Dichotomy Results for Fixed Point Counting in Boolean Dynamical Systems

We present dichotomy theorems regarding the computational complexity of counting fixed points in boolean (discrete) dynamical systems, i.e., finite discrete dynamical systems over the domain {0,1}. For a class F of boolean functions and a class G of graphs, an (F,G)-system is a boolean dynamical system with local transitions functions lying in F and graphs in G. We show that, if local transition functions are given by lookup tables, then the following complexity classification holds: Let F be a class of boolean functions closed under superposition and let G be a graph class closed under taking minors. If F contains all min-functions, all max-functions, or all self-dual and monotone functions, and G contains all planar graphs, then it is #P-complete to compute the number of fixed points in an (F,G)-system; otherwise it is computable in polynomial time. We also prove a dichotomy theorem for the case that local transition functions are given by formulas (over logical bases). This theorem has a significantly more complicated structure than the theorem for lookup tables. A corresponding theorem for boolean circuits coincides with the theorem for formulas.Comment: 16 pages, extended abstract presented at 10th Italian Conference on Theoretical Computer Science (ICTCS'2007

Tight lower bounds on the ambiguity of strong, total, associative, one-way functions

AbstractWe study the ambiguity, or “many-to-one”-ness, of two-argument, one-way functions that are strong (that is, hard to invert even if one of their arguments is given), total, and associative. Such powerful one-way functions are the basis of a cryptographic paradigm described by Rabi and Sherman (Inform. Process. Lett. 64(2) (1997) 239) and were shown by Hemaspaandra and Rothe (J. Comput. System Sci. 58(3) (1999) 648) to exist exactly if standard one-way functions exist.Rabi and Sherman (1997) show that no total, associative function defined over a universe having at least two elements is one-to-one. We show that if P≠UP, then, for every d∈N+, there is an O(log1dn)-to-one, strong, total, associative, one-way function σd. We argue that this bound is tight in the sense that any total, associative function having similar properties to σd but not necessarily strong or one-way must have at least the same order of magnitude of ambiguity as σd has. We demonstrate that the techniques used in proving the above-stated results easily apply to other classes of total, associative functions.We provide a complete characterization for the existence of strong, total, associative, one-way functions whose ambiguity approaches the lower bounds we provide. We say a language is in PolylogP if there exists a polynomial-time Turing machine M accepting the language such that for some d∈R+ it holds that M has on each string x at most O(logdn) accepting paths, where n=|x|. We show that P≠PolylogP if and only for some d∈R+ there exists an O(logdn)-to-one, strong, total, associative, one-way function

Cluster Computing and the Power of Edge Recognition

We study the robustness--the invariance under definition changes--of the cluster class CL#P [HHKW05]. This class contains each #P function that is computed by a balanced Turing machine whose accepting paths always form a cluster with respect to some length-respecting total order with efficient adjacency checks. The definition of CL#P is heavily influenced by the defining paper's focus on (global) orders. In contrast, we define a cluster class, CLU#P, to capture what seems to us a more natural model of cluster computing. We prove that the naturalness is costless: CL#P = CLU#P. Then we exploit the more natural, flexible features of CLU#P to prove new robustness results for CL#P and to expand what is known about the closure properties of CL#P. The complexity of recognizing edges--of an ordered collection of computation paths or of a cluster of accepting computation paths--is central to this study. Most particularly, our proofs exploit the power of unique discovery of edges--the ability of nondeterministic functions to, in certain settings, discover on exactly one (in some cases, on at most one) computation path a critical piece of information regarding edges of orderings or clusters

Tuning the Diversity of Open-Ended Responses from the Crowd

Crowdsourcing can solve problems that current fully automated systems cannot. Its effectiveness depends on the reliability, accuracy, and speed of the crowd workers that drive it. These objectives are frequently at odds with one another. For instance, how much time should workers be given to discover and propose new solutions versus deliberate over those currently proposed? How do we determine if discovering a new answer is appropriate at all? And how do we manage workers who lack the expertise or attention needed to provide useful input to a given task? We present a mechanism that uses distinct payoffs for three possible worker actions---propose,vote, or abstain---to provide workers with the necessary incentives to guarantee an effective (or even optimal) balance between searching for new answers, assessing those currently available, and, when they have insufficient expertise or insight for the task at hand, abstaining. We provide a novel game theoretic analysis for this mechanism and test it experimentally on an image---labeling problem and show that it allows a system to reliably control the balance betweendiscovering new answers and converging to existing ones

Twitter Job/Employment Corpus: A Dataset of Job-Related Discourse Built with Humans in the Loop

We present the Twitter Job/Employment Corpus, a collection of tweets annotated by a humans-in-the-loop supervised learning framework that integrates crowdsourcing contributions and expertise on the local community and employment environment. Previous computational studies of job-related phenomena have used corpora collected from workplace social media that are hosted internally by the employers, and so lacks independence from latent job-related coercion and the broader context that an open domain, general-purpose medium such as Twitter provides. Our new corpus promises to be a benchmark for the extraction of job-related topics and advanced analysis and modeling, and can potentially benefit a wide range of research communities in the future

Plane Decompositions as Tools for Approximation

Tree decompositions were developed by Robertson and Seymour [21]. Since then algorithms have been developed to solve intractable problems efficiently for graphs of bounded tree width. In this paper we extend tree decompositions to allow cycles to exist in the decomposition graph; we call these new decompositions plane decompositions because we require that the decomposition graph be planar. First, we give some background material about tree decompositions and an overview of algorithms both for decompositions and for approximations of planar graphs. Then, we give our plane decomposition definition and an algorithm that uses this decomposition to approximate the size of the maximum independent set of the underlying graph in polynomial time