184 research outputs found
Average-energy games
Two-player quantitative zero-sum games provide a natural framework to
synthesize controllers with performance guarantees for reactive systems within
an uncontrollable environment. Classical settings include mean-payoff games,
where the objective is to optimize the long-run average gain per action, and
energy games, where the system has to avoid running out of energy.
We study average-energy games, where the goal is to optimize the long-run
average of the accumulated energy. We show that this objective arises naturally
in several applications, and that it yields interesting connections with
previous concepts in the literature. We prove that deciding the winner in such
games is in NP inter coNP and at least as hard as solving mean-payoff games,
and we establish that memoryless strategies suffice to win. We also consider
the case where the system has to minimize the average-energy while maintaining
the accumulated energy within predefined bounds at all times: this corresponds
to operating with a finite-capacity storage for energy. We give results for
one-player and two-player games, and establish complexity bounds and memory
requirements.Comment: In Proceedings GandALF 2015, arXiv:1509.0685
The Complexity of Change
Many combinatorial problems can be formulated as "Can I transform
configuration 1 into configuration 2, if certain transformations only are
allowed?". An example of such a question is: given two k-colourings of a graph,
can I transform the first k-colouring into the second one, by recolouring one
vertex at a time, and always maintaining a proper k-colouring? Another example
is: given two solutions of a SAT-instance, can I transform the first solution
into the second one, by changing the truth value one variable at a time, and
always maintaining a solution of the SAT-instance? Other examples can be found
in many classical puzzles, such as the 15-Puzzle and Rubik's Cube.
In this survey we shall give an overview of some older and more recent work
on this type of problem. The emphasis will be on the computational complexity
of the problems: how hard is it to decide if a certain transformation is
possible or not?Comment: 28 pages, 6 figure
Completeness Results for Parameterized Space Classes
The parameterized complexity of a problem is considered "settled" once it has
been shown to lie in FPT or to be complete for a class in the W-hierarchy or a
similar parameterized hierarchy. Several natural parameterized problems have,
however, resisted such a classification. At least in some cases, the reason is
that upper and lower bounds for their parameterized space complexity have
recently been obtained that rule out completeness results for parameterized
time classes. In this paper, we make progress in this direction by proving that
the associative generability problem and the longest common subsequence problem
are complete for parameterized space classes. These classes are defined in
terms of different forms of bounded nondeterminism and in terms of simultaneous
time--space bounds. As a technical tool we introduce a "union operation" that
translates between problems complete for classical complexity classes and for
W-classes.Comment: IPEC 201
Structure of computations in parallel complexity classes
Issued as Annual report, and Final project report, Project no. G-36-67
Hardness of Approximation in PSPACE and Separation Results for Pebble Games
We consider the pebble game on DAGs with bounded fan-in introduced in
[Paterson and Hewitt '70] and the reversible version of this game in [Bennett
'89], and study the question of how hard it is to decide exactly or
approximately the number of pebbles needed for a given DAG in these games. We
prove that the problem of eciding whether ~pebbles suffice to reversibly
pebble a DAG is PSPACE-complete, as was previously shown for the standard
pebble game in [Gilbert, Lengauer and Tarjan '80]. Via two different graph
product constructions we then strengthen these results to establish that both
standard and reversible pebbling space are PSPACE-hard to approximate to within
any additive constant. To the best of our knowledge, these are the first
hardness of approximation results for pebble games in an unrestricted setting
(even for polynomial time). Also, since [Chan '13] proved that reversible
pebbling is equivalent to the games in [Dymond and Tompa '85] and [Raz and
McKenzie '99], our results apply to the Dymond--Tompa and Raz--McKenzie games
as well, and from the same paper it follows that resolution depth is
PSPACE-hard to determine up to any additive constant. We also obtain a
multiplicative logarithmic separation between reversible and standard pebbling
space. This improves on the additive logarithmic separation previously known
and could plausibly be tight, although we are not able to prove this. We leave
as an interesting open problem whether our additive hardness of approximation
result could be strengthened to a multiplicative bound if the computational
resources are decreased from polynomial space to the more common setting of
polynomial time
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