Download (.ps) Complexity and Information: Measuring Emergence, Self-organization, and Homeostasis at Multiple Scales
Co-authored with Carlos Gershenson. Submitted to Complexity.
Concepts used in the scientific study of complex systems have become so widespread that their use and abuse has led to... more Concepts used in the scientific study of complex systems have become so widespread that their use and abuse has led to ambiguity and confusion in their meaning. In this paper we use information theory to provide abstract and concise measures of complexity, emergence, self-organization, and homeostasis. The purpose is to clarify the meaning of these concepts with the aid of the proposed formal measures. In a simplified version of the measures (focussing on the information produced by a system), emergence becomes the opposite of self-organization, while complexity represents their balance. We use computational experiments on random Boolean networks and elementary cellular automata to illustrate our measures at multiple scales.
Complexity of Social Network Anonymization
by Sean Chester
Ssocial network privacy paper co-authored with Bruce M. Kapron, Gautam Srivastava, and Venkatesh Srinivasan. To appear in an upcoming issue of the Springer journal Social Network Analysis and Mining (SNAM).
With an abundance of social network data being released, the need
to protect sensitive information within these... more
With an abundance of social network data being released, the need
to protect sensitive information within these networks has become an impor-
tant concern of data publishers. To achieve this objective, various notions of
k-anonymization have been proposed for social network graphs. In this paper
we focus on the complexity of optimization problems that arise from trying
to anonymize graphs, establishing that optimally k-anonymizing the label se-
quences of edge-labeled graphs is intractable. We show how this result implies
intractability for other notions of k-anonymization in literature.
We also consider the case of bipartite social network graphs which arise
from the representation of distinct entities, such as movies and viewers, pa-
tients and drugs, or products and customers. Within this setting we demon-
strate that, although k-anonymizing edge-labeled graphs is intractable for
k ≥ 3, polynomial time algorithms exist for arbitrary bipartite graphs when
k = 2 and for unlabeled bipartite graphs irrespective of the value of k.
Finally, in this paper we extend the study of attribute disclosure within
the context of social networks by defining t-closeness, a measure of how effec-
tively an adversary can determine sensitive information about members of a
k-anonymous social network.
Tractable competence
In the study of cognitive processes, limitations on computational resources (computing time and memory space) are... more
In the study of cognitive processes, limitations on computational resources (computing time and memory space) are usually considered to be beyond the scope of a theory of competence, and to be exclusively relevant to the study of performance. Starting from considerations derived from the theory of
computational complexity, in this paper I argue that there are good reasons for claiming that some aspects of resource limitations pertain to the domain of a theory of competence.
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Parameterizing by the Number of Numbers
Michael R. Fellows, Serge Gaspers, and Frances A. Rosamond.
Parameterizing by the Number of Numbers.
Theory of Computing Systems.
The usefulness of parameterized algorithmics has often depended on what Niedermeier has called "the art of... more The usefulness of parameterized algorithmics has often depended on what Niedermeier has called "the art of problem parameterization". In this paper we introduce and explore a novel but general form of parameterization: the number of numbers. Several classic numerical problems, such as Subset Sum, Partition, 3-Partition, Numerical 3-Dimensional Matching, and Numerical Matching with Target Sums, have multisets of integers as input. We initiate the study of parameterizing these problems by the number of distinct integers in the input. We rely on an FPT result for Integer Linear Programming Feasibility to show that all the abovementioned problems are fixed-parameter tractable when parameterized in this way. In various applied settings, problem inputs often consist in part of multisets of integers or multisets of weighted objects (such as edges in a graph, or jobs to be scheduled). Such number-of-numbers parameterized problems often reduce to subproblems about transition systems of various kinds, parameterized by the size of the system description. We consider several core problems of this kind relevant to number-of-numbers parameterization. Our main hardness result considers the problem: given a non-deterministic Mealy machine M (a finite state automaton outputting a letter on each transition), an input word x, and a census requirement c for the output word specifying how many times each letter of the output alphabet should be written, decide whether there exists a computation of M reading x that outputs a word y that meets the requirement c. We show that this problem is hard for W[1]. If the question is whether there exists an input word x such that a computation of M on x outputs a word that meets c, the problem becomes fixed-parameter tractable.
k-Gap Interval Graphs
Fedor V. Fomin, Serge Gaspers, Petr Golovach, Karol Suchan, Stefan Szeider, Erik Jan van Leeuwen, Martin Vatshelle, and Yngve Villanger.
k-Gap Interval Graphs.
