Optimal Sparsification for Some Binary CSPs Using Low-degree Polynomials

This paper analyzes to what extent it is possible to efficiently reduce the number of clauses in NP-hard satisfiability problems, without changing the answer. Upper and lower bounds are established using the concept of kernelization. Existing results show that if NP is not contained in coNP/poly, no efficient preprocessing algorithm can reduce n-variable instances of CNF-SAT with d literals per clause, to equivalent instances with $O(n^{d-e})$ bits for any e > 0. For the Not-All-Equal SAT problem, a compression to size $\~O(n^{d-1})$ exists. We put these results in a common framework by analyzing the compressibility of binary CSPs. We characterize constraint types based on the minimum degree of multivariate polynomials whose roots correspond to the satisfying assignments, obtaining (nearly) matching upper and lower bounds in several settings. Our lower bounds show that not just the number of constraints, but also the encoding size of individual constraints plays an important role. For example, for Exact Satisfiability with unbounded clause length it is possible to efficiently reduce the number of constraints to n+1, yet no polynomial-time algorithm can reduce to an equivalent instance with $O(n^{2-e})$ bits for any e > 0, unless NP is a subset of coNP/poly.Comment: Updated the cross-composition in lemma 18 (minor update), since the previous version did NOT satisfy requirement 4 of lemma 18 (the proof of Claim 20 was incorrect

The development of the five mini-theories of self-determination theory: an historical overview, emerging trends, and future directions

Self-determination theory is a macro-theory of human motivation, emotion, and personality that has been under development for 40 years following the seminal work of Edward Deci and Richard Ryan. Self-determination theory (SDT; Deci & Ryan, 1985b, 2000; Niemiec, Ryan, & Deci, in press; Ryan & Deci, 2000; Vansteenkiste, Ryan, & Deci, 2008) has been advanced in a cumulative, research-driven manner, as new ideas have been naturally and steadily integrated into the theory following sufficient empirical support, which has helped SDT maintain its internal consistency. To use a metaphor, the development of SDT is similar to the construction of a puzzle. Over the years, new pieces have been added to the theory once their fit was determined. At present, dozens of scholars throughout the world continue to add their piece to the ‘‘SDT puzzle,’’ and hundreds of practitioners working with all age groups, and in various domains and cultures, have used SDT to inform their practice. Herein, we provide an historical overview of the development of the five mini-theories (viz., cognitive evaluation theory, organismic integration theory, causality orientations theory, basic needs theory, and goal content theory) that constitute SDT, discuss emerging trends within those mini-theories, elucidate similarities with and differences from other theoretical frameworks, and suggest directions for future researc