The Cϩϩ 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 Cϩϩ. The resulting derivative evaluation routines may be called from C/Cϩϩ, 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.

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 o(k) poly(m), where n is the circuit input length, m is the circuit size, and k is the parameter, unless the (t − 1)-st level W [t − 1] of the Whierarchy 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 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 [1] 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 ≥ 2), clique, and independent set, cannot be solved in time n 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 k poly(m) or O(n k).

Abstract. This paper examines a parameterized problem that we refer to as n − k Graph 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(knThe 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−k Clique Covering. The near linear-time kernelization algorithm that we present for n − k Clique 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.

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Measure-theoretic aspects of the <:-reducibility structure of the exponential time complexity classes E=DTIME(21inea') and E2 = DTIME(2Po'ynomia') are investigated. Particular attention is given to the complexity (measured by the size of complexity cores) and distribution (abundance in the sense of measure) of languages that are <:-hard for E and other complexity classes.Tight upper and lower bounds on the size of complexity cores of hard languages are derived. The upper bounds say that the <:-hard languages for E are unusuaIly simple, i n the sense that they have smaller complexity cores than most languages in E. It follows that the <:-complete languages for E form a measure 0 subset of E (and similarly in Ez). This latter fact is seen to be a special case of a more general theorem, namely, that every <:-degree (e.g., the degree of all <:-complete languages for N P ) has measure 0 in E and in E2.