Kei Uchizawa

TL;DR: It is proved that every such puzzle is solvable in O ( n 2 ) token swaps, and thus the problem of minimizing the number of token swaps to reach the target token placement can be solved exactly in polynomial time on complete bipartite graphs.

TL;DR: The precise computational complexities of all the three problems with regards to graph diameters are settled, and the FPT algorithms imply that all the problems can be solved in polynomial time for any graph with n vertices if |C|=O(logn).

TL;DR: This letter investigates a new complexity measure for threshold circuits, energy complexity, whose minimization yields computations with sparse activity, and proves that all computations by threshold circuits of polynomial size with entropy O(log n) can be restructured so that their energy complexity is reduced to a level near the entropy of circuit states.

TL;DR: It is proved that every puzzle is solvable in O(n 2) token swaps, and the problem of minimizing the number of token swaps to reach the target token placement can be solved exactly in polynomial time on complete bipartite graphs.

TL;DR: It is shown that there exists a trade-off among three complexity measures of threshold circuits: the energy complexity, size, and depth, which implies an exponential lower bound on the size of constant-depth threshold circuits with small energy complexity for a large class of Boolean functions.