The Quest for Strong Inapproximability Results with Perfect Completeness

The Unique Games Conjecture has pinned down the approximability of all constraint satisfaction problems (CSPs), showing that a natural semidefinite programming relaxation offers the optimal worst-case approximation ratio for any CSP. This elegant picture, however, does not apply for CSP instances that are perfectly satisfiable, due to the imperfect completeness inherent in the Unique Games Conjecture. This work is motivated by the pursuit of a better understanding of the approximability of perfectly satisfiable instances of CSPs. We prove that an “almost Unique” version of Label Cover can be approximated within a constant factor on satisfiable instances. Our main conceptual contribution is the formulation of a (hypergraph) version of Label Cover that we call V Label Cover . Assuming a conjecture concerning the inapproximability of V Label Cover on perfectly satisfiable instances, we prove the following implications: • There is an absolute constant c 0 such that for k ≥ 3, given a satisfiable instance of Boolean k -CSP, it is hard to find an assignment satisfying more than c 0 k 2 /2 k fraction of the constraints. • Given a k -uniform hypergraph, k ≥ 2, for all ε > 0, it is hard to tell if it is q -strongly colorable or has no independent set with an ε fraction of vertices, where q =⌈ k +√ k -1/2⌉. • Given a k -uniform hypergraph, k ≥ 3, for all ε > 0, it is hard to tell if it is ( k -1)-rainbow colorable or has no independent set with an ε fraction of vertices.

Warm-starting quantum optimization

There is an increasing interest in quantum algorithms for problems of integer programming and combinatorial optimization. Classical solvers for such problems employ relaxations, which replace binary variables with continuous ones, for instance in the form of higher-dimensional matrix-valued problems (semidefinite programming). Under the Unique Games Conjecture, these relaxations often provide the best performance ratios available classically in polynomial time. Here, we discuss how to warm-start quantum optimization with an initial state corresponding to the solution of a relaxation of a combinatorial optimization problem and how to analyze properties of the associated quantum algorithms. In particular, this allows the quantum algorithm to inherit the performance guarantees of the classical algorithm. We illustrate this in the context of portfolio optimization, where our results indicate that warm-starting the Quantum Approximate Optimization Algorithm (QAOA) is particularly beneficial at low depth. Likewise, Recursive QAOA for MAXCUT problems shows a systematic increase in the size of the obtained cut for fully connected graphs with random weights, when Goemans-Williamson randomized rounding is utilized in a warm start. It is straightforward to apply the same ideas to other randomized-rounding schemes and optimization problems.

2-Approximating Feedback Vertex Set in Tournaments

A tournament is a directed graph T such that every pair of vertices is connected by an arc. A feedback vertex set is a set S of vertices in T such that T − S is acyclic. We consider the Feedback Vertex Set problem in tournaments. Here, the input is a tournament T and a weight function w : V ( T ) → N, and the task is to find a feedback vertex set S in T minimizing w ( S ) = ∑ v∈S w ( v ). Rounding optimal solutions to the natural LP-relaxation of this problem yields a simple 3-approximation algorithm. This has been improved to 2.5 by Cai et al. [SICOMP 2000], and subsequently to 7/3 by Mnich et al. [ESA 2016]. In this article, we give the first polynomial time factor 2-approximation algorithm for this problem. Assuming the Unique Games Conjecture, this is the best possible approximation ratio achievable in polynomial time.

Three candidate plurality is stablest for small correlations

Using the calculus of variations, we prove the following structure theorem for noise-stable partitions: a partition of n-dimensional Euclidean space into m disjoint sets of fixed Gaussian volumes that maximise their noise stability must be $(m-1)$ -dimensional, if $m-1\leq n$ . In particular, the maximum noise stability of a partition of m sets in $\mathbb {R}^{n}$ of fixed Gaussian volumes is constant for all n satisfying $n\geq m-1$ . From this result, we obtain: (i) A proof of the plurality is stablest conjecture for three candidate elections, for all correlation parameters $\rho $ satisfying $0<\rho <\rho _{0}$ , where $\rho _{0}>0$ is a fixed constant (that does not depend on the dimension n), when each candidate has an equal chance of winning. (ii) A variational proof of Borell’s inequality (corresponding to the case $m=2$ ). The structure theorem answers a question of De–Mossel–Neeman and of Ghazi–Kamath–Raghavendra. Item (i) is the first proof of any case of the plurality is stablest conjecture of Khot-Kindler-Mossel-O’Donnell for fixed $\rho $ , with the case $\rho \to L1^{-}$ being solved recently. Item (i) is also the first evidence for the optimality of the Frieze–Jerrum semidefinite program for solving MAX-3-CUT, assuming the unique games conjecture. Without the assumption that each candidate has an equal chance of winning in (i), the plurality is stablest conjecture is known to be false.

Statistics of spatial rotations and projection directions considering molecular symmetry in 3D electron cryo-microscopy

Electron cryo-microscopy (cryoEM) three-dimensional (3D) reconstruction is based on estimations of orientations of projection images or 3D volumes. It is common that the macromolecules studied by cryoEM have molecular symmetry, which, unfortunately, has not been taken into consideration by any statistics for either spatial rotations or projection directions at this point. Meanwhile, there are growing needs to adopt advanced statistical methods, and further, modern machine learning techniques in cryoEM. Since those methodologies are built heavily upon statistical learning cornerstones, the absence of their domain-specific statistical justification limits their applications in cryoEM. In this research, based on the concept of non-unique-games (NUG), we propose two key statistical measurements, the mean and the variance, of both spatial rotations and projection directions when molecular symmetry is considered. Such methods are implemented in the open-source python package pySymStat.

Psychotherapeutic components of commercially available virtual reality games (Preprint)

BACKGROUND Immersive virtual reality (VR) games are increasingly becoming part of everyday life. Several studies support immersive VR technology as a treatment method for mental health problems. There is however minimal research into the feasibility, prevalence, and quality of commercially available VR games on commercial platforms as tools for treatment or add on to treatment of mental health problems. OBJECTIVE The aim of this study was to explore the prevalence, feasibility and quality of commercially available games related to psychotherapy on a commercially available platform. METHODS We performed a search for keywords related to diagnosis and treatment strategies of mental health problems. The search was performed during March 27th on STEAM (VR content and gaming platform). A usability scale was used as a tool to look at the interaction and usability of the games, the VR-UI-UX-8. The tool contains 8 statements about usability scored 0-10, 0 indicating “Not at all” and 10 indicating “very much so”. The score ranges from 0-80 with a higher score indicating worse usability. RESULTS In total, 516 hits were found, 371 unique games. After the games were reviewed, 83 games passed the inclusion criteria, were purchased and played. Majority of the games which were excluded were either not connected to mental health, contained violence, adult content or were in other ways irrelevant or inappropriate. The mean score for the games on the VR-UI-UX-8 was 16.5 (standard deviation 15.8) with a range from 0-68. Most relevant and feasible games were found in the search words meditation, mindfulness, and LSD. CONCLUSIONS Commercial platforms hold great potential for VR games with psychotherapeutic components. The platforms are only at the beginning of the development towards serious games, e-learning and psychotherapeutic treatments. Currently the quality and usability for clinical and at home applicability are scarce, but hold great potential.