233 research outputs found
On generating the irredundant conjunctive and disjunctive normal forms of monotone Boolean functions
AbstractLet f:{0,1}nâ{0,1} be a monotone Boolean function whose value at any point xâ{0,1}n can be determined in time t. Denote by c=âIâCâiâIxi the irredundant CNF of f, where C is the set of the prime implicates of f. Similarly, let d=âJâDâjâJxj be the irredundant DNF of the same function, where D is the set of the prime implicants of f. We show that given subsets CâČâC and DâČâD such that (CâČ,DâČ)â (C,D), a new term in (Câ§čCâČ)âȘ(Dâ§čDâČ) can be found in time O(n(t+n))+mo(logm), where m=|CâČ|+|DâČ|. In particular, if f(x) can be evaluated for every xâ{0,1}n in polynomial time, then the forms c and d can be jointly generated in incremental quasi-polynomial time. On the other hand, even for the class of â§,âš-formulae f of depth 2, i.e., for CNFs or DNFs, it is unlikely that uniform sampling from within the set of the prime implicates and implicants of f can be carried out in time bounded by a quasi-polynomial 2polylog(·) in the input size of f. We also show that for some classes of polynomial-time computable monotone Boolean functions it is NP-hard to test either of the conditions DâČ=D or CâČ=C. This provides evidence that for each of these classes neither conjunctive nor disjunctive irredundant normal forms can be generated in total (or incremental) quasi-polynomial time. Such classes of monotone Boolean functions naturally arise in game theory, networks and relay contact circuits, convex programming, and include a subset of â§,âš-formulae of depth 3
Assessing non-Markovian dynamics
We investigate what a snapshot of a quantum evolution - a quantum channel
reflecting open system dynamics - reveals about the underlying continuous time
evolution. Remarkably, from such a snapshot, and without imposing additional
assumptions, it can be decided whether or not a channel is consistent with a
time (in)dependent Markovian evolution, for which we provide computable
necessary and sufficient criteria. Based on these, a computable measure of
`Markovianity' is introduced. We discuss how the consistency with Markovian
dynamics can be checked in quantum process tomography. The results also clarify
the geometry of the set of quantum channels with respect to being solutions of
time (in)dependent master equations.Comment: 5 pages, RevTex, 2 figures. (Except from typesetting) version to be
published in the Physical Review Letter
On Relevant Equilibria in Reachability Games
We study multiplayer reachability games played on a finite directed graph
equipped with target sets, one for each player. In those reachability games, it
is known that there always exists a Nash equilibrium (NE) and a subgame perfect
equilibrium (SPE). But sometimes several equilibria may coexist such that in
one equilibrium no player reaches his target set whereas in another one several
players reach it. It is thus very natural to identify "relevant" equilibria. In
this paper, we consider different notions of relevant equilibria including
Pareto optimal equilibria and equilibria with high social welfare. We provide
complexity results for various related decision problems
On the Number of Iterations for Dantzig-Wolfe Optimization and Packing-Covering Approximation Algorithms
We give a lower bound on the iteration complexity of a natural class of
Lagrangean-relaxation algorithms for approximately solving packing/covering
linear programs. We show that, given an input with random 0/1-constraints
on variables, with high probability, any such algorithm requires
iterations to compute a
-approximate solution, where is the width of the input.
The bound is tight for a range of the parameters .
The algorithms in the class include Dantzig-Wolfe decomposition, Benders'
decomposition, Lagrangean relaxation as developed by Held and Karp [1971] for
lower-bounding TSP, and many others (e.g. by Plotkin, Shmoys, and Tardos [1988]
and Grigoriadis and Khachiyan [1996]). To prove the bound, we use a discrepancy
argument to show an analogous lower bound on the support size of
-approximate mixed strategies for random two-player zero-sum
0/1-matrix games
Mirror-Descent Methods in Mixed-Integer Convex Optimization
In this paper, we address the problem of minimizing a convex function f over
a convex set, with the extra constraint that some variables must be integer.
