Asymptotic Delsarte cliques in distance-regular graphs

We give a new bound on the parameter $\lambda$ (number of common neighbors of a pair of adjacent vertices) in a distance-regular graph $G$, improving and generalizing bounds for strongly regular graphs by Spielman (1996) and Pyber (2014). The new bound is one of the ingredients of recent progress on the complexity of testing isomorphism of strongly regular graphs (Babai, Chen, Sun, Teng, Wilmes 2013). The proof is based on a clique geometry found by Metsch (1991) under certain constraints on the parameters. We also give a simplified proof of the following asymptotic consequence of Metsch's result: if $k\mu = o(\lambda^2)$ then each edge of $G$ belongs to a unique maximal clique of size asymptotically equal to $\lambda$, and all other cliques have size $o(\lambda)$. Here $k$ denotes the degree and $\mu$ the number of common neighbors of a pair of vertices at distance 2. We point out that Metsch's cliques are "asymptotically Delsarte" when $k\mu = o(\lambda^2)$, so families of distance-regular graphs with parameters satisfying $k\mu = o(\lambda^2)$ are "asymptotically Delsarte-geometric."Comment: 10 page

P?=NP as minimization of degree 4 polynomial, integration or Grassmann number problem, and new graph isomorphism problem approaches

While the P vs NP problem is mainly approached form the point of view of discrete mathematics, this paper proposes reformulations into the field of abstract algebra, geometry, fourier analysis and of continuous global optimization - which advanced tools might bring new perspectives and approaches for this question. The first one is equivalence of satisfaction of 3-SAT problem with the question of reaching zero of a nonnegative degree 4 multivariate polynomial (sum of squares), what could be tested from the perspective of algebra by using discriminant. It could be also approached as a continuous global optimization problem inside $[0,1]^n$, for example in physical realizations like adiabatic quantum computers. However, the number of local minima usually grows exponentially. Reducing to degree 2 polynomial plus constraints of being in $\{0,1\}^n$, we get geometric formulations as the question if plane or sphere intersects with $\{0,1\}^n$. There will be also presented some non-standard perspectives for the Subset-Sum, like through convergence of a series, or zeroing of $\int_0^{2\pi} \prod_i \cos(\varphi k_i) d\varphi$ fourier-type integral for some natural $k_i$. The last discussed approach is using anti-commuting Grassmann numbers $\theta_i$, making $(A \cdot \textrm{diag}(\theta_i))^n$ nonzero only if $A$ has a Hamilton cycle. Hence, the P$\ne$NP assumption implies exponential growth of matrix representation of Grassmann numbers. There will be also discussed a looking promising algebraic/geometric approach to the graph isomorphism problem -- tested to successfully distinguish strongly regular graphs with up to 29 vertices.Comment: 19 pages, 8 figure

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

Generation of cubic graphs

We describe a new algorithm for the efficient generation of all non-isomorphic connected cubic graphs. Our implementation of this algorithm is more than 4 times faster than previous generators. The generation can also be efficiently restricted to cubic graphs with girth at least 4 or 5

A matrix representation of graphs and its spectrum as a graph invariant

We use the line digraph construction to associate an orthogonal matrix with each graph. From this orthogonal matrix, we derive two further matrices. The spectrum of each of these three matrices is considered as a graph invariant. For the first two cases, we compute the spectrum explicitly and show that it is determined by the spectrum of the adjacency matrix of the original graph. We then show by computation that the isomorphism classes of many known families of strongly regular graphs (up to 64 vertices) are characterized by the spectrum of this matrix. We conjecture that this is always the case for strongly regular graphs and we show that the conjecture is not valid for general graphs. We verify that the smallest regular graphs which are not distinguished with our method are on 14 vertices.Comment: 14 page

Quantum Fourier sampling, Code Equivalence, and the quantum security of the McEliece and Sidelnikov cryptosystems

The Code Equivalence problem is that of determining whether two given linear codes are equivalent to each other up to a permutation of the coordinates. This problem has a direct reduction to a nonabelian hidden subgroup problem (HSP), suggesting a possible quantum algorithm analogous to Shor's algorithms for factoring or discrete log. However, we recently showed that in many cases of interest---including Goppa codes---solving this case of the HSP requires rich, entangled measurements. Thus, solving these cases of Code Equivalence via Fourier sampling appears to be out of reach of current families of quantum algorithms. Code equivalence is directly related to the security of McEliece-type cryptosystems in the case where the private code is known to the adversary. However, for many codes the support splitting algorithm of Sendrier provides a classical attack in this case. We revisit the claims of our previous article in the light of these classical attacks, and discuss the particular case of the Sidelnikov cryptosystem, which is based on Reed-Muller codes

Graph Isomorphism and the Lasserre Hierarchy

In this paper we show lower bounds for a certain large class of algorithms solving the Graph Isomorphism problem, even on expander graph instances. Spielman [25] shows an algorithm for isomorphism of strongly regular expander graphs that runs in time exp(O(n^(1/3)) (this bound was recently improved to expf O(n^(1/5) [5]). It has since been an open question to remove the requirement that the graph be strongly regular. Recent algorithmic results show that for many problems the Lasserre hierarchy works surprisingly well when the underlying graph has expansion properties. Moreover, recent work of Atserias and Maneva [3] shows that k rounds of the Lasserre hierarchy is a generalization of the k-dimensional Weisfeiler-Lehman algorithm for Graph Isomorphism. These two facts combined make the Lasserre hierarchy a good candidate for solving graph isomorphism on expander graphs. Our main result rules out this promising direction by showing that even Omega(n) rounds of the Lasserre semidefinite program hierarchy fail to solve the Graph Isomorphism problem even on expander graphs.Comment: 22 pages, 3 figures, submitted to CC