Proceedings of the 10th Latin American Theoretical Informatics Symposium (LATIN 2012).
We initiate the study of a new parameterization of graph problems. In a multiple interval representation of a graph,... more We initiate the study of a new parameterization of graph problems. In a multiple interval representation of a graph, each vertex is associated to at least one interval of the real line, with an edge between two vertices if and only if an interval associated to one vertex has a nonempty intersection with an interval associated to the other vertex. A graph on n vertices is a k-gap interval graph if it has a multiple interval representation with at most n+k intervals in total. In order to scale up the nice algorithmic properties of interval graphs (where k=0), we parameterize graph problems by k, and find FPT algorithms for several problems, including Feedback Vertex Set, Dominating Set, Independent Set, Clique, Clique Cover, and Multiple Interval Transversal. The Coloring problem turns out to be W[1]-hard and we design an XP algorithm for the recognition problem.
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Replaceability and computational equivalence in finite distributive lattices
Theory of Computation Report 61, University of Warwick 1984
Notions of replaceability and computational equivalence are defined in an abstract algebraic setting, and investigated... more Notions of replaceability and computational equivalence are defined in an abstract algebraic setting, and investigated in detail for finite distributive lattices. It is shown that, when computing an element f of a finite distributive lattice D, the elements of D partition into classes of computationally equivalent elements, and define a quotient of D in which all intervals of the form [t /\ f, t \/ f] are boolean. This quotient is an abstract simplicial complex with respect to ordering by replaceability. Other results include generalisations and extensions of known theorems concerning replacement rules for monotone boolean networks.
Monotone boolean functions as combinatorially piecewise linear maps
Computer Science Research Report CS-RR-109, University of Warwick 1987 (presented at 4th British TCS Colloquium, Edinburgh 1988)
The class of combinatorially piecewise linear (cpl) maps was first introduced to solve a geometric problem concerning... more The class of combinatorially piecewise linear (cpl) maps was first introduced to solve a geometric problem concerning the representability of piecewise linear functions as pointwise maxima of minima of linear functions. Such maps correspond in a canonical fashion to monotone boolean functions. This paper describes how a monotone boolean function in n variables whose prime implicants and prime clauses are non-trivial defines a partition of the symmetric group on n symbols into a set of "singular cycles" representing relations between transpositions of adjacent symbols. Several possible approaches to the classification of such cycles are described, and some characteristic properties of singular cycles are identified. The potential computational significance of singular cycles is indicated with reference to new combinatorial models for monotone boolean formulae and circuits that arise directly from the appropriate theory of computational equivalence and replaceability. The prospects for application to monotone boolean function complexity are briefly examined, A catalogue of known relations is included as an Appendix.
Replacement in monotone Boolean networks: an algebraic perspective
Lecture Notes in Computer Science 181, Springer-Verlag 1984, 165-178
Replaceability and computational equivalence for monotone Boolean functions
Acta Informatica 22, 1985, 433-449
Computational Complexity of Semi-Stable Semantics In Abstract Argumentation Frameworks
Semi-stable semantics offer a further extension based formalism by which the concept of “collection of justified... more
Semi-stable semantics offer a further extension based formalism by which the concept of “collection of justified arguments” in abstract argumentation frameworks may be described. In contrast to the better known stable semantics, one advantage of semi-stability is that any finite argumentation framework always has at least one semi-stable extension. Although there has been some development of the formal logical theory of semi-stable semantics so that a number of computational properties of these extensions have been identified, with the exception of some algorithmic studies, more detailed investigation of computational complexity issues has been neglected. Our purpose in this article is to present a number of results on the complexity of some natural decision
questions for semi-stable semantics.
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The Parameterized Complexity of Local Consistency
Serge Gaspers and Stefan Szeider.
The Parameterized Complexity of Local Consistency.
Proceedings of the 17th International Conference on Principles and Practice of Constraint Programming (CP 2011),
Springer LNCS 6876, pp. 302-316.
