674 research outputs found
On space efficiency of algorithms working on structural decompositions of graphs
Dynamic programming on path and tree decompositions of graphs is a technique
that is ubiquitous in the field of parameterized and exponential-time
algorithms. However, one of its drawbacks is that the space usage is
exponential in the decomposition's width. Following the work of Allender et al.
[Theory of Computing, '14], we investigate whether this space complexity
explosion is unavoidable. Using the idea of reparameterization of Cai and
Juedes [J. Comput. Syst. Sci., '03], we prove that the question is closely
related to a conjecture that the Longest Common Subsequence problem
parameterized by the number of input strings does not admit an algorithm that
simultaneously uses XP time and FPT space. Moreover, we complete the complexity
landscape sketched for pathwidth and treewidth by Allender et al. by
considering the parameter tree-depth. We prove that computations on tree-depth
decompositions correspond to a model of non-deterministic machines that work in
polynomial time and logarithmic space, with access to an auxiliary stack of
maximum height equal to the decomposition's depth. Together with the results of
Allender et al., this describes a hierarchy of complexity classes for
polynomial-time non-deterministic machines with different restrictions on the
access to working space, which mirrors the classic relations between treewidth,
pathwidth, and tree-depth.Comment: An extended abstract appeared in the proceedings of STACS'16. The new
version is augmented with a space-efficient algorithm for Dominating Set
using the Chinese remainder theore
MALL proof equivalence is Logspace-complete, via binary decision diagrams
Proof equivalence in a logic is the problem of deciding whether two proofs
are equivalent modulo a set of permutation of rules that reflects the
commutative conversions of its cut-elimination procedure. As such, it is
related to the question of proofnets: finding canonical representatives of
equivalence classes of proofs that have good computational properties. It can
also be seen as the word problem for the notion of free category corresponding
to the logic.
It has been recently shown that proof equivalence in MLL (the multiplicative
with units fragment of linear logic) is PSPACE-complete, which rules out any
low-complexity notion of proofnet for this particular logic.
Since it is another fragment of linear logic for which attempts to define a
fully satisfactory low-complexity notion of proofnet have not been successful
so far, we study proof equivalence in MALL- (multiplicative-additive without
units fragment of linear logic) and discover a situation that is totally
different from the MLL case. Indeed, we show that proof equivalence in MALL-
corresponds (under AC0 reductions) to equivalence of binary decision diagrams,
a data structure widely used to represent and analyze Boolean functions
efficiently.
We show these two equivalent problems to be LOGSPACE-complete. If this
technically leaves open the possibility for a complete solution to the question
of proofnets for MALL-, the established relation with binary decision diagrams
actually suggests a negative solution to this problem.Comment: in TLCA 201
Restricted Space Algorithms for Isomorphism on Bounded Treewidth Graphs
The Graph Isomorphism problem restricted to graphs of bounded treewidth or
bounded tree distance width are known to be solvable in polynomial time
[Bod90],[YBFT99]. We give restricted space algorithms for these problems
proving the following results: - Isomorphism for bounded tree distance width
graphs is in L and thus complete for the class. We also show that for this kind
of graphs a canon can be computed within logspace. - For bounded treewidth
graphs, when both input graphs are given together with a tree decomposition,
the problem of whether there is an isomorphism which respects the
decompositions (i.e. considering only isomorphisms mapping bags in one
decomposition blockwise onto bags in the other decomposition) is in L. - For
bounded treewidth graphs, when one of the input graphs is given with a tree
decomposition the isomorphism problem is in LogCFL. - As a corollary the
isomorphism problem for bounded treewidth graphs is in LogCFL. This improves
the known TC1 upper bound for the problem given by Grohe and Verbitsky
[GroVer06].Comment: STACS conference 2010, 12 page
Pure Nash Equilibria: Hard and Easy Games
We investigate complexity issues related to pure Nash equilibria of strategic
games. We show that, even in very restrictive settings, determining whether a
game has a pure Nash Equilibrium is NP-hard, while deciding whether a game has
a strong Nash equilibrium is SigmaP2-complete. We then study practically
relevant restrictions that lower the complexity. In particular, we are
interested in quantitative and qualitative restrictions of the way each players
payoff depends on moves of other players. We say that a game has small
neighborhood if the utility function for each player depends only on (the
actions of) a logarithmically small number of other players. The dependency
structure of a game G can be expressed by a graph DG(G) or by a hypergraph
H(G). By relating Nash equilibrium problems to constraint satisfaction problems
(CSPs), we show that if G has small neighborhood and if H(G) has bounded
hypertree width (or if DG(G) has bounded treewidth), then finding pure Nash and
Pareto equilibria is feasible in polynomial time. If the game is graphical,
then these problems are LOGCFL-complete and thus in the class NC2 of highly
parallelizable problems
Efficient Algorithms for Membership in Boolean Hierarchies of Regular Languages
The purpose of this paper is to provide efficient algorithms that decide
membership for classes of several Boolean hierarchies for which efficiency (or
even decidability) were previously not known. We develop new forbidden-chain
characterizations for the single levels of these hierarchies and obtain the
following results: - The classes of the Boolean hierarchy over level
of the dot-depth hierarchy are decidable in (previously only the
decidability was known). The same remains true if predicates mod for fixed
are allowed. - If modular predicates for arbitrary are allowed, then
the classes of the Boolean hierarchy over level are decidable. - For
the restricted case of a two-letter alphabet, the classes of the Boolean
hierarchy over level of the Straubing-Th\'erien hierarchy are
decidable in . This is the first decidability result for this hierarchy. -
The membership problems for all mentioned Boolean-hierarchy classes are
logspace many-one hard for . - The membership problems for quasi-aperiodic
languages and for -quasi-aperiodic languages are logspace many-one complete
for
Stream Reasoning in Temporal Datalog
In recent years, there has been an increasing interest in extending
traditional stream processing engines with logical, rule-based, reasoning
capabilities. This poses significant theoretical and practical challenges since
rules can derive new information and propagate it both towards past and future
time points; as a result, streamed query answers can depend on data that has
not yet been received, as well as on data that arrived far in the past. Stream
reasoning algorithms, however, must be able to stream out query answers as soon
as possible, and can only keep a limited number of previous input facts in
memory. In this paper, we propose novel reasoning problems to deal with these
challenges, and study their computational properties on Datalog extended with a
temporal sort and the successor function (a core rule-based language for stream
reasoning applications)
Pseudorandomness for Regular Branching Programs via Fourier Analysis
We present an explicit pseudorandom generator for oblivious, read-once,
permutation branching programs of constant width that can read their input bits
in any order. The seed length is , where is the length of the
branching program. The previous best seed length known for this model was
, which follows as a special case of a generator due to
Impagliazzo, Meka, and Zuckerman (FOCS 2012) (which gives a seed length of
for arbitrary branching programs of size ). Our techniques
also give seed length for general oblivious, read-once branching
programs of width , which is incomparable to the results of
Impagliazzo et al.Our pseudorandom generator is similar to the one used by
Gopalan et al. (FOCS 2012) for read-once CNFs, but the analysis is quite
different; ours is based on Fourier analysis of branching programs. In
particular, we show that an oblivious, read-once, regular branching program of
width has Fourier mass at most at level , independent of the
length of the program.Comment: RANDOM 201
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