15088558
Some Results on Matchgates and Holographic Algorithms
We establish a 1-1 correspondence between Valiant's character theory of matchgate/matchcircuit [14] and his signature theory of planar-matchgate/matchgrid [16], thus unifying the two theories in expressibility. In [3], we had established a complete characterization of general matchgates, in terms of a set of useful Grassmann-Plucker identities. With this correspondence, we give a corresponding set of identities which completely characterizes planar-matchgates and their signatures. Applying this characterization we prove some negative results for holographic algorithms. On the positive side, we also give a polynomial time algorithm for a simultaneous node-edge deletion problem, using holographic algorithms. Finally we give characterizations of symmetric signatures realizable in the Hadamard basis.
43665408
Resolution of Hartmanis' Conjecture for NL-Hard Sparse Sets
We resolve a conjecture of Hartmanis from 1978 about sparse hard sets for nondeterministic logspace (NL). We show that there exists a sparse hard set S for NL under logspace many-one reductions if and only if NL = L (deterministic logspace).
6323016
Valiant's Holant Theorem and matchgate tensors
We propose matchgate tensors as a natural and proper language to develop Valiant’s new theory of Holographic Algorithms. We give a treatment of the central theorem in this theory— the Holant Theorem—in terms of matchgate tensors. Some generalizations are presented.
1980209
Basis Collapse in Holographic Algorithms
Abstract.Holographic algorithms are a novel approach to design polynomial time computations using linear superpositions. Most holographic algorithms are designed with basis vectors of dimension 2. Recently Valiant showed that a basis of dimension 4 can be used to solve in P an interesting (restrictive SAT) counting problem mod 7. This problem without modulo 7 is #P-complete, and counting mod 2 is NP-hard.We give a general collapse theorem for bases of dimension 4 to dimension 2 in the holographic algorithms framework. We also define an extension of holographic algorithms to allow more general support vectors. Finally we give a Basis Folding Theorem showing that in a natural setting the support vectors can be simulated by bases of dimension 2.
11342779
On Proving Circuit Lower Bounds against the Polynomial-Time Hierarchy: Positive and Negative Results
We consider the problem of proving circuit lower bounds against the polynomial-time hierarchy. We give both positive and negative results. For the positive side, for any fixed integer k > 0, we give an explicit Σ2p language, acceptable by a Σ2p-machine with running time O(nk2+k), that requires circuit size > nk. For the negative side, we propose a new stringent notion of relativization, and prove under this stringent relativization that every language in the polynomial-time hierarchy has polynomial circuit size. (For technical details, see also [CW03].)
474133
On Block-Wise Symmetric Signatures for Matchgates
In a tank having an internal float there is a standpipe coupled through the tank. The standpipe conducts liquid into the tank. Equipment for controlling the level of liquid in the tank includes a valve. This valve is operated by the float. A conduit depends from the valve. This conduit is longer than the valve and conducts liquid thereto. Also included is a pliable seal which mounted on the conduit at an end opposite the valve. This seal engages an interior surface of the conduit. The seal also has a bore sized to encircle and seal the standpipe. Thus, the standpipe may telescope into the conduit.
27123977
Classifying G-protein coupled receptors with bagging classification tree
G-protein coupled receptors (GPCRs) play a key role in different biological processes, such as regulation of growth, death and metabolism of cells. They are major therapeutic targets of numerous prescribed drugs. However, the ligand specificity of many receptors is unknown and there is little structural information available. Bioinformatics may offer one approach to bridge the gap between sequence data and functional knowledge of a receptor. In this paper, we use a bagging classification tree algorithm to predict the type of the receptor based on its amino acid composition. The prediction is performed for GPCR at the sub-family and sub-sub-family level. In a cross-validation test, we achieved an overall predictive accuracy of 91.1% for GPCR sub-family classification, and 82.4% for sub-sub-family classification. These results demonstrate the applicability of this relative simple method and its potential for improving prediction accuracy.
38252080
A New Transference Theorem in the Geometry of Numbers
We prove a new transference theorem in the geometry of numbers, giving optimal bounds relating the successive minima of a lattice with the minimal length of generating vectors of its dual. It generalizes the transference theorem due to Banaszczyk. The theorem is motivated by our efforts to improve Ajtai's connection factors in the connection of average-case to worst-case complexity of the shortest lattice vector problem. Our proofs are non-constructive, based on methods from harmonic analysis.
44495329
Enumeration on Nonseparable Planar Maps
Abstract This paper provides some functional equations satisfied by the generating functions for nonseparable rooted planar maps with the valency of root-vertex, the number of edges and the valency of root-faces of the maps as three parameters. But the solutions of these equations can only be obtained indirectly by considering some relations between nonseparable and general rooted planar maps. One of them is an answer to the open problem 6.1 in Liu (1983, Comb. Optim. CORR83-26, University of Waterloo).
5733190
Automating the drug scheduling of cancer chemotherapy via evolutionary computation
This paper presents the optimal control of drug scheduling in cancer chemotherapy using a distributed evolutionary computing software. Unlike conventional methods that often require gradient information or hybridization of different approaches in drug scheduling, the proposed evolutionary optimization methodology is simple and capable of automatically finding the near-optimal solutions for complex cancer chemotherapy problems. It is shown that different number of variable pairs in evolutionary representation for drug scheduling can be easily implemented via the software, since the computational workload is shared and distributed among multiple computers over the Internet. Simulation results show that the proposed evolutionary approach produces excellent control of drug scheduling in cancer chemotherapy, which are competitive or equivalent to the best solutions published in literature.