38,181 research outputs found
The Hardness of Finding Linear Ranking Functions for Lasso Programs
Finding whether a linear-constraint loop has a linear ranking function is an
important key to understanding the loop behavior, proving its termination and
establishing iteration bounds. If no preconditions are provided, the decision
problem is known to be in coNP when variables range over the integers and in
PTIME for the rational numbers, or real numbers. Here we show that deciding
whether a linear-constraint loop with a precondition, specifically with
partially-specified input, has a linear ranking function is EXPSPACE-hard over
the integers, and PSPACE-hard over the rationals. The precise complexity of
these decision problems is yet unknown. The EXPSPACE lower bound is derived
from the reachability problem for Petri nets (equivalently, Vector Addition
Systems), and possibly indicates an even stronger lower bound (subject to open
problems in VAS theory). The lower bound for the rationals follows from a novel
simulation of Boolean programs. Lower bounds are also given for the problem of
deciding if a linear ranking-function supported by a particular form of
inductive invariant exists. For loops over integers, the problem is PSPACE-hard
for convex polyhedral invariants and EXPSPACE-hard for downward-closed sets of
natural numbers as invariants.Comment: In Proceedings GandALF 2014, arXiv:1408.5560. I thank the organizers
of the Dagstuhl Seminar 14141, "Reachability Problems for Infinite-State
Systems", for the opportunity to present an early draft of this wor
On Termination of Integer Linear Loops
A fundamental problem in program verification concerns the termination of
simple linear loops of the form x := u ; while Bx >= b do {x := Ax + a} where x
is a vector of variables, u, a, and c are integer vectors, and A and B are
integer matrices. Assuming the matrix A is diagonalisable, we give a decision
procedure for the problem of whether, for all initial integer vectors u, such a
loop terminates. The correctness of our algorithm relies on sophisticated tools
from algebraic and analytic number theory, Diophantine geometry, and real
algebraic geometry. To the best of our knowledge, this is the first substantial
advance on a 10-year-old open problem of Tiwari (2004) and Braverman (2006).Comment: Accepted to SODA1
On Multiphase-Linear Ranking Functions
Multiphase ranking functions () were proposed as a means
to prove the termination of a loop in which the computation progresses through
a number of "phases", and the progress of each phase is described by a
different linear ranking function. Our work provides new insights regarding
such functions for loops described by a conjunction of linear constraints
(single-path loops). We provide a complete polynomial-time solution to the
problem of existence and of synthesis of of bounded depth
(number of phases), when variables range over rational or real numbers; a
complete solution for the (harder) case that variables are integer, with a
matching lower-bound proof, showing that the problem is coNP-complete; and a
new theorem which bounds the number of iterations for loops with
. Surprisingly, the bound is linear, even when the
variables involved change in non-linear way. We also consider a type of
lexicographic ranking functions, , more expressive than types
of lexicographic functions for which complete solutions have been given so far.
We prove that for the above type of loops, lexicographic functions can be
reduced to , and thus the questions of complexity of
detection and synthesis, and of resulting iteration bounds, are also answered
for this class.Comment: typos correcte
More Borda Count Variations for Project Assesment
We introduce and analyze the following variants of the Borda rule: median Borda rule,geometric Borda rule, Litvak’s method as well as methods based on forming linear combinations of entries in the preference outranking matrix. The properties we focus upon are the elimination of the Condorcet loser as well as several consistency-type criteria.Borda rule, median rule, Nash welfare function, outranking matrix, maximin rule, consistency
Contextual Centrality: Going Beyond Network Structures
Centrality is a fundamental network property which ranks nodes by their
structural importance. However, structural importance may not suffice to
predict successful diffusions in a wide range of applications, such as
word-of-mouth marketing and political campaigns. In particular, nodes with high
structural importance may contribute negatively to the objective of the
diffusion. To address this problem, we propose contextual centrality, which
integrates structural positions, the diffusion process, and, most importantly,
nodal contributions to the objective of the diffusion. We perform an empirical
analysis of the adoption of microfinance in Indian villages and weather
insurance in Chinese villages. Results show that contextual centrality of the
first-informed individuals has higher predictive power towards the eventual
adoption outcomes than other standard centrality measures. Interestingly, when
the product of diffusion rate and the largest eigenvalue is
larger than one and diffusion period is long, contextual centrality linearly
scales with eigenvector centrality. This approximation reveals that contextual
centrality identifies scenarios where a higher diffusion rate of individuals
may negatively influence the cascade payoff. Further simulations on the
synthetic and real-world networks show that contextual centrality has the
advantage of selecting an individual whose local neighborhood generates a high
cascade payoff when . Under this condition, stronger homophily
leads to higher cascade payoff. Our results suggest that contextual centrality
captures more complicated dynamics on networks and has significant implications
for applications, such as information diffusion, viral marketing, and political
campaigns
Pro’s and Con’s of a reverse-auction to evaluate conservation easements
Resource /Energy Economics and Policy,
Infinite-horizon choice functions
We analyze infinite-horizon choice functions within the setting of a simple technology. Efficiency and time consistency are characterized by stationary consumption and inheritance functions, as well as a transversality condition. In addition, we consider the equity axioms Suppes-Sen, Pigou-Dalton, and resource monotonicity. We show that Suppes-Sen and Pigou-Dalton imply that the consumption and inheritance functions are monotone with respect to time - thus justifying sustainability - while resource monotonicity implies that the consumption and inheritance functions are monotone with respect to the resource. Examples illustrate the characterization results.Intergenerational resource allocation, infinite-horizon choice
Spherical Preferences
We introduce and study the property of orthogonal independence, a restricted
additivity axiom applying when alternatives are orthogonal. The axiom requires
that the preference for one marginal change over another should be maintained
after each marginal change has been shifted in a direction that is orthogonal
to both.
We show that continuous preferences satisfy orthogonal independence if and
only if they are spherical: their indifference curves are spheres with the same
center, with preference being "monotone" either away or towards the center.
Spherical preferences include linear preferences as a special (limiting) case.
We discuss different applications to economic and political environments. Our
result delivers Euclidean preferences in models of spatial voting, quadratic
welfare aggregation in social choice, and expected utility in models of choice
under uncertainty
Creating Composite Indicators with DEA and Robustness Analysis: the case of the Technology Achievement Index
Composite indicators are regularly used for benchmarking countries’ performance, but equally often stir controversies about the unavoidable subjectivity that is connected with their construction. Data Envelopment Analysis helps to overcome some key limitations, viz., the undesirable dependence of final results from the preliminary normalization of sub-indicators, and, more cogently, from the subjective nature of the weights used for aggregating. Still, subjective decisions remain, and such modelling uncertainty propagates onto countries’ composite indicator values and relative rankings. Uncertainty and sensitivity analysis are therefore needed to assess robustness of final results and to analyze how much each individual source of uncertainty contributes to the output variance. The current paper reports on these issues, using the Technology Achievement Index as an illustration.factor is more important in explaining the observed progress.composite indicators, aggregation, weighting, Internal Market
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