An Ordered Approach to Solving Parity Games in Quasi Polynomial Time and Quasi Linear Space

Parity games play an important role in model checking and synthesis. In their paper, Calude et al. have shown that these games can be solved in quasi-polynomial time. We show that their algorithm can be implemented efficiently: we use their data structure as a progress measure, allowing for a backward implementation instead of a complete unravelling of the game. To achieve this, a number of changes have to be made to their techniques, where the main one is to add power to the antagonistic player that allows for determining her rational move without changing the outcome of the game. We provide a first implementation for a quasi-polynomial algorithm, test it on small examples, and provide a number of side results, including minor algorithmic improvements, a quasi bi-linear complexity in the number of states and edges for a fixed number of colours, and matching lower bounds for the algorithm of Calude et al

More consequences of falsifying SETH and the orthogonal vectors conjecture

The Strong Exponential Time Hypothesis and the OV-conjecture are two popular hardness assumptions used to prove a plethora of lower bounds, especially in the realm of polynomial-time algorithms. The OV-conjecture in moderate dimension States there is no > 0 for which an O(N2−ε) poly(D) time algorithm can decide whether there is a pair of orthogonal vectors in a given set of size N that contains D-dimensional binary vectors. We strengthen the evidence for these hardness assumptions. In particular, we show that if the OV-conjecture fails, then two problems for which we are far from obtaining even tiny improvements over exhaustive search would have surprisingly fast algorithms. If the OV conjecture is false, then there is a fixed > 0 such that: (1) For all d and all large enough k, there is a randomized algorithm that takes O(n(1−ε)k) time to solve the Zero-Weight-k-Clique and Min-Weight-k-Clique problems on d-hypergraphs with n vertices. As a consequence, the OV-conjecture is implied by the Weighted Clique conjecture. (2) For all c, the satisfiability of sparse TC1 circuits on n inputs (that is, circuits with cn wires, depth c log n, and negation, AND, OR, and threshold gates) can be computed in time O((2 −)n)