Blair D. Sullivan

Adiabatic quantum programming: minor embedding with hard faults

Adiabatic quantum programming defines the time-dependent mapping of a quantum algorithm into an underlying hardware or logical fabric. An essential step is embedding problem-specific information into the quantum logical fabric. We present algorithms for embedding arbitrary instances of the adiabatic quantum optimization algorithm into a square lattice of specialized unit cells. These methods extend with fabric growth while scaling linearly in time and quadratically in footprint. We also provide methods for handling hard faults in the logical fabric without invoking approximations to the original problem and illustrate their versatility through numerical studies of embeddability versus fault rates in square lattices of complete bipartite unit cells. The studies show that these algorithms are more resilient to faulty fabrics than naive embedding approaches, a feature which should prove useful in benchmarking the adiabatic quantum optimization algorithm on existing faulty hardware.

Cycles in dense digraphs

AbstractLet G be a digraph (without parallel edges) such that every directed cycle has length at least four; let β(G) denote the size of the smallest subset X ⊆ E(G) such that G∖X has no directed cycles, and let γ(G) be the number of unordered pairs {u, v} of vertices such that u, v are nonadjacent in G. It is easy to see that if γ(G) = 0 then β(G) = 0; what can we say about β(G) if γ(G) is bounded?We prove that in general β(G) ≤ γ(G). We conjecture that in fact β(G) ≤ ½γ(G) (this would be best possible if true), and prove this conjecture in two special cases: when V(G) is the union of two cliqueswhen the vertices of G can be arranged in a circle such that if distinct u, v, w are in clockwise order and uw is a (directed) edge, then so are both uv, vw.

On the Threshold of Intractability

We study the computational complexity of the graph modification problems Open image in new window and Open image in new window , adding and deleting as few edges as possible to transform the input into a threshold (or chain) graph. In this article, we show that both problems are Open image in new window -hard, resolving a conjecture by Natanzon, Shamir, and Sharan (2001). On the positive side, we show that these problems admit quadratic vertex kernels. Furthermore, we give a subexponential time parameterized algorithm solving Open image in new window in Open image in new window time, making it one of relatively few natural problems in this complexity class on general graphs. These results are of broader interest to the field of social network analysis, where recent work of Brandes (2014) posits that the minimum edit distance to a threshold graph gives a good measure of consistency for node centralities. Finally, we show that all our positive results extend to Open image in new window , as well as the completion and deletion variants of both problems.

Tree decompositions and social graphs

Abstract Recent work has established that large informatics graphs such as social and information networks have non-trivial tree-like structure when viewed at moderate size scales. Here, we present results from the first detailed empirical evaluation of the use of tree decomposition (TD) heuristics for structure identification and extraction in social graphs. Although TDs have historically been used in structural graph theory and scientific computing, we show that—even with existing TD heuristics developed for those very different areas—TD methods can identify interesting structure in a wide range of realistic informatics graphs. Our main contributions are the following: we show that TD methods can identify structures that correlate strongly with the core-periphery structure of realistic networks, even when using simple greedy heuristics; we show that the peripheral bags of these TDs correlate well with low-conductance communities (when they exist) found using local spectral computations; and we show that several types of large-scale “ground-truth” communities, defined by demographic metadata on the nodes of the network, are well-localized in the large-scale and/or peripheral structures of the TDs. Our other main contributions are the following: we provide detailed empirical results for TD heuristics on toy and synthetic networks to establish a baseline to understand better the behavior of the heuristics on more complex real-world networks; and we prove a theorem providing formal justification for the intuition that the only two impediments to low-distortion hyperbolic embedding are high tree-width and long geodesic cycles. Our results suggest future directions for improved TD heuristics that are more appropriate for realistic social graphs.

Hyperbolicity, Degeneracy, and Expansion of Random Intersection Graphs

We establish the conditions under which several algorithmically exploitable structural features hold for random intersection graphs, a natural model for many real-world networks where edges correspond to shared attributes. Specifically, we fully characterize the degeneracy of random intersection graphs, and prove that the model asymptotically almost surely produces graphs with hyperbolicity at least

Optimizing adiabatic quantum program compilation using a graph-theoretic framework

\log {n}

A Fast Parameterized Algorithm for Co-Path Set.

logn. Further, we prove that when degenerate, the graphs generated by this model belong to a bounded-expansion graph class with high probability, a property particularly suitable for the design of linear time algorithms.

A multi-level anomaly detection algorithm for time-varying graph data with interactive visualization

Low-treedepth colorings are an important tool for algorithms that exploit structure in classes of bounded expansion; they guarantee subgraphs that use few colors are guaranteed to have bounded treedepth. These colorings have an implicit tradeoff between the total number of colors used and the treedepth bound, and prior empirical work suggests that the former dominates the run time of existing algorithms in practice. We introduce