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In this paper, we introduce a cooperative game model arising from bin covering problem, called bin covering game, and discuss the computational complexity issues on the core and the approximate core of the game. Making use of duality theorem of linear programming, a sufficient and necessary condition on core nonemptiness is proposed. When the core is empty, a lower bound on the minimum taxrate of...
We give a subexponential time approximation algorithm for the Unique Games problem. The algorithms run in time that is exponential in an arbitrarily small polynomial of the input size, nε. The approximation guarantee depends on ε, but not on the alphabet size or the number of variables. We also obtain a subexponential algorithms with improved approximations for SMALL-SET EXPANSION and MULTICUT. For...
We present a new algorithm for Unique Games which is based on purely spectral techniques, in contrast to previous work in the area, which relies heavily on semidefinite programming (SDP). Given a highly satisfiable instance of Unique Games, our algorithm is able to recover a good assignment. The approximation guarantee depends only on the completeness of the game, and not on the alphabet size, while...
K Nearest Neighbor search has many applications including data mining, multi-media, image processing, and monitoring moving objects. In this paper, we study the problem of KNN over multi-valued objects. We aim to provide effective and efficient techniques to identify KNN sensitive to relative distributions of objects.We propose to use quantiles to summarize relative-distribution-sensitive K nearest...
In recent years, idempotent methods (specifically, max-plus methods) have been developed for solution of nonlinear control problems. It was thought that idempotent linearity of the associated semigroup was required for application of these techniques. It is now known that application of the max-plus distributive property allows one to apply the max-plus curse-of-dimensionality-free approach to stochastic...
In this paper, we consider the problem of measurement allocation in a spatially correlated sensor field. Our objective is to determine the probability of each sensor's being measured for improved observability; the sensor located at less correlated area should be assigned more probability. To this end, we quantify the level of correlation of each sensor through the mutual information criterion reflecting...
Linial, London and Rabinovich [16] and Aumann and Rabani [3] proved that the min-cut max-flow ratio for general maximum concurrent flow problems (when there are k commodities) is O(logfe). Here we attempt to derive a more general theory of Steiner cut and flow problems, and we prove bounds that are poly-logarithmic in k for a much broader class of multicommodity flow and cut problems. Our structural...
We prove that the problem of computing an Arrow-Debreu market equilibrium is PPAD-complete even when all traders use additively separable, piecewise-linear and concave utility functions. In fact, our proof shows that this market-equilibrium problem does not have a fully polynomial-time approximation scheme, unless every problem in PPAD is solvable in polynomial time.
In a landmark paper, Papadimitriou introduced a number of syntactic subclasses of TFNP based on proof styles that (unlike TFNP) admit complete problems. A recent series of results has shown that finding Nash equilibria is complete for PPAD, a particularly notable subclass of TFNP. A major goal of this work is to expand the universe of known PPAD-complete problems. We resolve the computational complexity...
This paper ties the line of work on algorithms that find an O(??(log n))-approximation to the SPARSEST CUT together with the line of work on algorithms that run in subquadratic time by using only single-commodity flows. We present an algorithm that simultaneously achieves both goals, finding an O(??(log (n)/??))-approximation using O(n?? logO(1) n) max-flows. The core of the algorithm is a stronger,...
With the work of Khot and Vishnoi as a starting point, we obtain integrality gaps for certain strong SDP relaxations of Unique Games. Specifically, we exhibit a Unique Games gap instance for the basic semidefinite program strengthened by all valid linear inequalities on the inner products of up to exp(??(log log n)1/4) vectors. For a stronger relaxation obtained from the basic semidefinite program...
We prove the following strong hardness result for learning: Given a distribution on labeled examples from the hypercube such that there exists a monomial (or conjunction) consistent with (1-??)-fraction of the examples, it is NP-hard to find a halfspace that is correct on ( 1/2 + ??)-fraction of the examples, for arbitrary constant ?? > 0. In learning theory terms, weak agnostic learning of monomials...
We study the complexity of rationalizing network formation. In this problem we fix an underlying model describing how selfish parties (the vertices) produce a graph by making individual decisions to form or not form incident edges. The model is equipped with a notion of stability (or equilibrium), and we observe a set of "snapshots" of graphs that are assumed to be stable. From this we would...
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...
We show a connection between the semidefinite relaxation of unique games and their behavior under parallel repetition. Specifically,denoting by val(G) the value of a two-prover unique game G, andby sdpval(G) the value of a natural semidefinite program to approximate val(G), we prove that for every l epsi N, if sdpval(G) ges 1-delta, then val(Gl) ges 1-radicsldelta. Here, Gl denotes the l-fold parallel...
The parallel repetition theorem states that for any two-prover game, with value 1 - isin (for, say, isin les 1/2), the value of the game repeated in parallel n times is at most (1 - isinc)Omega(n/s), where s is the answers' length (of the original game) and c is a universal constant. Several researchers asked wether this bound could be improved to (1 - isin)Omega(n/s); this question is usually referred...
In the kernel clustering problem we are given a large ntimesn positive semi-definite matrix A=(aij) with Sigmai,jn=1 aij=0 and a small ktimesk positivesemi-definite matrix B=bij. The goal is to find a partition S1,..Sk of {1,...n} which maximizes the quantity Sigmai,j=1k(Sigma(i,j)isinSitimesSj). We study the computational complexity of this generic clustering problem which originates in the...
We study the approximability of predicates on k variables from a domain [q], and give a new sufficient condition for such predicates to be approximation resistant under the unique games conjecture. Specifically, we show that a predicate P is approximation resistant if there exists a balanced pairwise independent distribution over [q]k whose support is contained in the set of satisfying assignments...
We investigate the notions of may- and must-approximation in Erratic Idealized Algol (a nondeterministic extension of Idealized Algol), and give explicit characterizations of both using its game model. Notably, must-approximation is captured by a novel preorder on nondeterministic strategies, whose definition is formulated in terms of winning regions in a reachability game. The game is played on traces...
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