On inefficient special cases of NP-complete problems

AbstractEvery intractable set A has a polynomial complexity core, a set H such that for any P-subset S of A or of Ā, S∩H is finite. A complexity core H of A is proper if H⊆A. It is shown here that if P≠NP, then every currently known (i.e., either invertibly paddable or k-creative) NP-complete set A and its complement Ā have proper polynomial complexity cores that are nonsparse and are accepted by deterministic machines in time 2cn for some constant c. Turning to the intractable class DEXT=∪c>0DTIME(2cn), it is shown that every set that is ⩽pm-complete for DEXT has an infinite proper polynomial complexity core that is nonsparse and recursive

Feedback Generation for Performance Problems in Introductory Programming Assignments

Providing feedback on programming assignments manually is a tedious, error prone, and time-consuming task. In this paper, we motivate and address the problem of generating feedback on performance aspects in introductory programming assignments. We studied a large number of functionally correct student solutions to introductory programming assignments and observed: (1) There are different algorithmic strategies, with varying levels of efficiency, for solving a given problem. These different strategies merit different feedback. (2) The same algorithmic strategy can be implemented in countless different ways, which are not relevant for reporting feedback on the student program. We propose a light-weight programming language extension that allows a teacher to define an algorithmic strategy by specifying certain key values that should occur during the execution of an implementation. We describe a dynamic analysis based approach to test whether a student's program matches a teacher's specification. Our experimental results illustrate the effectiveness of both our specification language and our dynamic analysis. On one of our benchmarks consisting of 2316 functionally correct implementations to 3 programming problems, we identified 16 strategies that we were able to describe using our specification language (in 95 minutes after inspecting 66, i.e., around 3%, implementations). Our dynamic analysis correctly matched each implementation with its corresponding specification, thereby automatically producing the intended feedback.Comment: Tech report/extended version of FSE 2014 pape

Mean-Variance Optimization in Markov Decision Processes

We consider finite horizon Markov decision processes under performance measures that involve both the mean and the variance of the cumulative reward. We show that either randomized or history-based policies can improve performance. We prove that the complexity of computing a policy that maximizes the mean reward under a variance constraint is NP-hard for some cases, and strongly NP-hard for others. We finally offer pseudopolynomial exact and approximation algorithms.Comment: A full version of an ICML 2011 pape

A Random Matrix Model of Adiabatic Quantum Computing

We present an analysis of the quantum adiabatic algorithm for solving hard instances of 3-SAT (an NP-complete problem) in terms of Random Matrix Theory (RMT). We determine the global regularity of the spectral fluctuations of the instantaneous Hamiltonians encountered during the interpolation between the starting Hamiltonians and the ones whose ground states encode the solutions to the computational problems of interest. At each interpolation point, we quantify the degree of regularity of the average spectral distribution via its Brody parameter, a measure that distinguishes regular (i.e., Poissonian) from chaotic (i.e., Wigner-type) distributions of normalized nearest-neighbor spacings. We find that for hard problem instances, i.e., those having a critical ratio of clauses to variables, the spectral fluctuations typically become irregular across a contiguous region of the interpolation parameter, while the spectrum is regular for easy instances. Within the hard region, RMT may be applied to obtain a mathematical model of the probability of avoided level crossings and concomitant failure rate of the adiabatic algorithm due to non-adiabatic Landau-Zener type transitions. Our model predicts that if the interpolation is performed at a uniform rate, the average failure rate of the quantum adiabatic algorithm, when averaged over hard problem instances, scales exponentially with increasing problem size.Comment: 9 pages, 7 figure

Exact and Efficient Simulation of Concordant Computation

Concordant computation is a circuit-based model of quantum computation for mixed states, that assumes that all correlations within the register are discord-free (i.e. the correlations are essentially classical) at every step of the computation. The question of whether concordant computation always admits efficient simulation by a classical computer was first considered by B. Eastin in quant-ph/1006.4402v1, where an answer in the affirmative was given for circuits consisting only of one- and two-qubit gates. Building on this work, we develop the theory of classical simulation of concordant computation. We present a new framework for understanding such computations, argue that a larger class of concordant computations admit efficient simulation, and provide alternative proofs for the main results of quant-ph/1006.4402v1 with an emphasis on the exactness of simulation which is crucial for this model. We include detailed analysis of the arithmetic complexity for solving equations in the simulation, as well as extensions to larger gates and qudits. We explore the limitations of our approach, and discuss the challenges faced in developing efficient classical simulation algorithms for all concordant computations.Comment: 16 page