6 research outputs found
Path-search in the pyramid and in other graphs
We are given an acyclic directed graph with one source, and a subset of its
edges which contains exactly one outgoing edge for every non-sink vertex. These
edges determine a unique path from the source to a sink. We can think of it as
a switch in every vertex, which determines which way the water arriving to that
vertex flows further. We are interested in determining either the sink the flow
arrives, or the whole path, with as few questions as possible. The questions we
can ask correspond to the vertices of the graph, and the answer describes the
switch, i.e. tells which outgoing edge is in our given subset. Originally the
problem was proposed by Soren Riis (who posed the question for pyramid graphs)
in the following more general form. We are given a natural number k, and k
questions can be asked in a round. The goal is to minimize the number of
rounds. We completely solve this problem for complete t-ary trees. Also, for
pyramid graphs we present some non-trivial partial results
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
Paths Beyond Local Search: A Nearly Tight Bound for Randomized Fixed-Point Computation
In 1983, Aldous proved that randomization can speedup local search. For
example, it reduces the query complexity of local search over [1:n]^d from
Theta (n^{d-1}) to O (d^{1/2}n^{d/2}). It remains open whether randomization
helps fixed-point computation. Inspired by this open problem and recent
advances on equilibrium computation, we have been fascinated by the following
question:
Is a fixed-point or an equilibrium fundamentally harder to find than a local
optimum? In this paper, we give a nearly-tight bound of Omega(n)^{d-1} on the
randomized query complexity for computing a fixed point of a discrete Brouwer
function over [1:n]^d. Since the randomized query complexity of global
optimization over [1:n]^d is Theta (n^{d}), the randomized query model over
[1:n]^d strictly separates these three important search problems: Global
optimization is harder than fixed-point computation, and fixed-point
computation is harder than local search. Our result indeed demonstrates that
randomization does not help much in fixed-point computation in the query model;
the deterministic complexity of this problem is Theta (n^{d-1})
How to trap a gradient flow
We consider the problem of finding an -approximate stationary
point of a smooth function on a compact domain of . In contrast
with dimension-free approaches such as gradient descent, we focus here on the
case where is finite, and potentially small. This viewpoint was explored in
1993 by Vavasis, who proposed an algorithm which, for any fixed finite
dimension , improves upon the oracle complexity of
gradient descent. For example for , Vavasis' approach obtains the
complexity . Moreover for he also proved a lower bound
of for deterministic algorithms (we extend this
result to randomized algorithms).
Our main contribution is an algorithm, which we call gradient flow trapping
(GFT), and the analysis of its oracle complexity. In dimension , GFT
closes the gap with Vavasis' lower bound (up to a logarithmic factor), as we
show that it has complexity
. In dimension
, we show a complexity of
, improving upon
Vavasis' . In higher dimensions, GFT has
the remarkable property of being a logarithmic parallel depth strategy, in
stark contrast with the polynomial depth of gradient descent or Vavasis'
algorithm. In this higher dimensional regime, the total work of GFT improves
quadratically upon the only other known polylogarithmic depth strategy for this
problem, namely naive grid search. We augment this result with another
algorithm, named \emph{cut and flow} (CF), which improves upon Vavasis'
algorithm in any fixed dimension.Comment: 25 pages, 5 figures. Added an improved algorithm for dimensions >
The Communication Complexity of Local Search
We study the following communication variant of local search. There is some
fixed, commonly known graph . Alice holds and Bob holds , both
are functions that specify a value for each vertex. The goal is to find a local
maximum of with respect to , i.e., a vertex for which
for every neighbor of .
Our main result is that finding a local maximum requires polynomial (in the
number of vertices) bits of communication. The result holds for the following
families of graphs: three dimensional grids, hypercubes, odd graphs, and degree
4 graphs. Moreover, we provide an \emph{optimal} communication bound of
for the hypercube, and for a constant dimensional greed,
where is the number of vertices in the graph.
We provide applications of our main result in two domains, exact potential
games and combinatorial auctions. First, we show that finding a pure Nash
equilibrium in -player -action exact potential games requires polynomial
(in ) communication. We also show that finding a pure Nash equilibrium in
-player -action exact potential games requires exponential (in )
communication.
The second domain that we consider is combinatorial auctions, in which we
prove that finding a local maximum in combinatorial auctions requires
exponential (in the number of items) communication even when the valuations are
submodular.
Each one of the results demonstrates an exponential separation between the
non-deterministic communication complexity and the randomized communication
complexity of a total search problem
The Query Complexity of Local Search and Brouwer in Rounds
We consider the query complexity of finding a local minimum of a function
defined on a graph, where at most rounds of interaction with the oracle are
allowed. Rounds model parallel settings, where each query takes resources to
complete and is executed on a separate processor. Thus the query complexity in
rounds informs how many processors are needed to achieve a parallel time of
.
We focus on the d-dimensional grid , where the dimension is a
constant, and consider two regimes for the number of rounds: constant and
polynomial in n. We give algorithms and lower bounds that characterize the
trade-off between the number of rounds of adaptivity and the query complexity
of local search.
When the number of rounds is constant, we show that the query complexity
of local search in rounds is , for both deterministic and randomized algorithms. When the number
of rounds is polynomial, i.e. for , the
randomized query complexity is for all . For and , we show
the same upper bound expression holds and give almost matching lower bounds.
The local search analysis also enables us to characterize the query complexity
of computing a Brouwer fixed point in rounds. Our proof technique for lower
bounding the query complexity in rounds may be of independent interest as an
alternative to the classical relational adversary method of Aaronson from the
fully adaptive setting