499 research outputs found
Coefficients of Sylvester's Denumerant
For a given sequence of positive integers, we consider
the combinatorial function that counts the nonnegative
integer solutions of the equation , where the right-hand side is a varying
nonnegative integer. It is well-known that is a
quasi-polynomial function in the variable of degree . In combinatorial
number theory this function is known as Sylvester's denumerant.
Our main result is a new algorithm that, for every fixed number , computes
in polynomial time the highest coefficients of the quasi-polynomial
as step polynomials of (a simpler and more explicit
representation). Our algorithm is a consequence of a nice poset structure on
the poles of the associated rational generating function for
and the geometric reinterpretation of some rational
generating functions in terms of lattice points in polyhedral cones. Our
algorithm also uses Barvinok's fundamental fast decomposition of a polyhedral
cone into unimodular cones. This paper also presents a simple algorithm to
predict the first non-constant coefficient and concludes with a report of
several computational experiments using an implementation of our algorithm in
LattE integrale. We compare it with various Maple programs for partial or full
computation of the denumerant.Comment: minor revision, 28 page
Robust and MaxMin Optimization under Matroid and Knapsack Uncertainty Sets
Consider the following problem: given a set system (U,I) and an edge-weighted
graph G = (U, E) on the same universe U, find the set A in I such that the
Steiner tree cost with terminals A is as large as possible: "which set in I is
the most difficult to connect up?" This is an example of a max-min problem:
find the set A in I such that the value of some minimization (covering) problem
is as large as possible.
In this paper, we show that for certain covering problems which admit good
deterministic online algorithms, we can give good algorithms for max-min
optimization when the set system I is given by a p-system or q-knapsacks or
both. This result is similar to results for constrained maximization of
submodular functions. Although many natural covering problems are not even
approximately submodular, we show that one can use properties of the online
algorithm as a surrogate for submodularity.
Moreover, we give stronger connections between max-min optimization and
two-stage robust optimization, and hence give improved algorithms for robust
versions of various covering problems, for cases where the uncertainty sets are
given by p-systems and q-knapsacks.Comment: 17 pages. Preliminary version combining this paper and
http://arxiv.org/abs/0912.1045 appeared in ICALP 201
Space--Time Tradeoffs for Subset Sum: An Improved Worst Case Algorithm
The technique of Schroeppel and Shamir (SICOMP, 1981) has long been the most
efficient way to trade space against time for the SUBSET SUM problem. In the
random-instance setting, however, improved tradeoffs exist. In particular, the
recently discovered dissection method of Dinur et al. (CRYPTO 2012) yields a
significantly improved space--time tradeoff curve for instances with strong
randomness properties. Our main result is that these strong randomness
assumptions can be removed, obtaining the same space--time tradeoffs in the
worst case. We also show that for small space usage the dissection algorithm
can be almost fully parallelized. Our strategy for dealing with arbitrary
instances is to instead inject the randomness into the dissection process
itself by working over a carefully selected but random composite modulus, and
to introduce explicit space--time controls into the algorithm by means of a
"bailout mechanism"
Equal-Subset-Sum Faster Than the Meet-in-the-Middle
In the Equal-Subset-Sum problem, we are given a set S of n integers and the problem is to decide if there exist two disjoint nonempty subsets A,B subseteq S, whose elements sum up to the same value. The problem is NP-complete. The state-of-the-art algorithm runs in O^*(3^(n/2)) <= O^*(1.7321^n) time and is based on the meet-in-the-middle technique. In this paper, we improve upon this algorithm and give O^*(1.7088^n) worst case Monte Carlo algorithm. This answers a question suggested by Woeginger in his inspirational survey.
Additionally, we analyse the polynomial space algorithm for Equal-Subset-Sum. A naive polynomial space algorithm for Equal-Subset-Sum runs in O^*(3^n) time. With read-only access to the exponentially many random bits, we show a randomized algorithm running in O^*(2.6817^n) time and polynomial space
Generic Decoding of Restricted Errors
Several recently proposed code-based cryptosystems base their security on a
slightly generalized version of the classical (syndrome) decoding problem.
Namely, in the so-called restricted (syndrome) decoding problem, the error
values stem from a restricted set. In this paper, we propose new generic
decoders, that are inspired by subset sum solvers and tailored to the new
setting. The introduced algorithms take the restricted structure of the error
set into account in order to utilize the representation technique efficiently.
This leads to a considerable decrease in the security levels of recently
published code-based cryptosystems
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