The circuit satisfiability problem has been intensively studied since Ryan Williams showed a connection between the problem and lower bounds for circuit complexity. In this letter, we present a #SAT algorithm for synchronous Boolean circuits of n inputs and s gates in time $2^{nleft(1 - rac{1}{2^{O(s/n)}} ight)}$ if s=o(n log n).

Let G=(V,E) be an unweighted simple graph. A distance-d independent set is a subset I ⊆ V such that dist(u, v) ≥ d for any two vertices u, v in I, where dist(u, v) is the distance between u and v. Then, Maximum Distance-d Independent Set problem requires to compute the size of a distance-d independent set with the maximum number of vertices. Even for a fixed integer d ≥ 3, this problem is NP-hard. In this paper, we design an exact exponential algorithm that calculates the size of a maximum distance-3 independent set in O(1.4143n) time.

The family of stable matching problems have been well-studied across a wide field of research areas, including economics, mathematics and computer science. In general, an instance of a stable matching problem is given by a set of participants who have expressed their preferences of each other, and asks to find a “stable” matching, that is, a pairing of the participants such that no unpaired participants prefer each other to their assigned partners. In the case of the Stable Roommates Problem (SR), it is known that given an even number n of participants, there might not exist a stable matching that pairs all of the participants, but there exist efficient algorithms to determine if this is possible or not, and if it is possible, produce such a matching. Common extensions of SR allow for the participants' preference lists to be incomplete, or include indifference. Allowing indifference in turn, gives rise to different possible definitions of stability, super, strong, and weak stability. While instances asking for super and strongly stable matching can be efficiently solved even if preference lists are incomplete, the case of weak stability is NP-complete. We examine a restricted case of indifference, introducing the concept of unranked entries. For this type of instances, we show that the problem of finding a weakly stable matching remains NP-complete even if each participant has a complete preference list with at most two unranked entries, or is herself unranked for up to three other participants. On the other hand, for instances where there are m acceptable pairs and there are in total k unranked entries in all of the participants' preference lists, we propose an O(2kn2)-time and polynomial space algorithm that finds a stable matching, or determines that none exists in the given instance.