Quantum and Randomized Lower Bounds for Local Search on Vertex-Transitive Graphs

Abstract

Abstract. We study the problem of 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 Ω(2 n/2−o(1) ) 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 Ω(2 n/2 /n 2 ) 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 Ω " √ N d log N "and Ω, respectively.

Almost Envy-Freeness with General Valuations

“…This is done via a reduction from local search on a class of graphs known as the Odd graphs, for which we prove an exponential lower bound. In combination with results due to Dinh and Russell [13] and Valencia-Pabon and Vera [30], this yields an analogous exponential lower bound for randomized algorithms. Dobzinski et al [14] also use a local search reduction to prove a lower bound on the number of value queries required to find a certain type of equilibrium in a simultaneous second price auction, for bidders with XOS (i.e., fractionally subadditive) valuations.…”

“…To complete the lower bound, we proved an exponential lower bound on the number of queries required to find a local maximum on K(2k+1, k). We used results from Dinh and Russell [13] and Valencia-Pabon and Vera [30] to obtain an exponential lower bound for randomized algorithms as well. Our EFX lower bounds hold even for two players with identical submodular valuations.…”

“…Our reduction from Local Search to EFX Allocation also yields an exponential lower bound for randomized algorithms for free, thanks to results due to Dinh and Russell [13] and Valencia-Pabon and Vera [30]. Let R[LS(G)] be the minimum number of queries required to solve Local Search on G by a randomized algorithm: the algorithm should output a local maximum with probability at least 2/3 (say) over its internal coin flips.…”

“…We now show how Theorem 4.2 can easily be used to find an EFX allocation for two players with general and possibly distinct valuations. 13 Our algorithm for this follows from the observation that any player can partition the goods into k bundles that are mutually EFX from her viewpoint, simply by computing the leximin++ solution with k copies of herself.…”

Almost Envy-Freeness with General Valuations

“…13 Our algorithm for this follows from the observation that any player can partition the goods into k bundles that are mutually EFX from her viewpoint, simply by computing the leximin++ solution with k copies of herself.…”