arXiv subjects

Christos H. Papadimitriou

Publications and source records attributed to Christos H. Papadimitriou.

23 records · Page 2Linked to original sources

arXiv2014

Unsupervised Learning through Prediction in a Model of Cortex

We propose a primitive called PJOIN, for "predictive join," which combines and extends the operations JOIN and LINK, which Valiant proposed as the basis of a computational theory of cortex. We show that PJOIN can be implemented in Valiant's model. We also show that, using PJOIN, certain reasonably complex learning and pattern matching tasks can be performed, in a way that involves phenomena which have been observed in cognition and the brain, namely memory-based prediction and downward traffic in the cortical hierarchy.

cs.NE↗

arXiv2011

The Complexity of the Homotopy Method, Equilibrium Selection, and Lemke-Howson Solutions

We show that the widely used homotopy method for solving fixpoint problems, as well as the Harsanyi-Selten equilibrium selection process for games, are PSPACE-complete to implement. Extending our result for the Harsanyi-Selten process, we show that several other homotopy-based algorithms for finding equilibria of games are also PSPACE-complete to implement. A further application of our techniques yields the result that it is PSPACE-complete to compute any of the equilibria that could be found via the classical Lemke-Howson algorithm, a complexity-theoretic strengthening of the result in [Savani and von Stengel]. These results show that our techniques can be widely applied and suggest that the PSPACE-completeness of implementing homotopy methods is a general principle.

cs.GT↗

arXiv2011

On Oblivious PTAS's for Nash Equilibrium

If a game has a Nash equilibrium with probability values that are either zero or Omega(1) then this equilibrium can be found exhaustively in polynomial time. Somewhat surprisingly, we show that there is a PTAS for the games whose equilibria are guaranteed to have small-O(1/n)-values, and therefore large-Omega(n)-supports. We also point out that there is a PTAS for games with sparse payoff matrices, which are known to be PPAD-complete to solve exactly. Both algorithms are of a special kind that we call oblivious: The algorithm just samples a fixed distribution on pairs of mixed strategies, and the game is only used to determine whether the sampled strategies comprise an eps-Nash equilibrium; the answer is yes with inverse polynomial probability. These results bring about the question: Is there an oblivious PTAS for Nash equilibrium in general games? We answer this question in the negative; our lower bound comes close to the quasi-polynomial upper bound of [Lipton, Markakis, Mehta 2003]. Another recent PTAS for anonymous games is also oblivious in a weaker sense appropriate for this class of games (it samples from a fixed distribution on unordered collections of mixed strategies), but its runtime is exponential in 1/eps. We prove that any oblivious PTAS for anonymous games with two strategies and three player types must have 1/eps^c in the exponent of the running time for some c>1/3, rendering the algorithm in [Daskalakis 2008] essentially optimal within oblivious algorithms. In contrast, we devise a poly(n) (1/eps)^O(log^2(1/eps)) non-oblivious PTAS for anonymous games with 2 strategies and any bounded number of player types. Our algorithm is based on the construction of a sparse (and efficiently computable) eps-cover of the set of all possible sums of n independent indicators, under the total variation distance. The size of the cover is poly(n) (1/ eps^{O(log^2 (1/eps))}.

cs.GT↗

arXiv2008

Discretized Multinomial Distributions and Nash Equilibria in Anonymous Games

We show that there is a polynomial-time approximation scheme for computing Nash equilibria in anonymous games with any fixed number of strategies (a very broad and important class of games), extending the two-strategy result of Daskalakis and Papadimitriou 2007. The approximation guarantee follows from a probabilistic result of more general interest: The distribution of the sum of n independent unit vectors with values ranging over {e1, e2, ...,ek}, where ei is the unit vector along dimension i of the k-dimensional Euclidean space, can be approximated by the distribution of the sum of another set of independent unit vectors whose probabilities of obtaining each value are multiples of 1/z for some integer z, and so that the variational distance of the two distributions is at most eps, where eps is bounded by an inverse polynomial in z and a function of k, but with no dependence on n. Our probabilistic result specifies the construction of a surprisingly sparse eps-cover -- under the total variation distance -- of the set of distributions of sums of independent unit vectors, which is of interest on its own right.

cs.GT↗

arXiv2007

The Connectivity of Boolean Satisfiability: Computational and Structural Dichotomies

Boolean satisfiability problems are an important benchmark for questions about complexity, algorithms, heuristics and threshold phenomena. Recent work on heuristics, and the satisfiability threshold has centered around the structure and connectivity of the solution space. Motivated by this work, we study structural and connectivity-related properties of the space of solutions of Boolean satisfiability problems and establish various dichotomies in Schaefer's framework. On the structural side, we obtain dichotomies for the kinds of subgraphs of the hypercube that can be induced by the solutions of Boolean formulas, as well as for the diameter of the connected components of the solution space. On the computational side, we establish dichotomy theorems for the complexity of the connectivity and st-connectivity questions for the graph of solutions of Boolean formulas. Our results assert that the intractable side of the computational dichotomies is PSPACE-complete, while the tractable side - which includes but is not limited to all problems with polynomial time algorithms for satisfiability - is in P for the st-connectivity question, and in coNP for the connectivity question. The diameter of components can be exponential for the PSPACE-complete cases, whereas in all other cases it is linear; thus, small diameter and tractability of the connectivity problems are remarkably aligned. The crux of our results is an expressibility theorem showing that in the tractable cases, the subgraphs induced by the solution space possess certain good structural properties, whereas in the intractable cases, the subgraphs can be arbitrary.

cs.CC↗