19,085 research outputs found
New Unconditional Hardness Results for Dynamic and Online Problems
There has been a resurgence of interest in lower bounds whose truth rests on
the conjectured hardness of well known computational problems. These
conditional lower bounds have become important and popular due to the painfully
slow progress on proving strong unconditional lower bounds. Nevertheless, the
long term goal is to replace these conditional bounds with unconditional ones.
In this paper we make progress in this direction by studying the cell probe
complexity of two conjectured to be hard problems of particular importance:
matrix-vector multiplication and a version of dynamic set disjointness known as
Patrascu's Multiphase Problem. We give improved unconditional lower bounds for
these problems as well as introducing new proof techniques of independent
interest. These include a technique capable of proving strong threshold lower
bounds of the following form: If we insist on having a very fast query time,
then the update time has to be slow enough to compute a lookup table with the
answer to every possible query. This is the first time a lower bound of this
type has been proven
PPP-Completeness with Connections to Cryptography
Polynomial Pigeonhole Principle (PPP) is an important subclass of TFNP with
profound connections to the complexity of the fundamental cryptographic
primitives: collision-resistant hash functions and one-way permutations. In
contrast to most of the other subclasses of TFNP, no complete problem is known
for PPP. Our work identifies the first PPP-complete problem without any circuit
or Turing Machine given explicitly in the input, and thus we answer a
longstanding open question from [Papadimitriou1994]. Specifically, we show that
constrained-SIS (cSIS), a generalized version of the well-known Short Integer
Solution problem (SIS) from lattice-based cryptography, is PPP-complete.
In order to give intuition behind our reduction for constrained-SIS, we
identify another PPP-complete problem with a circuit in the input but closely
related to lattice problems. We call this problem BLICHFELDT and it is the
computational problem associated with Blichfeldt's fundamental theorem in the
theory of lattices.
Building on the inherent connection of PPP with collision-resistant hash
functions, we use our completeness result to construct the first natural hash
function family that captures the hardness of all collision-resistant hash
functions in a worst-case sense, i.e. it is natural and universal in the
worst-case. The close resemblance of our hash function family with SIS, leads
us to the first candidate collision-resistant hash function that is both
natural and universal in an average-case sense.
Finally, our results enrich our understanding of the connections between PPP,
lattice problems and other concrete cryptographic assumptions, such as the
discrete logarithm problem over general groups
Faster all-pairs shortest paths via circuit complexity
We present a new randomized method for computing the min-plus product
(a.k.a., tropical product) of two matrices, yielding a faster
algorithm for solving the all-pairs shortest path problem (APSP) in dense
-node directed graphs with arbitrary edge weights. On the real RAM, where
additions and comparisons of reals are unit cost (but all other operations have
typical logarithmic cost), the algorithm runs in time
and is correct with high probability.
On the word RAM, the algorithm runs in time for edge weights in . Prior algorithms used either time for
various , or time for various
and .
The new algorithm applies a tool from circuit complexity, namely the
Razborov-Smolensky polynomials for approximately representing
circuits, to efficiently reduce a matrix product over the algebra to
a relatively small number of rectangular matrix products over ,
each of which are computable using a particularly efficient method due to
Coppersmith. We also give a deterministic version of the algorithm running in
time for some , which utilizes the
Yao-Beigel-Tarui translation of circuits into "nice" depth-two
circuits.Comment: 24 pages. Updated version now has slightly faster running time. To
appear in ACM Symposium on Theory of Computing (STOC), 201
Evolutionary Approaches to Minimizing Network Coding Resources
We wish to minimize the resources used for network coding while achieving the
desired throughput in a multicast scenario. We employ evolutionary approaches,
based on a genetic algorithm, that avoid the computational complexity that
makes the problem NP-hard. Our experiments show great improvements over the
sub-optimal solutions of prior methods. Our new algorithms improve over our
previously proposed algorithm in three ways. First, whereas the previous
algorithm can be applied only to acyclic networks, our new method works also
with networks with cycles. Second, we enrich the set of components used in the
genetic algorithm, which improves the performance. Third, we develop a novel
distributed framework. Combining distributed random network coding with our
distributed optimization yields a network coding protocol where the resources
used for coding are optimized in the setup phase by running our evolutionary
algorithm at each node of the network. We demonstrate the effectiveness of our
approach by carrying out simulations on a number of different sets of network
topologies.Comment: 9 pages, 6 figures, accepted to the 26th Annual IEEE Conference on
Computer Communications (INFOCOM 2007
Solving Bit-Vector Equations of Fixed and Non-Fixed Size
This report is concerned with solving equations on fixed andnon-fixed size bit-vector terms. We define an equational transformationsystem for solving equations on terms where all sizes ofbit-vectors and extraction positions are known. This transformationsystem suggests a generalization for dealing with bit-vectorsof unknown size and unknown extraction positions. Both solversadhere to the principle of splitting bit-vectors only on demand,thereby making them quite effective in practice
- …