22,129 research outputs found
A Polynomial Time Algorithm for Lossy Population Recovery
We give a polynomial time algorithm for the lossy population recovery
problem. In this problem, the goal is to approximately learn an unknown
distribution on binary strings of length from lossy samples: for some
parameter each coordinate of the sample is preserved with probability
and otherwise is replaced by a `?'. The running time and number of
samples needed for our algorithm is polynomial in and for
each fixed . This improves on algorithm of Wigderson and Yehudayoff that
runs in quasi-polynomial time for any and the polynomial time
algorithm of Dvir et al which was shown to work for by
Batman et al. In fact, our algorithm also works in the more general framework
of Batman et al. in which there is no a priori bound on the size of the support
of the distribution. The algorithm we analyze is implicit in previous work; our
main contribution is to analyze the algorithm by showing (via linear
programming duality and connections to complex analysis) that a certain matrix
associated with the problem has a robust local inverse even though its
condition number is exponentially small. A corollary of our result is the first
polynomial time algorithm for learning DNFs in the restriction access model of
Dvir et al
A Polynomial-time Algorithm for Outerplanar Diameter Improvement
The Outerplanar Diameter Improvement problem asks, given a graph and an
integer , whether it is possible to add edges to in a way that the
resulting graph is outerplanar and has diameter at most . We provide a
dynamic programming algorithm that solves this problem in polynomial time.
Outerplanar Diameter Improvement demonstrates several structural analogues to
the celebrated and challenging Planar Diameter Improvement problem, where the
resulting graph should, instead, be planar. The complexity status of this
latter problem is open.Comment: 24 page
An Exact Quantum Polynomial-Time Algorithm for Simon's Problem
We investigate the power of quantum computers when they are required to
return an answer that is guaranteed to be correct after a time that is
upper-bounded by a polynomial in the worst case. We show that a natural
generalization of Simon's problem can be solved in this way, whereas previous
algorithms required quantum polynomial time in the expected sense only, without
upper bounds on the worst-case running time. This is achieved by generalizing
both Simon's and Grover's algorithms and combining them in a novel way. It
follows that there is a decision problem that can be solved in exact quantum
polynomial time, which would require expected exponential time on any classical
bounded-error probabilistic computer if the data is supplied as a black box.Comment: 12 pages, LaTeX2e, no figures. To appear in Proceedings of the Fifth
Israeli Symposium on Theory of Computing and Systems (ISTCS'97
A Polynomial Time Algorithm for Spatio-Temporal Security Games
An ever-important issue is protecting infrastructure and other valuable
targets from a range of threats from vandalism to theft to piracy to terrorism.
The "defender" can rarely afford the needed resources for a 100% protection.
Thus, the key question is, how to provide the best protection using the limited
available resources. We study a practically important class of security games
that is played out in space and time, with targets and "patrols" moving on a
real line. A central open question here is whether the Nash equilibrium (i.e.,
the minimax strategy of the defender) can be computed in polynomial time. We
resolve this question in the affirmative. Our algorithm runs in time polynomial
in the input size, and only polylogarithmic in the number of possible patrol
locations (M). Further, we provide a continuous extension in which patrol
locations can take arbitrary real values. Prior work obtained polynomial-time
algorithms only under a substantial assumption, e.g., a constant number of
rounds. Further, all these algorithms have running times polynomial in M, which
can be very large
- …