On Isoperimetric Profiles and Computational Complexity

The isoperimetric profile of a graph is a function that measures, for an integer k, the size of the smallest edge boundary over all sets of vertices of size k. We observe a connection between isoperimetric profiles and computational complexity. We illustrate this connection by an example from communication complexity, but our main result is in algebraic complexity. We prove a sharp super-polynomial separation between monotone arithmetic circuits and monotone arithmetic branching programs. This shows that the classical simulation of arithmetic circuits by arithmetic branching programs by Valiant, Skyum, Berkowitz, and Rackoff (1983) cannot be improved, as long as it preserves monotonicity. A key ingredient in the proof is an accurate analysis of the isoperimetric profile of finite full binary trees. We show that the isoperimetric profile of a full binary tree constantly fluctuates between one and almost the depth of the tree

The Query Complexity of Correlated Equilibria

We consider the complexity of finding a correlated equilibrium of an $n$-player game in a model that allows the algorithm to make queries on players' payoffs at pure strategy profiles. Randomized regret-based dynamics are known to yield an approximate correlated equilibrium efficiently, namely, in time that is polynomial in the number of players $n$. Here we show that both randomization and approximation are necessary: no efficient deterministic algorithm can reach even an approximate correlated equilibrium, and no efficient randomized algorithm can reach an exact correlated equilibrium. The results are obtained by bounding from below the number of payoff queries that are needed

Computationally efficient solution to the Cahn–Hilliard equation: Adaptive implicit time schemes, mesh sensitivity analysis and the 3D isoperimetric problem

We present an efficient numerical framework for analyzing spinodal decomposition described by the Cahn–Hilliard equation. We focus on the analysis of various implicit time schemes for two and three dimensional problems. We demonstrate that significant computational gains can be obtained by applying embedded, higher order Runge–Kutta methods in a time adaptive setting. This allows accessing time-scales that vary by five orders of magnitude. In addition, we also formulate a set of test problems that isolate each of the sub-processes involved in spinodal decomposition: interface creation and bulky phase coarsening. We analyze the error fluctuations using these test problems on the split form of the Cahn–Hilliard equation solved using the finite element method with basis functions of different orders. Any scheme that ensures at least four elements per interface satisfactorily captures both sub-processes. Our findings show that linear basis functions have superior error-to-cost properties. This strategy – coupled with a domain decomposition based parallel implementation – let us notably augment the efficiency of a numerical Cahn–Hillard solver, and open new venues for its practical applications, especially when three dimensional problems are considered. We use this framework to address the isoperimetric problem of identifying local solutions in the periodic cube in three dimensions. The framework is able to generate all five hypothesized candidates for the local solution of periodic isoperimetric problem in 3D – sphere, cylinder, lamella, doubly periodic surface with genus two (Lawson surface) and triply periodic minimal surface (P Schwarz surface)

Explicit convergence bounds for Metropolis Markov chains: isoperimetry, spectral gaps and profiles

We derive the first explicit bounds for the spectral gap of a random walk Metropolis algorithm on $R^d$ for any value of the proposal variance, which when scaled appropriately recovers the correct $d^{-1}$ dependence on dimension for suitably regular invariant distributions. We also obtain explicit bounds on the ${\rm L}^2$-mixing time for a broad class of models. In obtaining these results, we refine the use of isoperimetric profile inequalities to obtain conductance profile bounds, which also enable the derivation of explicit bounds in a much broader class of models. We also obtain similar results for the preconditioned Crank--Nicolson Markov chain, obtaining dimension-independent bounds under suitable assumptions

Medial Axis Isoperimetric Profiles

Recently proposed as a stable means of evaluating geometric compactness, the isoperimetric profile of a planar domain measures the minimum perimeter needed to inscribe a shape with prescribed area varying from 0 to the area of the domain. While this profile has proven valuable for evaluating properties of geographic partitions, existing algorithms for its computation rely on aggressive approximations and are still computationally expensive. In this paper, we propose a practical means of approximating the isoperimetric profile and show that for domains satisfying a "thick neck" condition, our approximation is exact. For more general domains, we show that our bound is still exact within a conservative regime and is otherwise an upper bound. Our method is based on a traversal of the medial axis which produces efficient and robust results. We compare our technique with the state-of-the-art approximation to the isoperimetric profile on a variety of domains and show significantly tighter bounds than were previously achievable.Comment: Code and supplemental available here: https://github.com/pzpzpzp1/isoperimetric_profil

The complexity of solution concepts in Lipschitz games

Nearly a decade ago, Azrieli and Shmaya introduced the class of λ-Lipschitz games in which every player’s payoff function is λ-Lipschitz with respect to the actions of the other players. They showed via the probabilistic method that n-player Lipschitz games with m strategies per player have pure -approximate Nash equilibria, for ≥ λ√8n log(2mn). They left open, however, the question of how hard it is to find such an equilibrium. In this work, we develop an efficient reduction from more general games to Lipschitz games. We use this reduction to study both the query and computational complexity of algorithms finding λ-approximate pure Nash equilibria of λ-Lipschitz games and related classes. We show a query lower bound exponential in nλ/ against randomized algorithms finding - approximatepure Nash equilibria of n-player, λ-Lipschitz games. We additionally present the first PPAD-completeness result for finding pure Nash equilibria in a class of finite, non-Bayesian games (we show this for λ-Lipschitz polymatrix games for suitable pairs of values and λ) in which both the proof of PPAD-hardness and the proof of containment in PPAD require novel approaches (in fact, our approach implies containment in PPAD for any class of Lipschitz games in which payoffs from mixed-strategy profiles can be deterministically computed), and present a definition of “randomized PPAD”. We define and subsequently analyze the class of “Multi-Lipschitz games”, a generalization of Lipschitz games involving player-specific Lipschitz parameters in which the value of interest appears to be the average of the individual Lipschitz parameters. We discuss a dichotomy of the deterministic query complexity of finding -approximate Nash equilibria of general games and, subsequently, a query lower bound for λ-Lipschitz games in which any non-trivial value of requires exponentially-many queries to achieve. We examine which parts of this extend to the concepts of approximate correlated and coarse correlated equilibria, and in the process generalize the edge-isoperimetric inequalities to generalizations of the hypercube. Finally, we improve the block update algorithm presented by Goldberg and Marmolejo to break the potential boundary of a 0.75-approximation factor, presenting a randomized algorithm achieving a 0.7368-approximate Nash equilibrium making polynomially-many profile queries of an n-player 1/n−1 -Lipschitz game with an unbounded number of actions

Holographic Complexity Growth Rate in Horndeski Theory

Based on the context of complexity = action (CA) conjecture, we calculate the holographic complexity of AdS black holes with planar and spherical topologies in Horndeski theory. We find that the rate of change of holographic complexity for neutral AdS black holes saturates the Lloyd's bound. For charged black holes, we find that there exists only one horizon and thus the corresponding holographic complexity can't be expressed as the difference of some thermodynamical potential between two horizons as that of Reissner-Nordstrom AdS black hole in Einstein-Maxwell theory. However, the Lloyd's bound is not violated for charged AdS black hole in Horndeski theory.Comment: 20 pages, 6 figures, references added, typos correcte