We give faster exponential-time randomised approximation algorithms for counting problems where polynomial-time approximation is unavailable and exact exponential-time counting remains expensive. For general n-vertex graphs, our independent-set counter runs in O^{∗}(1.1869ⁿ) time, improving the previous O^*(1.2041ⁿ) general-graph bound. For n-variable #2-SAT, we obtain an O^*(1.2373ⁿ)-time approximation algorithm, narrowly below Wahlström’s currently cited O^*(1.2377ⁿ) variable-parameter exact bound. The new algorithmic point is to take the square root after decomposition. For a single bounded unweighted self-reduction with f(x) positive leaves and recursion-compatible upper bound b(x), an enumerate-or-sample estimator gives an (ε,δ)-approximation in O^*(√{b(x)} ε^{-2}log(1/δ)) time. After preprocessing decomposes an input into many bounded cores, the combined estimator pays O^*(√{∑_i b_i(x_i)} ε^{-2} log (1/δ)) , rather than estimating the cores separately at cost ∑_i √{b_i(x_i)}. The same conversion improves the bases for counting maximal cliques, minimal separators, and perfect matchings in subcubic graphs. Bounded unweighted self-reductions provide the formal language; at the level of counting classes, the resulting unweighted formulation has the same Karp closure as TotP. With explicit recursion-tree access, the framework yields black-box quantum speed-ups.

We study liquid democracy, a collective decision making paradigm that allows voters to transitively delegate their votes, through an algorithmic lens. In our model, there are two alternatives, one correct and one incorrect, and we are interested in the probability that the majority opinion is correct. Our main question is whether there exist delegation mechanisms that are guaranteed to outperform direct voting, in the sense of being always at least as likely, and sometimes more likely, to make a correct decision. Even though we assume that voters can only delegate their votes to better-informed voters, we show that local delegation mechanisms, which only take the local neighborhood of each voter as input (and, arguably, capture the spirit of liquid democracy), cannot provide the foregoing guarantee. By contrast, we design a non-local delegation mechanism that does provably outperform direct voting under mild assumptions about voters.

The Lazy Shortest Path (LazySP) class consists of motion-planning algorithms that only evaluate edges along candidate shortest paths between the source and target. These algorithms were designed to minimize the number of edge evaluations in settings where edge evaluation dominates the running time of the algorithm such as manipulation in cluttered environments and planning for robots in surgical settings; but how close to optimal are LazySP algorithms in terms of this objective? Our main result is an analytical upper bound, in a probabilistic model, on the number of edge evaluations required by LazySP algorithms; a matching lower bound shows that these algorithms are asymptotically optimal in the worst case.

We study liquid democracy, a collective decision making paradigm that allows voters to transitively delegate their votes, through an algorithmic lens. In our model, there are two alternatives, one correct and one incorrect, and we are interested in the probability that the majority opinion is correct. Our main question is whether there exist delegation mechanisms that are guaranteed to outperform direct voting, in the sense of being always at least as likely, and sometimes more likely, to make a correct decision. Even though we assume that voters can only delegate their votes to better-informed voters, we show that local delegation mechanisms, which only take the local neighborhood of each voter as input (and, arguably, capture the spirit of liquid democracy), cannot provide the foregoing guarantee. By contrast, we design a non-local delegation mechanism that does provably outperform direct voting under mild assumptions about voters.

Sequential allocation is a simple allocation mechanism in which agents are given pre-specified turns in which they take one item among those that are still available. It has long been known that sequential allocation is not strategyproof. This raises the question of the complexity of computing a preference report that yields a higher utility than the truthful preference. We show that the problem is NP-complete for one manipulating agent with additive utilities and several non-manipulating agents. In doing so, we correct a wrong claim made in a previous paper. We then give two additional results. First, we present a polynomial-time algorithm for optimal manipulation when the manipulator has additive binary utilities. Second, we consider a stronger notion of manipulation whereby the untruthful outcome yields more utility than the truthful outcome for all utilities consistent with the ordinal preferences; for this notion, we show that a manipulation, if any, can be computed in polynomial time.

We consider the well-studied cake cutting problem in which the goal is to find an envy-free allocation based on queries from n agents. The problem has received attention in computer science, mathematics, and economics. It has been a major open problem whether there exists a discrete and bounded envy-free protocol. We resolve the problem by proposing a discrete and bounded envy-free protocol for any number of agents. The maximum number of queries required by the protocol is nnnnnn. Even if we do not run our protocol to completion, it can find in at most nn+1queries an envy-free partial allocation of the cake in which each agent gets at least 1/n of the value of the whole cake.

We consider the well-studied cake cutting problem in which the goal is to identify an envy-free allocation based on a minimal number of queries from the agents. The problem has attracted considerable attention within various branches of computer science, mathematics, and economics. Although, the elegant Selfridge-Conway envy-free protocol for three agents has been known since 1960, it has been a major open problem to obtain a bounded envy-free protocol for more than three agents. The problem has been termed the central open problem in cake cutting. We solve this problem by proposing a discrete and bounded envy-free protocol for four agents.

Balanced knockout tournaments are one of the most common formats for sports competitions, and are also used in elections and decision-making. We consider the computational problem of finding the optimal draw for a particular player in such a tournament. The problem has generated considerable research within AI in recent years. We prove that checking whether there exists a draw in which a player wins is NP-complete, thereby settling an outstanding open problem. Our main result has a number of interesting implications on related counting and approximation problems. We present a memoization-based algorithm for the problem that is faster than previous approaches. Moreover, we highlight two natural cases that can be solved in polynomial time. All of our results also hold for the more general problem of counting the number of draws in which a given player is the winner.