181,538 research outputs found
The Search for the Maximum of a Polynomial
AbstractFor a real polynomialf(X) ofKvariables the problem of finding maxXâRKf(X) is investigated by reducing it to that of searching for the real roots of the univariate polynomial F(z):=âj(zâf(Îj)), where the product is extended over all the critical points Îjoff(X). Employment of the Hermite method of separation of real solutions of an algebraic equation system permits one to construct along withF(z) its Sturm series, and to restore the coordinates of the corresponding critical point. The problem of finding the maxfin the set defined by the real polynomial inequalityG(X)â„0 is also discussed
Improved approximation for 3-dimensional matching via bounded pathwidth local search
One of the most natural optimization problems is the k-Set Packing problem,
where given a family of sets of size at most k one should select a maximum size
subfamily of pairwise disjoint sets. A special case of 3-Set Packing is the
well known 3-Dimensional Matching problem. Both problems belong to the Karp`s
list of 21 NP-complete problems. The best known polynomial time approximation
ratio for k-Set Packing is (k + eps)/2 and goes back to the work of Hurkens and
Schrijver [SIDMA`89], which gives (1.5 + eps)-approximation for 3-Dimensional
Matching. Those results are obtained by a simple local search algorithm, that
uses constant size swaps.
The main result of the paper is a new approach to local search for k-Set
Packing where only a special type of swaps is considered, which we call swaps
of bounded pathwidth. We show that for a fixed value of k one can search the
space of r-size swaps of constant pathwidth in c^r poly(|F|) time. Moreover we
present an analysis proving that a local search maximum with respect to O(log
|F|)-size swaps of constant pathwidth yields a polynomial time (k + 1 +
eps)/3-approximation algorithm, improving the best known approximation ratio
for k-Set Packing. In particular we improve the approximation ratio for
3-Dimensional Matching from 3/2 + eps to 4/3 + eps.Comment: To appear in proceedings of FOCS 201
Dynamic Ordered Sets with Exponential Search Trees
We introduce exponential search trees as a novel technique for converting
static polynomial space search structures for ordered sets into fully-dynamic
linear space data structures.
This leads to an optimal bound of O(sqrt(log n/loglog n)) for searching and
updating a dynamic set of n integer keys in linear space. Here searching an
integer y means finding the maximum key in the set which is smaller than or
equal to y. This problem is equivalent to the standard text book problem of
maintaining an ordered set (see, e.g., Cormen, Leiserson, Rivest, and Stein:
Introduction to Algorithms, 2nd ed., MIT Press, 2001).
The best previous deterministic linear space bound was O(log n/loglog n) due
Fredman and Willard from STOC 1990. No better deterministic search bound was
known using polynomial space.
We also get the following worst-case linear space trade-offs between the
number n, the word length w, and the maximal key U < 2^w: O(min{loglog n+log
n/log w, (loglog n)(loglog U)/(logloglog U)}). These trade-offs are, however,
not likely to be optimal.
Our results are generalized to finger searching and string searching,
providing optimal results for both in terms of n.Comment: Revision corrects some typoes and state things better for
applications in subsequent paper
Linear Time LexDFS on Cocomparability Graphs
Lexicographic depth first search (LexDFS) is a graph search protocol which
has already proved to be a powerful tool on cocomparability graphs.
Cocomparability graphs have been well studied by investigating their
complements (comparability graphs) and their corresponding posets. Recently
however LexDFS has led to a number of elegant polynomial and near linear time
algorithms on cocomparability graphs when used as a preprocessing step [2, 3,
11]. The nonlinear runtime of some of these results is a consequence of
complexity of this preprocessing step. We present the first linear time
algorithm to compute a LexDFS cocomparability ordering, therefore answering a
problem raised in [2] and helping achieve the first linear time algorithms for
the minimum path cover problem, and thus the Hamilton path problem, the maximum
independent set problem and the minimum clique cover for this graph family
Reduced search space multiple shift maximum element sequential matrix diagonalisation algorithm
The Multiple Shift Maximum Element Sequential Matrix Diagonalisation (MSME-SMD) algorithm is a powerful but costly method for performing approximate polynomial eigenvalue decomposition (PEVD) for space-time covariance-type matrices encountered in e.g. broadband array processing. This paper discusses a newly developed search method that restricts the order growth within the MSME-SMD algorithm. In addition to enhanced control of the polynomial degree of the paraunitary and parahermitian factors in this decomposition, the new search method is also computationally less demanding as fewer elements are searched compared to the original while the excellent diagonalisation of MSME-SMD is maintained
Sphere-constrained ML detection for frequency-selective channels
The maximum-likelihood (ML) sequence detection problem for channels with memory is investigated. The Viterbi algorithm (VA) provides an exact solution. Its computational complexity is linear in the length of the transmitted sequence, but exponential in the channel memory length. On the other hand, the sphere decoding (SD) algorithm also solves the ML detection problem exactly, and has expected complexity which is a low-degree polynomial (often cubic) in the length of the transmitted sequence over a wide range of signal-to-noise ratios. We combine the sphere-constrained search strategy of SD with the dynamic programming principles of the VA. The resulting algorithm has the worst-case complexity determined by the VA, but often significantly lower expected complexity
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