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### Some provably hard crossing number problems

Daniel Bienstock
1991 Discrete & Computational Geometry
We also study the problem of drawing a graph with polygonal edges, to achieve the (unrestricted) minimum number of crossings.  ...  This paper presents a connection between the problem of drawing a graph with the minimum number of edge crossings, and the theory of arrangements of pseudolines, a topic well-studied by combinatorialists  ...  Some Provably Hard Crossing Number Problems 445 Theorem 1 . 1 Let A be an arrangement of n pseudolines.  ...

### Optimal Register Allocation in Polynomial Time [chapter]

Philipp Klaus Krause
2013 Lecture Notes in Computer Science
a bound on the number of goto labels per function.  ...  An implementation is already the default register allocator in most backends of a mainstream cross-compiler for embedded systems.  ...  However, even a substantially simplified version of the register allocation problem is W[SAT]-and co-W[SAT]-hard when parametrized by the number of registers even for tw(G) = 2 [26] .  ...

### Page 4744 of Mathematical Reviews Vol. , Issue 92i [page]

1992 Mathematical Reviews
Martin Skoviera (CS-KMSK-TC) 92i:05081 05C10 05C85 Bienstock, Daniel (1-CLMB-I) Some provably hard crossing number problems. Discrete Comput. Geom. 6 (1991), no. 5, 443-459.  ...  graph the rectilinear crossing number equals the crossing number is also NP-hard.  ...

### A Mathematical Programming Approach to Marker-Assisted Gene Pyramiding [chapter]

Stefan Canzar, Mohammed El-Kebir
2011 Lecture Notes in Computer Science
The task is to schedule crossings of individuals such that the number of generations, the number of crossings, and the required populations size are minimized.  ...  We present for the first time a mathematical model for the general problem variant and show that the problem is N P-hard and even hard to approximate.  ...  The NP-hardness proof involves only the number of crossings; as for the number of generations, the same reduction can be applied. The hardness with respect to the population size remains open.  ...

### An Efficient Lattice-Based Signature Scheme with Provably Secure Instantiation [chapter]

Sedat Akleylek, Nina Bindel, Johannes Buchmann, Juliane Krämer, Giorgia Azzurra Marson
2016 Lecture Notes in Computer Science
To this end, we provide a tight security reduction for the new scheme from the ring learning with errors problem which allows for provably secure and efficient instantiations.  ...  Both come with a security reduction from a lattice problem and have high performance.  ...  computationally hard problem.  ...

### BBK* (Branch and Bound Over K*): A Provable and Efficient Ensemble-Based Protein Design Algorithm to Optimize Stability and Binding Affinity Over Large Sequence Spaces

Adegoke A. Ojewole, Jonathan D. Jou, Vance G. Fowler, Bruce R. Donald
2018 Journal of Computational Biology
Ka for a combinatorial number of sequences, and avoids SS computation for all provably suboptimal sequences.  ...  Thus, to our knowledge, BBK* is the first provable, ensemble-based CPD algorithm to run in time sublinear in the number of sequences.  ...  Mark Hallen and Pablo Gainza for helpful discussions and for providing useful protein-ligand binding problems; Dr. Jeffrey Martin for assisting with software optimizations; Jack Holland, Dr.  ...

### Efficient and provably secure key agreement [chapter]

N. Alexandris, M. Burmester, V. Chrissikopoulos, D. Peppes
1996 IFIP Advances in Information and Communication Technology
We propose a variant of the Diffie-Hellman key agreement protocol which is provably secure and efficient.  ...  In particular, we consider two models for provable security, one based on probabilistic encryption, the other on zero-knowledge.  ...  J. respectively ). * It is not clear if this problem is as hard as the Discrete Logarithm problem, but it is generally regarded as a hard problem. ** In can be shown that if the MTI protocol was zero-knowledge  ...

