2 research outputs found
Quantum and Randomized Lower Bounds for Local Search on Vertex-Transitive Graphs
We study the problem of \emph{local search} on a graph. Given a real-valued
black-box function f on the graph's vertices, this is the problem of
determining a local minimum of f--a vertex v for which f(v) is no more than f
evaluated at any of v's neighbors. In 1983, Aldous gave the first strong lower
bounds for the problem, showing that any randomized algorithm requires
queries to determine a local minima on the
n-dimensional hypercube. The next major step forward was not until 2004 when
Aaronson, introducing a new method for query complexity bounds, both
strengthened this lower bound to and gave an analogous
lower bound on the quantum query complexity. While these bounds are very
strong, they are known only for narrow families of graphs (hypercubes and
grids). We show how to generalize Aaronson's techniques in order to give
randomized (and quantum) lower bounds on the query complexity of local search
for the family of vertex-transitive graphs. In particular, we show that for any
vertex-transitive graph G of N vertices and diameter d, the randomized and
quantum query complexities for local search on G are
and , respectively
Almost Envy-Freeness with General Valuations
The goal of fair division is to distribute resources among competing players
in a "fair" way. Envy-freeness is the most extensively studied fairness notion
in fair division. Envy-free allocations do not always exist with indivisible
goods, motivating the study of relaxed versions of envy-freeness. We study the
envy-freeness up to any good (EFX) property, which states that no player
prefers the bundle of another player following the removal of any single good,
and prove the first general results about this property. We use the leximin
solution to show existence of EFX allocations in several contexts, sometimes in
conjunction with Pareto optimality. For two players with valuations obeying a
mild assumption, one of these results provides stronger guarantees than the
currently deployed algorithm on Spliddit, a popular fair division website.
Unfortunately, finding the leximin solution can require exponential time. We
show that this is necessary by proving an exponential lower bound on the number
of value queries needed to identify an EFX allocation, even for two players
with identical valuations. We consider both additive and more general
valuations, and our work suggests that there is a rich landscape of problems to
explore in the fair division of indivisible goods with different classes of
player valuations.Comment: Accepted to SODA 201