67 research outputs found
Challenges in computational lower bounds
We draw two incomplete, biased maps of challenges in computational complexity
lower bounds
Separating NOF communication complexity classes RP and NP
We provide a non-explicit separation of the number-on-forehead communication
complexity classes RP and NP when the number of players is up to \delta log(n)
for any \delta<1. Recent lower bounds on Set-Disjointness [LS08,CA08] provide
an explicit separation between these classes when the number of players is only
up to o(loglog(n))
Sublinear Communication Protocols for Multi-Party Pointer Jumping and a Related Lower Bound
We study the one-way number-on-the-forehead (NOF) communication complexity of
the -layer pointer jumping problem with vertices per layer. This classic
problem, which has connections to many aspects of complexity theory, has seen a
recent burst of research activity, seemingly preparing the ground for an
lower bound, for constant . Our first result is a surprising
sublinear -- i.e., -- upper bound for the problem that holds for , dashing hopes for such a lower bound. A closer look at the protocol
achieving the upper bound shows that all but one of the players involved are
collapsing, i.e., their messages depend only on the composition of the layers
ahead of them. We consider protocols for the pointer jumping problem where all
players are collapsing. Our second result shows that a strong
lower bound does hold in this case. Our third result is another upper bound
showing that nontrivial protocols for (a non-Boolean version of) pointer
jumping are possible even when all players are collapsing. Our lower bound
result uses a novel proof technique, different from those of earlier lower
bounds that had an information-theoretic flavor. We hope this is useful in
further study of the problem
Hardness of Exact Distance Queries in Sparse Graphs Through Hub Labeling
A distance labeling scheme is an assignment of bit-labels to the vertices of
an undirected, unweighted graph such that the distance between any pair of
vertices can be decoded solely from their labels. An important class of
distance labeling schemes is that of hub labelings, where a node
stores its distance to the so-called hubs , chosen so that for
any there is belonging to some shortest
path. Notice that for most existing graph classes, the best distance labelling
constructions existing use at some point a hub labeling scheme at least as a
key building block. Our interest lies in hub labelings of sparse graphs, i.e.,
those with , for which we show a lowerbound of
for the average size of the hubsets.
Additionally, we show a hub-labeling construction for sparse graphs of average
size for some , where is the
so-called Ruzsa-Szemer{\'e}di function, linked to structure of induced
matchings in dense graphs. This implies that further improving the lower bound
on hub labeling size to would require a
breakthrough in the study of lower bounds on , which have resisted
substantial improvement in the last 70 years. For general distance labeling of
sparse graphs, we show a lowerbound of , where is the communication complexity of the
Sum-Index problem over . Our results suggest that the best achievable
hub-label size and distance-label size in sparse graphs may be
for some
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