Matching Is as Easy as the Decision Problem, in the NC Model

Is matching in NC, i.e., is there a deterministic fast parallel algorithm for it? This has been an outstanding open question in TCS for over three decades, ever since the discovery of randomized NC matching algorithms [KUW85, MVV87]. Over the last five years, the theoretical computer science community has launched a relentless attack on this question, leading to the discovery of several powerful ideas. We give what appears to be the culmination of this line of work: An NC algorithm for finding a minimum-weight perfect matching in a general graph with polynomially bounded edge weights, provided it is given an oracle for the decision problem. Consequently, for settling the main open problem, it suffices to obtain an NC algorithm for the decision problem. We believe this new fact has qualitatively changed the nature of this open problem. All known efficient matching algorithms for general graphs follow one of two approaches: given by Edmonds [Edm65] and Lov\'asz [Lov79]. Our oracle-based algorithm follows a new approach and uses many of the ideas discovered in the last five years. The difficulty of obtaining an NC perfect matching algorithm led researchers to study matching vis-a-vis clever relaxations of the class NC. In this vein, recently Goldwasser and Grossman [GG15] gave a pseudo-deterministic RNC algorithm for finding a perfect matching in a bipartite graph, i.e., an RNC algorithm with the additional requirement that on the same graph, it should return the same (i.e., unique) perfect matching for almost all choices of random bits. A corollary of our reduction is an analogous algorithm for general graphs.Comment: Appeared in ITCS 202

Finding Periodic Apartments : A Computational Study of Hyperbolic Buildings

This thesis presents a computational study of a fundamental open conjecture in geometric group theory using an intricate combination of Boolean Satisfiability and orderly generation. In particular, we focus on Gromov’s subgroup conjecture (GSC), which states that “each one-ended hyperbolic group contains a subgroup isomorphic to the fundamental group of a closed surface of genus at least 2”. Several classes of groups have been shown to satisfy GSC, but the status of non-right-angled groups with regard to GSC is presently unknown, and may provide counterexamples to the conjecture. With this in mind Kangaslampi and Vdovina constructed 23 such groups utilizing the theory of hyperbolic buildings [International Journal of Algebra and Computation, vol. 20, no. 4, pp. 591–603, 2010], and ran an exhaustive computational analysis of surface subgroups of genus 2 arising from so-called periodic apartments [Experimental Mathematics, vol. 26, no. 1, pp. 54–61, 2017]. While they were able to rule out 5 of the 23 groups as potential counterexamples to GSC, they reported that their computational approach does not scale to genera higher than 2. We extend the work of Kangaslampi and Vdovina by developing two new approaches to analyzing the subgroups arising from periodic apartments in the 23 groups utilizing different combinations of SAT solving and orderly generation. We develop novel SAT encodings and a specialized orderly algorithm for the approaches, and perform an exhaustive analysis (over the 23 groups) of the genus 3 subgroups arising from periodic apartments. With the aid of massively parallel computation we also exhaust the case of genus 4. As a result we rule out 4 additional groups as counterexamples to GSC leaving 14 of the 23 groups for further inspection. In addition to this our approach provides an independent verification of the genus 2 results reported by Kangaslampi and Vdovina

Alternating sign matrices and domino tilings

We introduce a family of planar regions, called Aztec diamonds, and study the ways in which these regions can be tiled by dominoes. Our main result is a generating function that not only gives the number of domino tilings of the Aztec diamond of order $n$ but also provides information about the orientation of the dominoes (vertical versus horizontal) and the accessibility of one tiling from another by means of local modifications. Several proofs of the formula are given. The problem turns out to have connections with the alternating sign matrices of Mills, Robbins, and Rumsey, as well as the square ice model studied by Lieb

Error correction of parity-encoding-based annealing through post-readout decoding

Lechner, Hauke, and Zoller proposed a parity-encoded spin-embedding scheme for quantum annealing (QA) with all-to-all connectivity to avoid the issue of limited connectivity in near-term QA hardware and to enable the implementation thereof using only geometrically local interactions between spins fabricated on the planar substrate. Nevertheless, the redundant encoding of logical information, i.e., using a large number of spins to embed the logical information, increases the computational cost and reduces the efficiency. In this study, we show through Monte Carlo simulation that this redundant encoding may be exploited to solve the problems of the inefficiency and computational cost of the parity-encoded scheme by incorporating appropriate decoding, namely classical post-processing, of the spins to retrieve the logical information. Our findings open up the possibility of parity-encoded schemes for realizing the QA with near-term quantum technologies.Comment: 34 pages, 20 figures with minor corrections, the bibliography according to the PRX forma

A discrete, unitary, causal theory of quantum gravity

A discrete model of Lorentzian quantum gravity is proposed. The theory is completely background free, containing no reference to absolute space, time, or simultaneity. The states at one slice of time are networks in which each vertex is labelled with two arrows, which point along an adjacent edge, or to the vertex itself. The dynamics is specified by a set of unitary replacement rules, which causally propagate the local degrees of freedom. The inner product between any two states is given by a sum over histories. Assuming it converges (or can be Abel resummed), this inner product is proven to be hermitian and fully gauge-degenerate under spacetime diffeomorphisms. At least for states with a finite past, the inner product is also positive. This allows a Hilbert space of physical states to be constructed.Comment: 38 pages, 9 figures, v3 added to exposition and references, v4 expanded prospects sectio