David W. Juedes

Algorithm 755: ADOL-C: a package for the automatic differentiation of algorithms written in C/C++

The C++ package ADOL-C described here facilitates the evaluation of first and higher derivatives of vector functions that are defined by computer programs written in C or C++. The resulting derivat...The C++ package ADOL-C described here facilitates the evaluation of first and higher derivatives of vector functions that are defined by computer programs written in C or C++. The resulting derivative evaluation routines may be called from C/C++, Fortran, or any other language that can be linked with C. The numerical values of derivative vectors are obtained free of truncation errors at a small multiple of the run-time and randomly accessed memory of the given function evaluation program. Derivative matrices are obtained by columns or rows. For solution curves defined by ordinary differential equations, special routines are provided that evaluate the Taylor coefficient vectors and their Jacobians with respect to the current state vector. The derivative calculations involve a possibly substantial (but always predictable) amount of data that are accessed strictly sequentially and are therefore automatically paged out to external files.

Tight lower bounds for certain parameterized NP-hard problems

Based on the framework of parameterized complexity theory, we derive tight lower bounds on the computational complexity for a number of well-known NP-hard problems. We start by proving a general result, namely that the parameterized weighted satisfiability problem on depth-t circuits cannot be solved in time n/sup o(k)/poly(m), where n is the circuit input length, m is the circuit size, and k is the parameter, unless the (t - l)-st level W[t

On the existence of subexponential parameterized algorithms

1] of the W-hierarchy collapses to FPT. By refining this technique, we prove that a group of parameterized NP-hard problems, including weighted SAT, dominating set, hitting set, set cover, and feature set, cannot be solved in time n/sup o(k)/poly(m), where n is the size of the universal set from which the k elements are to be selected and m is the instance size, unless the first level W[l] of the W-hierarchy collapses to FPT. We also prove that another group of parameterized problems which includes weighted q-SAT (for any fixed q /spl ges/ 2), clique, and independent set, cannot be solved in time n/sup o(k)/ unless all search problems in the syntactic class SNP, introduced by Papadimitriou and Yannakakis, are solvable in subexponential time. Note that all these parameterized problems have trivial algorithms of running time either n/sup k/ poly(m) or O(n/sup k/).

Linear kernels in linear time, or how to save k colors in O ( n 2 ) steps

The existence of subexponential-time parameterized algorithms is examined for various parameterized problems solvable in time O(2o(k)p(n)). It is shown that for each t ≥ 1, there are parameterized problems in FPT for which the existence of O(2o(k)p(n))-time parameterized algorithms implies the collapse of W[t] to FPT. Evidence is demonstrated that Max-SNP-hard optimization problems do not admit subexponential-time parameterized algorithms. In particular, it is shown that each Max-SNP-complete problem is solvable in time O(2o(k)p(n)) if and only if 3-SAT ∈ DTIME(2o(n)). These results are also applied to show evidence for the non-existence of O(2o(√k)p(n))-time parameterized algorithms for a number of other important problems such as Dominating Set, Vertex Cover, and Independent Set on planar graph instances.

The Complexity and Distribution of Hard Problems

This paper examines a parameterized problem that we refer to as n–kGraph Coloring, i.e., the problem of determining whether a graph G with n vertices can be colored using n–k colors. As the main result of this paper, we show that there exists a O(kn2 +k2 + 23.8161k)=O(n2) algorithm for n–kGraph Coloring for each fixed k. The core technique behind this new parameterized algorithm is kernalization via maximum (and certain maximal) matchings. The core technical content of this paper is a near linear-time kernelization algorithm for n–kClique Covering. The near linear-time kernelization algorithm that we present for n–kClique Covering produces a linear size (3k–3) kernel in O(k(n+m)) steps on graphs with n vertices and m edges. The algorithm takes an instance 〈G,k 〉 of Clique Covering that asks whether a graph G can be covered using |V|–k cliques and reduces it to the problem of determining whether a graph G′=(V′,E′) of size ≤ 3k–3 can be covered using |V′| – k′ cliques. We also present a similar near linear-time algorithm that produces a 3k kernel for Vertex Cover. This second kernelization algorithm is the crown reduction rule.

Weak completeness in E and E 2

Measure-theoretic aspects of the

Evolutionary computation techniques for multiple sequence alignment

\leq^{\rm P}_{\rm m}

Computational depth and reducibility

-reducibility structure of the exponential time complexity classes E=DTIME(

Subexponential Parameterized Algorithms Collapse the W-Hierarchy

2^{\rm linear}

The complexity and distribution of hard problems

) and