Computational complexity of reconstruction and isomorphism testing for designs and line graphs

Graphs with high symmetry or regularity are the main source for experimentally hard instances of the notoriously difficult graph isomorphism problem. In this paper, we study the computational complexity of isomorphism testing for line graphs of $t$-$(v,k,\lambda)$ designs. For this class of highly regular graphs, we obtain a worst-case running time of $O(v^{\log v + O(1)})$ for bounded parameters $t,k,\lambda$. In a first step, our approach makes use of the Babai--Luks algorithm to compute canonical forms of $t$-designs. In a second step, we show that $t$-designs can be reconstructed from their line graphs in polynomial-time. The first is algebraic in nature, the second purely combinatorial. For both, profound structural knowledge in design theory is required. Our results extend earlier complexity results about isomorphism testing of graphs generated from Steiner triple systems and block designs.Comment: 12 pages; to appear in: "Journal of Combinatorial Theory, Series A

Quartic Curves and Their Bitangents

A smooth quartic curve in the complex projective plane has 36 inequivalent representations as a symmetric determinant of linear forms and 63 representations as a sum of three squares. These correspond to Cayley octads and Steiner complexes respectively. We present exact algorithms for computing these objects from the 28 bitangents. This expresses Vinnikov quartics as spectrahedra and positive quartics as Gram matrices. We explore the geometry of Gram spectrahedra and we find equations for the variety of Cayley octads. Interwoven is an exposition of much of the 19th century theory of plane quartics.Comment: 26 pages, 3 figures, added references, fixed theorems 4.3 and 7.8, other minor change

The pragmatic proof: hypermedia API composition and execution

Machine clients are increasingly making use of the Web to perform tasks. While Web services traditionally mimic remote procedure calling interfaces, a new generation of so-called hypermedia APIs works through hyperlinks and forms, in a way similar to how people browse the Web. This means that existing composition techniques, which determine a procedural plan upfront, are not sufficient to consume hypermedia APIs, which need to be navigated at runtime. Clients instead need a more dynamic plan that allows them to follow hyperlinks and use forms with a preset goal. Therefore, in this paper, we show how compositions of hypermedia APIs can be created by generic Semantic Web reasoners. This is achieved through the generation of a proof based on semantic descriptions of the APIs' functionality. To pragmatically verify the applicability of compositions, we introduce the notion of pre-execution and post-execution proofs. The runtime interaction between a client and a server is guided by proofs but driven by hypermedia, allowing the client to react to the application's actual state indicated by the server's response. We describe how to generate compositions from descriptions, discuss a computer-assisted process to generate descriptions, and verify reasoner performance on various composition tasks using a benchmark suite. The experimental results lead to the conclusion that proof-based consumption of hypermedia APIs is a feasible strategy at Web scale.Peer ReviewedPostprint (author's final draft

Switching for Small Strongly Regular Graphs

We provide an abundance of strongly regular graphs (SRGs) for certain parameters $(n, k, \lambda, \mu)$ with $n < 100$. For this we use Godsil-McKay (GM) switching with a partition of type $4,n-4$ and Wang-Qiu-Hu (WQH) switching with a partition of type $3^2,n-6$. In most cases, we start with a highly symmetric graph which belongs to a finite geometry. To our knowledge, most of the obtained graphs are new. For all graphs, we provide statistics about the size of the automorphism group. We also find the recently discovered Kr\v{c}adinac partial geometry, therefore finding a third method of constructing it.Comment: 15 page

Entanglement-assisted quantum low-density parity-check codes

This paper develops a general method for constructing entanglement-assisted quantum low-density parity-check (LDPC) codes, which is based on combinatorial design theory. Explicit constructions are given for entanglement-assisted quantum error-correcting codes (EAQECCs) with many desirable properties. These properties include the requirement of only one initial entanglement bit, high error correction performance, high rates, and low decoding complexity. The proposed method produces infinitely many new codes with a wide variety of parameters and entanglement requirements. Our framework encompasses various codes including the previously known entanglement-assisted quantum LDPC codes having the best error correction performance and many new codes with better block error rates in simulations over the depolarizing channel. We also determine important parameters of several well-known classes of quantum and classical LDPC codes for previously unsettled cases.Comment: 20 pages, 5 figures. Final version appearing in Physical Review