Models and Algorithms for Graph Watermarking

We introduce models and algorithmic foundations for graph watermarking. Our frameworks include security definitions and proofs, as well as characterizations when graph watermarking is algorithmically feasible, in spite of the fact that the general problem is NP-complete by simple reductions from the subgraph isomorphism or graph edit distance problems. In the digital watermarking of many types of files, an implicit step in the recovery of a watermark is the mapping of individual pieces of data, such as image pixels or movie frames, from one object to another. In graphs, this step corresponds to approximately matching vertices of one graph to another based on graph invariants such as vertex degree. Our approach is based on characterizing the feasibility of graph watermarking in terms of keygen, marking, and identification functions defined over graph families with known distributions. We demonstrate the strength of this approach with exemplary watermarking schemes for two random graph models, the classic Erd\H{o}s-R\'{e}nyi model and a random power-law graph model, both of which are used to model real-world networks

Graph Isomorphism is not AC^0 reducible to Group Isomorphism

We give a new upper bound for the Group and Quasigroup Isomorphism problems when the input structures are given explicitly by multiplication tables. We show that these problems can be computed by polynomial size nondeterministic circuits of unbounded fan-in with $O(loglog n)$ depth and $O(log^2 n)$ nondeterministic bits, where $n$ is the number of group elements. This improves the existing upper bound from cite{Wolf 94} for the problems. In the previous upper bound the circuits have bounded fan-in but depth $O(log^2 n)$ and also $O(log^2 n)$ nondeterministic bits. We then prove that the kind of circuits from our upper bound cannot compute the Parity function. Since Parity is AC0 reducible to Graph Isomorphism, this implies that Graph Isomorphism is strictly harder than Group or Quasigroup Isomorphism under the ordering defined by AC0 reductions

Graph- versus Vector-Based Analysis of a Consensus Protocol

The Paxos distributed consensus algorithm is a challenging case-study for standard, vector-based model checking techniques. Due to asynchronous communication, exhaustive analysis may generate very large state spaces already for small model instances. In this paper, we show the advantages of graph transformation as an alternative modelling technique. We model Paxos in a rich declarative transformation language, featuring (among other things) nested quantifiers, and we validate our model using the GROOVE model checker, a graph-based tool that exploits isomorphism as a natural way to prune the state space via symmetry reductions. We compare the results with those obtained by the standard model checker Spin on the basis of a vector-based encoding of the algorithm.Comment: In Proceedings GRAPHITE 2014, arXiv:1407.767

On the complexity of isomorphism problems for tensors, groups, and polynomials IV: linear-length reductions and their applications

Many isomorphism problems for tensors, groups, algebras, and polynomials were recently shown to be equivalent to one another under polynomial-time reductions, prompting the introduction of the complexity class TI (Grochow & Qiao, ITCS '21; SIAM J. Comp., '23). Using the tensorial viewpoint, Grochow & Qiao (CCC '21) then gave moderately exponential-time search- and counting-to-decision reductions for a class of $p$-groups. A significant issue was that the reductions usually incurred a quadratic increase in the length of the tensors involved. When the tensors represent $p$-groups, this corresponds to an increase in the order of the group of the form $|G|^{\Theta(\log |G|)}$, negating any asymptotic gains in the Cayley table model. In this paper, we present a new kind of tensor gadget that allows us to replace those quadratic-length reductions with linear-length ones, yielding the following consequences: 1. Combined with the recent breakthrough $|G|^{O((\log |G|)^{5/6})}$-time isomorphism-test for $p$-groups of class 2 and exponent $p$ (Sun, STOC '23), our reductions extend this runtime to $p$-groups of class $c$ and exponent $p$ where $c<p$. 2. Our reductions show that Sun's algorithm solves several TI-complete problems over $F_p$, such as isomorphism problems for cubic forms, algebras, and tensors, in time $p^{O(n^{1.8} \log p)}$. 3. Polynomial-time search- and counting-to-decision reduction for testing isomorphism of $p$-groups of class $2$ and exponent $p$ in the Cayley table model. This answers questions of Arvind and T\'oran (Bull. EATCS, 2005) for this group class, thought to be one of the hardest cases of Group Isomorphism. 4. If Graph Isomorphism is in P, then testing equivalence of cubic forms and testing isomorphism of algebra over a finite field $F_q$ can both be solved in time $q^{O(n)}$, improving from the brute-force upper bound $q^{O(n^2)}$

On the Lattice Distortion Problem

We introduce and study the \emph{Lattice Distortion Problem} (LDP). LDP asks how "similar" two lattices are. I.e., what is the minimal distortion of a linear bijection between the two lattices? LDP generalizes the Lattice Isomorphism Problem (the lattice analogue of Graph Isomorphism), which simply asks whether the minimal distortion is one. As our first contribution, we show that the distortion between any two lattices is approximated up to a $n^{O(\log n)}$ factor by a simple function of their successive minima. Our methods are constructive, allowing us to compute low-distortion mappings that are within a $2^{O(n \log \log n/\log n)}$ factor of optimal in polynomial time and within a $n^{O(\log n)}$ factor of optimal in singly exponential time. Our algorithms rely on a notion of basis reduction introduced by Seysen (Combinatorica 1993), which we show is intimately related to lattice distortion. Lastly, we show that LDP is NP-hard to approximate to within any constant factor (under randomized reductions), by a reduction from the Shortest Vector Problem.Comment: This is the full version of a paper that appeared in ESA 201