Recently, several algorithms have been suggested for solving the discrete logarithm problem in the Jacobians of high-genus hyperelliptic curves over finite fields. Some of them have a provable subexponential running t...
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Recently, several algorithms have been suggested for solving the discrete logarithm problem in the Jacobians of high-genus hyperelliptic curves over finite fields. Some of them have a provable subexponential running time and are using the fact that smooth reduced ideals are sufficiently dense. We explicitly show how these density results can be derived. All proofs are purely combinatorial and do not exploit analytic properties of generating functions.
The discrete logarithm problem in various finite abelian groups is the basis for some well known public key cryptosystems. Recently, real quadratic congruence function fields were used to construct a public key distri...
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The discrete logarithm problem in various finite abelian groups is the basis for some well known public key cryptosystems. Recently, real quadratic congruence function fields were used to construct a public key distribution system. The security of this public key system is based on the difficulty of a discrete logarithm problem in these fields. In this paper, we present a probabilistic algorithm with subexponential running time that computes such discrete logarithms in real quadratic congruence function fields of sufficiently large genus. This algorithm is a generalization of similar algorithms for real quadratic number fields.
We present subexponential parameterized algorithms on planar graphs for a family of problems of the following shape: given a graph, find a connected (induced) subgraph with bounded maximum degree and with maximum numb...
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We present subexponential parameterized algorithms on planar graphs for a family of problems of the following shape: given a graph, find a connected (induced) subgraph with bounded maximum degree and with maximum number of edges (or vertices). These problems are natural generalisations of the Longest Path problem. Our approach uses bidimensionality theory combined with novel dynamic programming techniques over branch decompositions of the input graph. These techniques can be applied to a more general family of problems that deal with finding connected subgraphs under certain degree constraints. (C) 2010 Elsevier B.V. All rights reserved.
We present subexponential parameterized algorithms on planar graphs for a family of problems that consist in, given a graph G, finding a connected (induced) subgraph H with bounded maximum degree, while maximising the...
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The 3-coloring problem is well known to be NP-complete. It is also well known that it remains NP-complete when the input is restricted to graphs with diameter 4. Moreover, assuming the Exponential Time Hypothesis (ETH...
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ISBN:
(纸本)9783642358432
The 3-coloring problem is well known to be NP-complete. It is also well known that it remains NP-complete when the input is restricted to graphs with diameter 4. Moreover, assuming the Exponential Time Hypothesis (ETH), 3-coloring can not be solved in time 2(o)((n)) on graphs with n vertices and diameter at most 4. In spite of the extensive studies of the 3-coloring problem with respect to several basic parameters, the complexity status of this problem on graphs with small diameter, i.e. with diameter at most 2, or at most 3, has been a longstanding and challenging open question. In this paper we investigate graphs with small diameter. For graphs with diameter at most 2, we provide the first subexponential algorithm for 3-coloring, with complexity 2(O)(root(n log n)). Furthermore we present a subclass of graphs with diameter 2 that admits a polynomial algorithm for 3-coloring. For graphs with diameter at most 3, we establish the complexity of 3-coloring, even for the case of triangle-free graphs. Namely we prove that for every epsilon is an element of vertical bar 0, 1), 3-coloring is NP-complete on triangle-free graphs of diameter 3 and radius 2 with n vertices and minimum degree delta - Theta(n(epsilon)). Moreover, assuming ETH, we use three different amplification techniques of our hardness results, in order to obtain for every epsilon is an element of[0, 1) subexponential asymptotic lower bounds for the complexity of 3-coloring on triangle-free graphs with diameter 3 and minimum degree delta = Theta(n(epsilon)). Finally, we provide a 3-coloring algorithm with running time 2(O(min{delta Delta, n/delta log delta})) for arbitrary graphs with diameter 3, where n is the number of vertices and delta (resp. Delta) is the minimum (resp. maximum) degree of the input graph. To the best of our knowledge, this algorithm is the first subexponential algorithm for graphs with delta = w(1) and for graphs with delta = O(1) and Delta = o(n). Due to the above lower bounds of the c
The Tutte polynomial is a notoriously hard graph invariant, and efficient algorithms for it are known only for a few special graph classes, like for those of bounded tree-width. The notion of clique-width extends the ...
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ISBN:
(数字)9783540314684
ISBN:
(纸本)3540310002
The Tutte polynomial is a notoriously hard graph invariant, and efficient algorithms for it are known only for a few special graph classes, like for those of bounded tree-width. The notion of clique-width extends the definition of cograhs (graphs without induced P-4), and it is a more general notion than that of tree-width. We show a subexponential algorithm (running in time exp O(n(2/3))) for computing the Tutte polynomial on cographs. The algorithm can be extended to a subexponential algorithm computing the Tutte polynomial on on all graphs of bounded clique-width. In fact, our algorithm computes the more general U-polynomial.
Bidimensionality theory was introduced by Demaine et al. [JACM 2005] as a framework to obtain algorithmic results for hard problems on minor closed graph classes. The theory has been successfully applied to yield sube...
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ISBN:
(纸本)9781611972108
Bidimensionality theory was introduced by Demaine et al. [JACM 2005] as a framework to obtain algorithmic results for hard problems on minor closed graph classes. The theory has been successfully applied to yield subexponential time parameterized algorithms, EPTASs and linear kernels for many problems on families of graphs excluding a fixed graph H as a minor. In this paper we use several of the key ideas from Bidimensionality to give a new generic approach to design EPTASs and subexponential time parameterized algorithms for problems on classes of graphs which are not minor closed, but instead exhibit a geometric structure. In particular we present EPTASs and subexponential time parameterized algorithms for Feedback Vertex Set, Vertex Cover, Connected Vertex Cover, on map graphs and unit disk graphs, PTASs for Diamond Hitting Set on map graphs and unit disk graphs, and a PTAS and a subexponential time algorithm for Cycle Packing on unit disk graphs. To the best of our knowledge, these results were previously unknown, with the exception of the EPTAS and a subexponential time parameterized algorithm on unit disk graphs for Vertex Cover, which were obtained by Marx [ESA 2005] and Alber and Fiala [J. algorithms 2004], respectively. Our results are based on the recent decomposition theorems proved by Fomin et al. in [SODA 2011] and novel grid-excluding theorems in unit disk and map graphs without large cliques. Our algorithms work directly on the input graph and do not require the geometric representations of the input graph. We also show that our approach can not be extended in its full generality to more general classes of geometric graphs, such as intersection graphs of unit balls in R~d, d ≥ 3. Specifically, we prove that Feedback Vertex Set on unit- ball graphs in R3 neither admits PTASs unless P=NP, nor subexponential time algorithms unless the Exponential Time Hypothesis fails. Additionally, we show that the decomposition theorems which our approach is based on,
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