth.

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arXiv2023

Rainbow Cycle Number and EFX Allocations: (Almost) Closing the Gap

Recently, some studies on the fair allocation of indivisible goods notice a connection between a purely combinatorial problem called the Rainbow Cycle problem and a fairness notion known as $\efx$: assuming that the rainbow cycle number for parameter $d$ (i.e. $\rainbow(d)$) is $O(d^β\log^γd)$, we can find a $(1-ε)$-$\efx$ allocation with $O_ε(n^{\fracβ{β+1}}\log^{\fracγ{β+1}} n)$ number of discarded goods \cite{chaudhury2021improving}. The best upper bound on $\rainbow(d)$ is improved in a series of works to $O(d^4)$ \cite{chaudhury2021improving}, $O(d^{2+o(1)})$ \cite{berendsohn2022fixed}, and finally to $O(d^2)$ \cite{Akrami2022}.\footnote{We refer to the note at the end of the introduction for a short discussion on the result of \cite{Akrami2022}.} Also, via a simple observation, we have $\rainbow(d) \in Ω(d)$ \cite{chaudhury2021improving}. In this paper, we introduce another problem in extremal combinatorics. For a parameter $\ell$, we define the rainbow path degree and denote it by $\ech(\ell)$. We show that any lower bound on $\ech(\ell)$ yields an upper bound on $\rainbow(d)$. Next, we prove that $\ech(\ell) \in Ω(\ell^2/\log n)$ which yields an almost tight upper bound of $\rainbow(d) \in Ω(d \log d)$. This in turn proves the existence of $(1-ε)$-$\efx$ allocation with $O_ε(\sqrt{n \log n})$ number of discarded goods. In addition, for the special case of the Rainbow Cycle problem that the edges in each part form a permutation, we improve the upper bound to $\rainbow(d) \leq 2d-4$. We leverage $\ech(\ell)$ to achieve this bound. Our conjecture is that the exact value of $\ech(\ell) $ is $ \lfloor \frac{\ell^2}{2} \rfloor -1$. We provide some experiments that support this conjecture. Assuming this conjecture is correct, we have $\rainbow(d) \in Θ(d)$.

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