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Search indexed NASA NTRS and DOE OSTI research on propulsion, heat transfer, battery materials and energy systems. Follow report and document links to the original sources.
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Here, we study the structure of r-uniform hypergraphs containing no Berge cycles of length at least k for k ≤ r, and determine that such hypergraphs have some special substructure. In particular we determine the extremal number of such hypergraphs, giving an affirmative answer to the conjectured value when k = r and giving a simple solution to a recent result of Kostochka-Luo when k < r.
Intelligent partitioning models are commonly used for efficient parallelization of irregular applications on distributed systems. These models usually aim to minimize a single communication cost metric, which is either related to communication volume or message count. However, both volume- and message-related metrics should be taken into account during partitioning for a more efficient parallelization. There are only a few works that consider both of them and they usually address each in separate phases of a two-phase approach. In this work, we propose a recursive hypergraph bipartitioning framework that reduces the total volume and total message count in a single phase. In this framework, the standard hypergraph models, nets of which already capture the bandwidth cost, are augmented with message nets. The message nets encode the message count so that minimizing conventional cutsize captures the minimization of bandwidth and latency costs together. Our model provides a more accurate representation of the overall communication cost by incorporating both the bandwidth and the latency components into the partitioning objective. The use of the widely-adopted successful recursive bipartitioning framework provides the flexibility of using any existing hypergraph partitioner. In conclusion, the experiments on instances from different domains show that our model on the average achieves up to 52 percent reduction in total message count and hence results in 29 percent reduction in parallel running time compared to the model that considers only the total volume.
In this paper, we discuss our experience in implementing homology computation, in particular Betti number calculation in Chapel hypergraph Library (CHGL). Given a dataset represented as a hypergraph, a Betti number for a particular dimension $k$ indicates how many $k$-dimensional `voids' are present in the dataset. Computing Betti number involves various array-centric and linear algebra operations. We demonstrate that implementing these operations in Chapel is both concise and intuitive. In addition, we show that Chapel provides language constructs for implementing parallel and distributed execution of the linear algebra kernels with minimal effort. Syntactically, Chapel provides succinctness of Python, while delivering comparable and better performance than C++-based and Julia-based packages for calculating Betti numbers respectively.
Programmable self-assembly enables the construction of complex molecular, supramolecular, and crystalline architectures from well-designed building blocks. In this work, we introduce a hypergraph-based formalism, Blocks & Bonds (B&B), which generalizes classical chemical graph theory by incorporating directed and multicolored interactions, internal symmetries, and hierarchical organization. Within this framework, we develop the Structure Code (SC), a compact and versatile language for describing self-assembled architectures. We define a Kolmogorov-style structural complexity as the total information content of SC, obtained through its tokenization and Shannon information assignment. Complementing this encoding-based measure, we introduce a much simpler quantity, the compositional complexity, which depends only on the number and cumulative usage of block and bond types in the construction set. A central result of this work is a strong empirical correlation between the token-based structural complexity and the compositional complexity across all examined systems. Owing to this agreement, the compositional complexity emerges as the most practical and broadly applicable measure: it is easy to compute, requires no explicit encoding, and yet closely tracks the actual information content of structurally diverse architectures. Applications to molecular systems (ethylene glycol and glucose), DNA-origami lattices, and crystalline assemblies show that B&B hypergraphs provide a unified, scalable, and information-efficient representation of structural organization, naturally capturing symmetry, modularity, and stereochemistry. This framework establishes a quantitative foundation for complexity-aware classification and inverse design of programmable matter.
Constraint satisfaction problems are an important area of computer science. Many of these problems are in the complexity class NP which is exponentially hard for all known methods, both for worst cases and often typical. Fundamentally, the lack of any guided local minimum escape method ensures the hardness of both exact and approximate optimization classically, but the intuitive mechanism for approximation hardness in quantum algorithms based on Hamiltonian time evolution is poorly understood. We explore this question using the prototypically hard MAX-3-XORSAT problem class. We conclude that the mechanisms for quantum exact and approximation hardness are fundamentally distinct. We qualitatively identify why traditional methods such as quantum adiabatic optimization are not good approximation algorithms. We propose a new spectral folding optimization method that does not suffer from these issues and study it analytically and numerically. We consider random rank-3 hypergraphs including extremal planted solution instances, where the ground state satisfies an anomalously high fraction of constraints compared to truly random problems. We show that, if we define the energy to be $E = N_{unsat}-N_{sat}$, then spectrally folded quantum optimization will return states with energy $E \leq A E_{GS}$ (where $E_{GS}$ is the ground state energy) in polynomial time, where conservatively, $A \simeq 0.6$. We thoroughly benchmark variations of spectrally folded quantum optimization for random classically approximation-hard (planted solution) instances in simulation, and find performance consistent with this prediction. We do not claim that this approximation guarantee holds for all possible hypergraphs, though our algorithm's mechanism can likely generalize widely. These results suggest that quantum computers are more powerful for approximate optimization than had been previously assumed.
