arXiv subjects

Javad Maheri

Publications and source records attributed to Javad Maheri.

4 recordsLinked to original sources

arXiv2026

Fundamental Limits of Hypergraph Edge Partitioning under Independent Edge Sampling

Hypergraph edge partitioning is a central problem in theoretical and applied computer science, with broad impact on distributed computation, communications, optimization, and machine learning. In this setting, one is given a collection of hyperedges -- each consisting of up to $d$ vertices from a ground set of size $n$ -- and seeks to assign these hyperedges across $N$ partitions so as to minimize, for example, the vertex footprint, i.e., the maximum number of vertices that appear in any partition. We here identify the fundamental limits of hypergraph edge partitioning -- optimized over all conceivable algorithms -- for a broad class of probabilistic hypergraph models where each hyperedge may appear independently with \emph{its own} probability; a model sufficiently general to encompass well-known models such as the Degree-Corrected or Mixed-Membership models, the Hypergraph Stochastic Block model, the Latent-Space/Geometric or Kernel Models, and others. By pairing our deterministic partitioner with a new converse, we first show that, for any $n,d$, and under the very mild condition of $N \leq \binom{\lfloor\sqrt{\frac{nd}{2}}\rfloor}{d}$, as long as the hyperedge set $\mathbf{X}$ satisfies $|\mathbf{X}| \gtrsim n N \log N$, then with probability at least $1-2/3n^z$, no algorithm can provide a footprint $π_{\mathbf{X}}$ less than $$π^{\bigstar}_{\mathbf{X}} = \frac{1}{2\sqrt{2}}\frac{n}{N^{1/d}}. $$ We then show that our hypergraph partitioner comes to within a small constant factor from $π^{\bigstar}_{\mathbf{X}}$, for each $\mathbf{X}$. This optimality captures dense and sparse hypergraphs alike (with sizes down to linear in $n$), and it additionally entails a near-optimally balanced allocation of hyperedges across partitions.

cs.IT↗

arXiv2026

Order Optimal Task Allocation in Distributed Computing via Interweaved Cliques

We consider a distributed computing system in which a master node coordinates $N$ workers to evaluate a function over $n$ input files, where this function accepts general decomposition. In particular, we focus on the general case where the requested function admits a $d$-uniform decomposition, meaning that it can be decomposed into a set of subfunctions that each depends on a unique $d$-tuple of the $n$ files. Our objective is to design file and task allocations that minimize the worst-case communication from the master to any worker and the worst-case computational load across workers. We first show that the optimal file and task allocation with minimum communication and computation costs admits a natural characterization within combinatorial design theory: it corresponds to a Steiner system $S(t, k, v)$ with $t=d$, $v=n$, and $k \approx \frac{n}{N^{1/d}}$. However, Steiner systems are known to exist only for very restricted parameter regimes. To overcome this limitation, we propose the information-theoretic-inspired \emph{Interweaved Clique (IC) design}, a universal and deterministic allocation framework that relaxes the strict structure of Steiner systems by allowing slight variations in worker file loads. Although slightly suboptimal, the IC design achieves a communication cost within a constant factor $4e$ from our converse, while also maintaining an order-optimal computation cost, thus allowing this work to derive the fundamental scaling laws of this general distributed computing problem for a large range of parameters.

cs.IT↗

arXiv2026

Universal and Asymptotically Optimal Data and Task Allocation in Distributed Computing

We study the joint minimization of communication and computation costs in distributed computing, where a master node coordinates $N$ workers to evaluate a function over a library of $n$ files. Assuming that the function is decomposed into an arbitrary subfunction set $\mathbf{X}$, with each subfunction depending on $d$ input files, renders our distributed computing problem into a $d$-uniform hypergraph edge partitioning problem wherein the edge set (subfunction set), defined by $d$-wise dependencies between vertices (files) must be partitioned across $N$ disjoint groups (workers). The aim is to design a file and subfunction allocation, corresponding to a partition of $\mathbf{X}$, that minimizes the communication cost $π_{\mathbf{X}}$, representing the maximum number of distinct files per server, while also minimizing the computation cost $δ_{\mathbf{X}}$ corresponding to a maximal worker subfunction load. For a broad range of parameters, we propose a deterministic allocation solution, the \emph{Interweaved-Cliques (IC) design}, whose information-theoretic-inspired interweaved clique structure simultaneously achieves order-optimal communication and computation costs, for a large class of decompositions $\mathbf{X}$. This optimality is derived from our achievability and converse bounds, which reveal -- under reasonable assumptions on the density of $\mathbf{X}$ -- that the optimal scaling of the communication cost takes the form $n/N^{1/d}$, revealing that our design achieves the order-optimal \textit{partitioning gain} that scales as $N^{1/d}$, while also achieving an order-optimal computation cost. Interestingly, this order optimality is achieved in a deterministic manner, and very importantly, it is achieved blindly from $\mathbf{X}$, therefore enabling multiple desired functions to be computed without reshuffling files.

cs.IT↗

arXiv2025

Constructing Hamiltonian Decompositions of Complete $k$-Uniform Hypergraphs

Motivated by the wide-ranging applications of Hamiltonian decompositions in distributed computing, coded caching, routing, resource allocation, load balancing, and fault tolerance, our work presents a comprehensive design for Hamiltonian decompositions of complete $k$-uniform hypergraphs $K_n^k$. Building upon the resolution of the long-standing conjecture of the existence of Hamiltonian decompositions of complete hypergraphs, a problem that was resolved using existence-based methods, our contribution goes beyond the previous explicit designs, which were confined to the specific cases of $k=2$ and $k=3$, by providing explicit designs for all $k$ and $n$ prime, allowing for a broad applicability of Hamiltonian decompositions in various settings.

cs.IT↗