### Crossing numbers of graphs, lower bound techniques and algorithms: A survey [chapter]

Farhad Shahrokhi, László A. Székely, Imrich Vrt'o
1995 Lecture Notes in Computer Science
We give a survey of recent techniques for deriving lower bounds and algorithms for constructing upper bounds for several variations of the crossing number problem.  ...  In particular, the only existing algorithm for computing provably near-optimal solutions to the planar crossing number problem has been provided by the VLSI community [7] .  ...  There have not been too many results dealing with the crossing number problem on a surface, perhaps due to the difficulty of the problem which had made the existing tools inadequate.  ...

### A branch-and-cut approach to the crossing number problem

Christoph Buchheim, Markus Chimani, Dietmar Ebner, Carsten Gutwenger, Michael Jünger, Gunnar W. Klau, Petra Mutzel, René Weiskircher
2008 Discrete Optimization
Finding the crossing number is NP-hard for general graphs and no practical algorithm for its computation has been published so far.  ...  We present an integer linear programming formulation that is based on a reduction of the general problem to a restricted version of the crossing number problem in which each edge may be crossed at most  ...  They show that The general crossing minimization problem is NP-hard [18] .  ...

### Finding the capacity of a quantized binary-input DMC

Brian M. Kurkoski, Hideki Yagi
2012 2012 IEEE International Symposium on Information Theory Proceedings
problems!  ...   VLSI implementation: numbers are converted to bits -quantization  Power consumption, cost, etc. of receivers increases with the number of bits  Broad goal: Reduce the number of bits without sacrificing  ...  Conclusion Channel quantization is important for reducing complexity of receivers  Maximization of mutual information is a highly suitable metric  These concave optimization problems are NP-Hard Easier  ...

### Parameterized Algorithms and Data Reduction for Safe Convoy Routing

René Van Bevern, Till Fluschnik, Oxana Yu. Tsidulko, Michael Wagner
2018 Algorithmic Approaches for Transportation Modeling, Optimization, and Systems
In contrast, we show that the problem in general graphs is hard to preprocess.  ...  We study the problem in two types of transportation networks: graphs with small crossing number, as formed by road networks, and tree-like graphs, as formed by waterways.  ...  In contrast, we observed SSP to be a problem for which provably effective polynomialtime data reduction is rather hard to obtain (Theorems 2.14 and 3.10).  ...

### Counting Objects with a Combination of Horizontal and Overhead Sensors

Erik Halvorson, Ronald Parr
2009 The international journal of robotics research
We describe several results furthering the understanding of this problem: 1) A hardness result showing that computing a tight lower bound is intractable, 2) A greedy algorithm for maximizing the number  ...  Counting the number of distinct objects within a region is a basic problem in the field of surveillance, with a wide array of possible uses.  ...  [15] , it is possible to offer some better bounds for this case: • The Lower Bound remains the number of provably occupied polygons.  ...

### An Experimental Study of Crossing Minimization Heuristics [chapter]

Carsten Gutwenger, Petra Mutzel
2004 Lecture Notes in Computer Science
We present an extensive experimental study of heuristics for crossing minimization.  ...  The heuristics are based on the planarization approach, so far the most successful framework for crossing minimization.  ...  We do know, however, that the crossing number problem and several of its variants are NP-hard [13, 3] .  ...

### The Constrained Crossing Minimization Problem [chapter]

Petra Mutzel, Thomas Ziegler
1999 Lecture Notes in Computer Science
number of crossings between the walks.  ...  This is the first time that practical instances of the constrained crossing minimization problem can be solved to provable optimality.  ...  Unfortunately, the problem of minimizing the number of crossings in a drawing is NP-hard ( [2] ) and so far no practically efficient algorithm exists, even for small nontrivial graphs.  ...

### Another Look at "Provable Security". II [chapter]

Neal Koblitz, Alfred Menezes
2006 Lecture Notes in Computer Science
We give some examples to show the subtlety and difficulty of this question.  ...  Acknowledgments We would like to thank Andrey Sidorenko for his valuable comments on pseudorandom bit generators and Bart Preneel for answering our queries about the provable security of MAC algorithms  ...  numbers) that is widely believed to be very hard.  ...
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