Following a newly introduced approach by Rasetti and Merelli we investigate the possibility to extract topological information about the space where interacting systems are modelled. From the statistical datum of their observable quantities, like the correlation functions, we show how to reconstruct the activities of their constitutive parts which embed the topological information. The procedure is implemented on a class of polymer models on hypergraphs with hard-core interactions. We show that the model fulfils a set of iterative relations for the partition function that generalise those introduced by Heilmann and Lieb for the monomer–dimer case. After translating those relations into structural identities for the correlation functions we use them to test the precision and the robustness of the inverse problem. Finally the possible presence of a further interaction of peer-to-peer type is considered and a criterion to discover it is identified.
In this report we study the percolation critical surface of the kagome lattice in which each triangle is allowed an arbitrary connectivity. Using the method of critical polynomials, we find points along this critical surface to high precision. This kagome hypergraph contains many unsolved problems as special cases, including bond percolation on the kagome and (3, 12 2 ) lattices, and site percolation on the hexagonal, or honeycomb, lattice, as well as a single point for which there is an exact solution. We are able to compute enough points along the critical surface to find a very accurate fit, essentially a Taylor series about the exact point, that allows estimations of the critical point of any system that lies on the surface to precision rivaling Monte Carlo and traditional techniques of similar accuracy. We find also that this system sheds light on some of the surprising aspects of the method of critical polynomials, such as why it is so accurate for certain problems, like the kagome and (3, 12 2 ) lattices. The bond percolation critical points of these lattices can be found to 17 and 18 digits, respectively, because they are in close proximity, in a sense that can be made quantitative, to the exact point on the critical surface. We also discuss in detail a parallel implementation of the method which we use here for a few calculations.
The minimum positive co-degree of a nonempty r-graph H, denoted $δ^+_{r-1}$ (H), is the maximum k such that if S is an (r-1)-set contained in a hyperedge of H, then is contained in at least distinct hyperedges of H. Given an r-graph F, we introduce the positive co-degree Turán number co + ex(n, F) as the maximum positive co-degree $δ^+_{r-1}$ (H) over all n-vertex r-graphs H that do not contain F as a subhypergraph. In this paper, we concentrate on the behavior of co + ex(n, F) for 3-graphs F. In particular, we determine asymptotics and bounds for several well-known concrete 3-graphs F (e.g. $K^-_4$ and the Fano plane). Here, we also show that, for r-graphs, the limit γ + (F)≔ lim$_{n→∞}$ $\frac{co^+ex(n, F)}{n}$ exists, and “jumps” from 0 to 1/r, that is, it never takes on values in the interval . Moreover, we characterize which r-graphs F have γ + (F) = 0. Our motivation comes primarily from the study of (ordinary) co-degree Turán numbers where a number of results have been proved that inspire our results.
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Most problems within and beyond the scientific domain can be framed into one of the following three levels of complexity of function approximation. Type 1: Approximate an unknown function given input/output data. Type 2: Consider a collection of variables and functions, some of which are unknown, indexed by the nodes and hyperedges of a hypergraph (a generalized graph where edges can connect more than two vertices). Given partial observations of the variables of the hypergraph (satisfying the functional dependencies imposed by its structure), approximate all the unobserved variables and unknown functions. Type 3: Expanding on Type 2, if the hypergraph structure itself is unknown, use partial observations of the variables of the hypergraph to discover its structure and approximate its unknown functions. These hypergraphs offer a natural platform for organizing, communicating, and processing computational knowledge. While most scientific problems can be framed as the data-driven discovery of unknown functions in a computational hypergraph whose structure is known (Type 2), many require the data-driven discovery of the structure (connectivity) of the hypergraph itself (Type 3). We introduce an interpretable Gaussian Process (GP) framework for such (Type 3) problems that does not require randomization of the data, access to or control over its sampling, or sparsity of the unknown functions in a known or learned basis. Its polynomial complexity, which contrasts sharply with the super-exponential complexity of causal inference methods, is enabled by the nonlinear ANOVA capabilities of GPs used as a sensing mechanism.