We investigate the parameterized complexity of deciding whether a constraint network is k-consistent. We show that,... more
We investigate the parameterized complexity of deciding whether a constraint network is k-consistent. We show that, parameterized by k, the problem is complete for the complexity class co-W[2]. As secondary parameters we consider the maximum domain size d and the maximum number l of constraints in which a variable occurs. We show that parameterized by k + d, the problem drops down one complexity level and becomes co-W[1]-complete. Parameterized by k + d + l the problem drops down one more level and becomes fixed-parameter tractable. We further show that the same complexity classification applies to strong k-consistency, directional k-consistency, and strong directional k-consistency.
Our results establish a super-polynomial separation between input size and time complexity. Thus we strengthen the known lower bounds on time complexity of k-consistency that are based on input size.
Parameterizing by the Number of Numbers
Michael R. Fellows, Serge Gaspers, and Frances A. Rosamond.
Parameterizing by the number of numbers.
5th International Symposium on Parameterized and Exact Computation (IPEC 2010),
Springer LNCS 6478, 123–134, 2010.
The usefulness of parameterized algorithmics has often depended on what Niedermeier has called "the art of... more The usefulness of parameterized algorithmics has often depended on what Niedermeier has called "the art of problem parameterization". In this paper we introduce and explore a novel but general form of parameterization: the number of numbers. Several classic numerical problems, such as Subset Sum, Partition, 3-Partition, Numerical 3-Dimensional Matching, and Numerical Matching with Target Sums, have multisets of integers as input. We initiate the study of parameterizing these problems by the number of distinct integers in the input. We rely on an FPT result for Integer Linear Programming Feasibility to show that all the above-mentioned problems are fixed-parameter tractable when parameterized in this way. In various applied settings, problem inputs often consist in part of multisets of integers or multisets of weighted objects (such as edges in a graph, or jobs to be scheduled). Such number-of-numbers parameterized problems often reduce to subproblems about transition systems of various kinds, parameterized by the size of the system description. We consider several core problems of this kind relevant to number-of-numbers parameterization. Our main hardness result considers the problem: given a non-deterministic Mealy machine M (a finite state automaton outputting a letter on each transition), an input word x, and a census requirement c for the output word specifying how many times each letter of the output alphabet should be written, decide whether there exists a computation of M reading x that outputs a word y that meets the requirement c. We show that this problem is hard for W[1]. If the question is whether there exists an input word x such that a computation of M on x outputs a word that meets c, the problem becomes fixed-parameter tractable.
Complexity of splits reconstruction for low-degree trees
Serge Gaspers, Mathieu Liedloff, Maya J. Stein, and Karol Suchan. Complexity of Splits Reconstruction for low-degree trees.
37th International Workshop on Graph-Theoretic Concepts in Computer Science (WG 2011),
Springer LNCS 6986, pp. 167-178.
Given a vertex-weighted tree T, the split of an edge xy in T is min{s_x, s_y} where s_x (respectively, s_y) is the sum... more
Given a vertex-weighted tree T, the split of an edge xy in T is min{s_x, s_y} where s_x (respectively, s_y) is the sum of all weights of vertices that are closer to x than to y (respectively, closer to y than to x) in T. Given a set of weighted vertices V and a multiset of splits S, we consider the problem of constructing a tree on V whose splits correspond to S. The problem is known to be NP-complete, even when all vertices have unit weight and the maximum vertex degree of T is required to be no more than 4. We show that
- the problem is strongly NP-complete when T is required to be a path. For this variant we exhibit an algorithm that runs in polynomial time when the number of distinct vertex weights is constant. We also show that
- the problem is NP-complete when all vertices have unit weight and the maximum degree of T is required to be no more than 3, and
- it remains NP-complete when all vertices have unit weight and T is required to be a caterpillar with unbounded hair length and maximum degree at most 3.
Finally, we shortly discuss the problem when the vertex weights are not given but can be freely chosen by an algorithm.
The considered problem is related to building libraries of chemical compounds used for drug design and discovery. In these inverse problems, the goal is to generate chemical compounds having desired structural properties, as there is a strong correlation between structural properties, such as the Wiener index, which is closely connected to the considered problem, and biological activity.