Convergence Rate of Riemannian Hamiltonian Monte Carlo and Faster Polytope Volume Computation

We give the first rigorous proof of the convergence of Riemannian Hamiltonian Monte Carlo, a general (and practical) method for sampling Gibbs distributions. Our analysis shows that the rate of convergence is bounded in terms of natural smoothness parameters of an associated Riemannian manifold. We then apply the method with the manifold defined by the log barrier function to the problems of (1) uniformly sampling a polytope and (2) computing its volume, the latter by extending Gaussian cooling to the manifold setting. In both cases, the total number of steps needed is O^{*}(mn^{\frac{2}{3}}), improving the state of the art. A key ingredient of our analysis is a proof of an analog of the KLS conjecture for Gibbs distributions over manifolds

A polynomial time knot polynomial

We present the strongest known knot invariant that can be computed effectively (in polynomial time).Comment: Typos fixed, length reduced for publication in PAM

Ultimate Polynomial Time

The class $\mathcal{UP}$ of `ultimate polynomial time' problems over $\mathbb C$ is introduced; it contains the class $\mathcal P$ of polynomial time problems over $\mathbb C$. The $\tau$-Conjecture for polynomials implies that $\mathcal{UP}$ does not contain the class of non-deterministic polynomial time problems definable without constants over $\mathbb C$. This latest statement implies that $\mathcal P \ne \mathcal{NP}$ over $\mathbb C$. A notion of `ultimate complexity' of a problem is suggested. It provides lower bounds for the complexity of structured problems

Polynomial-time word problems

We find polynomial-time solutions to the word problem for free-by-cyclic groups, the word problem for automorphism groups of free groups, and the membership problem for the handlebody subgroup of the mapping class group. All of these results follow from observing that automorphisms of the free group strongly resemble straight line programs, which are widely studied in the theory of compressed data structures. In an effort to be self-contained we give a detailed exposition of the necessary results from computer science.Comment: 25 pages, 6 figure

Insignificant Choice Polynomial Time

In the late 1980s Gurevich conjectured that there is no logic capturing PTIME, where logic has to be understood in a very general way comprising computation models over structures. In this article we first refute Gurevich's conjecture. For this we extend the seminal research of Blass, Gurevich and Shelah on {\em choiceless polynomial time} (CPT), which exploits deterministic Abstract State Machines (ASMs) supporting unbounded parallelism to capture the choiceless fragment of PTIME. CPT is strictly included in PTIME. We observe that choice is unavoidable, but that a restricted version suffices, which guarantees that the final result is independent from the choice. Such a version of polynomially bounded ASMs, which we call {\em insignificant choice polynomial time} (ICPT) will indeed capture PTIME. Even more, insignificant choice can be captured by ASMs with choice restricted to atoms such that a {\em local insignificance condition} is satisfied. As this condition can be expressed in the logic of non-deterministic ASMs, we obtain a logic capturing PTIME. Furthermore, using inflationary fixed-points we can capture problems in PTIME by fixed-point formulae in a fragment of the logic of non-deterministic ASMs plus inflationary fixed-points. We use this result for our second contribution showing that PTIME differs from NP. For the proof we build again on the research on CPT first establishing a limitation on permutation classes of the sets that can be activated by an ICPT computation. We then prove an inseparability theorem, which characterises classes of structures that cannot be separated by the logic. In particular, this implies that SAT cannot be decided by an ICPT computation.Comment: 69 page

Polynomial-time kernel reductions

In the framework of computational complexity and in an effort to define a more natural reduction for problems of equivalence, we investigate the recently introduced kernel reduction, a reduction that operates on each element of a pair independently. This paper details the limitations and uses of kernel reductions. We show that kernel reductions are weaker than many-one reductions and provide conditions under which complete problems exist. Ultimately, the number and size of equivalence classes can dictate the existence of a kernel reduction. We leave unsolved the unconditional existence of a complete problem under polynomial-time kernel reductions for the standard complexity classes

On Polynomial Time Computable Numbers

It will be shown that the polynomial time computable numbers form a field, and especially an algebraically closed field.Comment: 19 page

Ramanujan Graphs in Polynomial Time

The recent work by Marcus, Spielman and Srivastava proves the existence of bipartite Ramanujan (multi)graphs of all degrees and all sizes. However, that paper did not provide a polynomial time algorithm to actually compute such graphs. Here, we provide a polynomial time algorithm to compute certain expected characteristic polynomials related to this construction. This leads to a deterministic polynomial time algorithm to compute bipartite Ramanujan (multi)graphs of all degrees and all sizes

Multicommodity Flow in Polynomial Time

The multicommodity flow problem is NP-hard already for two commodities over bipartite graphs. Nonetheless, using our recent theory of n-fold integer programming and extensions developed herein, we are able to establish the surprising polynomial time solvability of the problem in two broad situations

Controlling qubit networks in polynomial time

Future quantum devices often rely on favourable scaling with respect to the system components. To achieve desirable scaling, it is therefore crucial to implement unitary transformations in an efficient manner. We develop an upper bound for the minimum time required to implement a unitary transformation on a generic qubit network in which each of the qubits is subject to local time dependent controls. The set of gates is characterized that can be implemented in a time that scales at most polynomially in the number of qubits. Furthermore, we show how qubit systems can be concatenated through controllable two body interactions, making it possible to implement the gate set efficiently on the combined system. Finally a system is identified for which the gate set can be implemented with fewer controls. The considered model is particularly important, since it describes electron-nuclear spin interactions in NV centers