This problem, even when f is a piecewise linear function, is NP-hard. We study
an algorithmic approach to this problem, postponing its hardness to the
realization of an oracle. If this oracle can be realized in polynomial time,
then the problem can be solved in polynomial time as well. For problems with
two integer variables, we show that the oracle can be implemented efficiently,
that is, in O(ln(B)) approximate minimizations of f over the continuous
variables, where B is a known bound on the absolute value of the integer
variables.Our algorithm can be adapted to find the second best point of a
purely integer convex optimization problem in two dimensions, and more
generally its k-th best point. This observation allows us to formulate a
finite-time algorithm for mixed-integer convex optimization
Polynomial Delay Algorithm for Listing Minimal Edge Dominating sets in Graphs
The Transversal problem, i.e, the enumeration of all the minimal transversals
of a hypergraph in output-polynomial time, i.e, in time polynomial in its size
and the cumulated size of all its minimal transversals, is a fifty years old
open problem, and up to now there are few examples of hypergraph classes where
the problem is solved. A minimal dominating set in a graph is a subset of its
vertex set that has a non empty intersection with the closed neighborhood of
every vertex. It is proved in [M. M. Kant\'e, V. Limouzy, A. Mary, L. Nourine,
On the Enumeration of Minimal Dominating Sets and Related Notions, In Revision
2014] that the enumeration of minimal dominating sets in graphs and the
enumeration of minimal transversals in hypergraphs are two equivalent problems.
Hoping this equivalence can help to get new insights in the Transversal
problem, it is natural to look inside graph classes. It is proved independently
and with different techniques in [Golovach et al. - ICALP 2013] and [Kant\'e et
al. - ISAAC 2012] that minimal edge dominating sets in graphs (i.e, minimal
dominating sets in line graphs) can be enumerated in incremental
output-polynomial time. We provide the first polynomial delay and polynomial
space algorithm that lists all the minimal edge dominating sets in graphs,
answering an open problem of [Golovach et al. - ICALP 2013]. Besides the
result, we hope the used techniques that are a mix of a modification of the
well-known Berge's algorithm and a strong use of the structure of line graphs,
are of great interest and could be used to get new output-polynomial time
algorithms.Comment: proofs simplified from previous version, 12 pages, 2 figure
Optimal Reachability in Divergent Weighted Timed Games
Weighted timed games are played by two players on a timed automaton equipped
with weights: one player wants to minimise the accumulated weight while
reaching a target, while the other has an opposite objective. Used in a
reactive synthesis perspective, this quantitative extension of timed games
allows one to measure the quality of controllers. Weighted timed games are
notoriously difficult and quickly undecidable, even when restricted to
non-negative weights. Decidability results exist for subclasses of one-clock
games, and for a subclass with non-negative weights defined by a semantical
restriction on the weights of cycles. In this work, we introduce the class of
divergent weighted timed games as a generalisation of this semantical
restriction to arbitrary weights. We show how to compute their optimal value,
yielding the first decidable class of weighted timed games with negative
weights and an arbitrary number of clocks. In addition, we prove that
divergence can be decided in polynomial space. Last, we prove that for untimed
games, this restriction yields a class of games for which the value can be
computed in polynomial time
Tropically convex constraint satisfaction
A semilinear relation S is max-closed if it is preserved by taking the
componentwise maximum. The constraint satisfaction problem for max-closed
semilinear constraints is at least as hard as determining the winner in Mean
Payoff Games, a notorious problem of open computational complexity. Mean Payoff
Games are known to be in the intersection of NP and co-NP, which is not known
for max-closed semilinear constraints. Semilinear relations that are max-closed
and additionally closed under translations have been called tropically convex
in the literature. One of our main results is a new duality for open tropically
convex relations, which puts the CSP for tropically convex semilinaer
constraints in general into NP intersected co-NP. This extends the
