5,267 research outputs found
Settling Some Open Problems on 2-Player Symmetric Nash Equilibria
Over the years, researchers have studied the complexity of several decision
versions of Nash equilibrium in (symmetric) two-player games (bimatrix games).
To the best of our knowledge, the last remaining open problem of this sort is
the following; it was stated by Papadimitriou in 2007: find a non-symmetric
Nash equilibrium (NE) in a symmetric game. We show that this problem is
NP-complete and the problem of counting the number of non-symmetric NE in a
symmetric game is #P-complete.
In 2005, Kannan and Theobald defined the "rank of a bimatrix game"
represented by matrices (A, B) to be rank(A+B) and asked whether a NE can be
computed in rank 1 games in polynomial time. Observe that the rank 0 case is
precisely the zero sum case, for which a polynomial time algorithm follows from
von Neumann's reduction of such games to linear programming. In 2011, Adsul et.
al. obtained an algorithm for rank 1 games; however, it does not solve the case
of symmetric rank 1 games. We resolve this problem
On kernels, defaults and even graphs
Extensions in prerequisite-free, disjunction-free default theories have been shown to be in direct correspondence with kernels of directed graphs; hence default theories without odd cycles always have a ``standard'' kind of an extension. We show that, although all ``standard'' extensions can be enumerated explicitly, several other problems remain intractable for such theories: Telling whether a non-standard extension exists, enumerating all extensions, and finding the minimal standard extension. We also present a new graph-theoretic algorithm, based on vertex feedback sets, for enumerating all extensions of a general prerequisite-free, disjunction-free default theory (possibly with odd cycles). The algorithm empirically performs well for quite large theories
Critical slowing down in polynomial time algorithms
Combinatorial optimization algorithms which compute exact ground state
configurations in disordered magnets are seen to exhibit critical slowing down
at zero temperature phase transitions. Using arguments based on the physical
picture of the model, including vanishing stiffness on scales beyond the
correlation length and the ground state degeneracy, the number of operations
carried out by one such algorithm, the push-relabel algorithm for the random
field Ising model, can be estimated. Some scaling can also be predicted for the
2D spin glass.Comment: 4 pp., 3 fig
Swampland Criteria and Constraints on Inflation in a Gravity Theory
In this paper, we worked in the framework of an inflationary theory,
in the presence of a canonical scalar field. More specifically, the
gravity. The values of the dimensionless
parameters and are taken to be and . The motivation for that study was the striking similarities
between the slow-roll parameters of the inflationary model used in this work
and the ones obtained by the rescaled Einstein-Hilbert gravity inflation
. We examined a variety of potentials to determine if they agree
with the current Planck Constraints. In addition, we checked whether these
models satisfy the Swampland Criteria and we specified the exact region of the
parameter space that produces viable results for each model. As we mention in
Section IV the inflationary theory used in this work can not produce a
positive which implies that the stochastic gravitational wave background
will not be detectable.Comment: IJMPD Accepte
A Direct Reduction from k-Player to 2-Player Approximate Nash Equilibrium
We present a direct reduction from k-player games to 2-player games that
preserves approximate Nash equilibrium. Previously, the computational
equivalence of computing approximate Nash equilibrium in k-player and 2-player
games was established via an indirect reduction. This included a sequence of
works defining the complexity class PPAD, identifying complete problems for
this class, showing that computing approximate Nash equilibrium for k-player
games is in PPAD, and reducing a PPAD-complete problem to computing approximate
Nash equilibrium for 2-player games. Our direct reduction makes no use of the
concept of PPAD, thus eliminating some of the difficulties involved in
following the known indirect reduction.Comment: 21 page
Quenched Random Graphs
Spin models on quenched random graphs are related to many important
optimization problems. We give a new derivation of their mean-field equations
that elucidates the role of the natural order parameter in these models.Comment: 9 pages, report CPTH-A264.109
Qualitative Analysis of Partially-observable Markov Decision Processes
We study observation-based strategies for partially-observable Markov
decision processes (POMDPs) with omega-regular objectives. An observation-based
strategy relies on partial information about the history of a play, namely, on
the past sequence of observations. We consider the qualitative analysis
problem: given a POMDP with an omega-regular objective, whether there is an
observation-based strategy to achieve the objective with probability~1
(almost-sure winning), or with positive probability (positive winning). Our
main results are twofold. First, we present a complete picture of the
computational complexity of the qualitative analysis of POMDP s with parity
objectives (a canonical form to express omega-regular objectives) and its
subclasses. Our contribution consists in establishing several upper and lower
bounds that were not known in literature. Second, we present optimal bounds
(matching upper and lower bounds) on the memory required by pure and randomized
observation-based strategies for the qualitative analysis of POMDP s with
parity objectives and its subclasses
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