Proving circuit lower bounds in uniform classes is one of the grand challenges in computational complexity theory. Particularly, it is well known that proving superpolynomial circuit lower bounds in NP resolves the longstanding conjecture NP≠P. Towards the final goal, a lot of work have been dedicated to approaches proving circuit lower bounds in high classes. This tutorial article overviews those proof techniques developed for circuit lower bounds in higher classes than NP such as PH, ZPPNP, MAEXP, PP, Promise-MA, NEXP, and so forth.
A threshold circuit is a combinatorial circuit consisting of logic gates computing linear threshold functions. A threshold circuit is one of the most well-studied computational models in circuit complexity theory, and is commonly viewed as an abstract model of a neural network in the brain. In this paper, we investigate threshold circuits from the viewpoint of a biologically-inspired complexity measure, called energy complexity. Following basic definitions and examples of threshold circuits, we observe three lower bound results for threshold circuits of bounded energy.
In this survey, we consider arborescences in directed graphs. The concept of arborescences is a directed analogue of a spanning tree in an undirected graph, and one of the most fundamental concepts in graph theory and combinatorial optimization. This survey has two aims: we first show recent developments in the research on arborescences, and then give introduction of abstract concepts (e.g., matroids), and algorithmic techniques (e.g., primal-dual method) through well-known results for arborescences. In the first half of this survey, we consider the minimum-cost arborescence problem. The goal of this problem is to find a minimum-cost arborescence rooted at a designated vertex, where a matroid and a primal-dual method play important roles. In the second half of this survey, we study the arborescence packing problem. The goal of this problem is to find arc-disjoint arborescences rooted at a designated vertex, where the min-max theorem by Edmonds plays an important role.