corresponding complexity result for scheduling under and-or precedence
constraints, or equivalently the max-atoms problem. To this end, we present a
characterization of max-closed semilinear relations in terms of syntactically
restricted first-order logic, and another characterization in terms of a finite
set of relations L that allow primitive positive definitions of all other
relations in the class. We also present a subclass of max-closed constraints
where the CSP is in P; this class generalizes the class of max-closed
constraints over finite domains, and the feasibility problem for max-closed
linear inequalities. Finally, we show that the class of max-closed semilinear
constraints is maximal in the sense that as soon as a single relation that is
not max-closed is added to L, the CSP becomes NP-hard.Comment: 29 pages, 2 figure
Polyhedral Analysis using Parametric Objectives
The abstract domain of polyhedra lies at the heart of many program analysis techniques. However, its operations can be expensive, precluding their application to polyhedra that involve many variables. This paper describes a new approach to computing polyhedral domain operations. The core of this approach is an algorithm to calculate variable elimination (projection) based on parametric linear programming. The algorithm enumerates only non-redundant inequalities of the projection space, hence permits anytime approximation of the output
The impact of land use on occurrence of urban heat islands in Slovenia
Urbani toplotni otoki so rezultat antropogenega delovanja v mestnih obmoÄjih in predstavljajo temperaturno razliko med urbano in ruralno krajino. Kot urbane toplotne otoke smo v tej raziskavi doloÄili tista obmoÄja, znotraj katerih je temperatura povrĆĄja najtoplejĆĄe Äetrtine leta viĆĄja vsaj za 0,1 °C od temperature okolice, velikost obmoÄja pa je veÄja od 50 ha. Analizirali smo urbane toplotne otoke razliÄnih velikosti in razliÄnih rab tal (kmetijska, pozidana, zelena in gozdna raba tal, vodna telesa). Obravnavali smo vpliv rabe tal (deleĆŸ gozdov, deleĆŸ pozidanih zemljiĆĄÄ in fragmentiranost gozdnih povrĆĄin) na intenziteto urbanih toplotnih otokov na dveh ravneh. Vpliv rabe tal na intenziteto urbanega toplotnega otoka smo obravnavali za celotno obmoÄje Slovenije ter tista obmoÄja sosednjih drĆŸav, od koder urbani toplotni otoki segajo na njeno obmoÄje. V prostorski analizi pojavljanja urbanih toplotnih otokov smo uporabili podatke o povpreÄni temperaturi najtoplejĆĄe Äetrtine leta, ki izhaja iz satelitskih posnetkov MODIS ter evropske podatke o rabi tal CORINE LAND COVER (CLC). Ugotovili smo, da ima deleĆŸ gozdov znotraj urbanega toplotnega otoka 1. reda ĆĄibek vpliv na intenziteto urbanih toplotnih otokov, vpliv deleĆŸa gozdov na intenziteto urbanega toplotnega otoka 2. reda pa je veÄji (r2 = 0,38p < 0,001). Intenziteta urbanih toplotnih otokov 1. in 2. reda statistiÄno znaÄilno naraĆĄÄa z deleĆŸem pozidanih povrĆĄin (r2 = 0,35p < 0,001). Intenziteta urbanih toplotnih otokov se z veÄanjem gostote gozdnega roba in veÄanjem fragmentiranosti gozdnih povrĆĄin znotraj obmoÄja urbanega toplotnega otoka niĆŸa.Urban heat islands are the result of anthropogenic activity in urban areas and represent a temperature difference between urban and rural area. Urban heat islands in this study are determined as areas, within the temperature of the surface in the warmest quarter of the year is higher than 0,1 °C and the area is larger than 50 ha. We analysed urban heat islands of different sizes and different land uses (agricultural, built-up, green, forest areas and water bodies). We analysed the impact of land use (forest share, share of built-up areas, forest fragmentation) on the intensity of urban heat islands on two levels. The impact of land use on the intensity of the urban heat island was considered for the entire territory of Slovenia. The method of work was based on the analysis of the European land use map CORINE LAND COVER (CLC) and the satellite images MODIS of the average temperature of the warmest quarter of the year. We found, that share of forest within urban heat islands of level 1, has a weak influence on the intensity of urban heat islandsthe influence of forest share on level 2 is higher (r2 = 0,38p < 0,001). The intensity of urban heat islands of both levels is statistically significantly increasing with the share of built-up areas (r2 = 0,35p < 0,001). Intensity of urban heat islands increases by increasing the edge density and by increasing the fragmentation of forest areas within the urban heat island
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