Quantum algorithms for hidden nonlinear structures

Attempts to find new quantum algorithms that outperform classical computation have focused primarily on the nonabelian hidden subgroup problem, which generalizes the central problem solved by Shor's factoring algorithm. We suggest an alternative generalization, namely to problems of finding hidden nonlinear structures over finite fields. We give examples of two such problems that can be solved efficiently by a quantum computer, but not by a classical computer. We also give some positive results on the quantum query complexity of finding hidden nonlinear structures.Comment: 13 page

The Duality Gap for Two-Team Zero-Sum Games

We consider multiplayer games in which the players fall in two teams of size k, with payoffs equal within, and of opposite sign across, the two teams. In the classical case of k=1, such zero-sum games possess a unique value, independent of order of play, due to the von Neumann minimax theorem. However, this fails for all k>1; we can measure this failure by a duality gap, which quantifies the benefit of being the team to commit last to its strategy. In our main result we show that the gap equals 2(1-2^{1-k}) for m=2 and 2(1-m^{-(1-o(1))k}) for m>2, with m being the size of the action space of each player. At a finer level, the cost to a team of individual players acting independently while the opposition employs joint randomness is 1-2^{1-k} for k=2, and 1-m^{-(1-o(1))k} for m>2. This class of multiplayer games, apart from being a natural bridge between two-player zero-sum games and general multiplayer games, is motivated from Biology (the weak selection model of evolution) and Economics (players with shared utility but poor coordination)

The duality gap for two-team zero-sum games

We consider multiplayer games in which the players fall in two teams of size k, with payoffs equal within, and of opposite sign across, the two teams. In the classical case of k = 1, such zero-sum games possess a unique value, independent of order of play. However, this fails for all k > 1; we can measure this failure by a duality gap, which quantifies the benefit of being the team to commit last to its strategy. We show that the gap equals 2(1−2^(1−k)) for m = 2 and 2(1−m^(−(1−o(1))k)) for m > 2, with m being the size of the action space of each player. Extensions hold also for different-size teams and players with various-size action spaces. We further study the effect of exchanging order of commitment among individual players (not only among the entire teams). The class of two-team zero-sum games is motivated from the weak selection model of evolution, and from considering teams such as firms in which independent players (ideally) have shared utility

The duality gap for two-team zero-sum games

We consider multiplayer games in which the players fall in two teams of size k, with payoffs equal within, and of opposite sign across, the two teams. In the classical case of k = 1, such zero-sum games possess a unique value, independent of order of play. However, this fails for all k > 1; we can measure this failure by a duality gap, which quantifies the benefit of being the team to commit last to its strategy. We show that the gap equals 2(1−2^(1−k)) for m = 2 and 2(1−m^(−(1−o(1))k)) for m > 2, with m being the size of the action space of each player. Extensions hold also for different-size teams and players with various-size action spaces. We further study the effect of exchanging order of commitment among individual players (not only among the entire teams). The class of two-team zero-sum games is motivated from the weak selection model of evolution, and from considering teams such as firms in which independent players (ideally) have shared utility

Random Polynomial Time is Equal to Semi-Random Polynomial Time

We prove that any one-sided error random polynomial time (RP) algorithm can be simulated with a semi-random source at no more than polynomial factor loss in efficiency. i.e. RP=SRP. This contrasts with the fact that a semi-random source is too weak to simulate fair coin flips [SV]

The Two-Processor Scheduling Problem is in Random NC

An efficient parallel algorithm $(RNC^{2})$ for the two-processor scheduling problem is presented. An interesting feature of this algorithm is that it finds a highest level first schedule: such a schedule defines a lexicographically first solution to this problem in a natural way. A key ingredient of the algorithm is a generalization of a theorem of Tutte which establishes a one-to-one correspondence between the bases of the Tutte matrix of a graph and the sets of matches nodes in maximum matchings in the graph

Quantum State Description Complexity (Invited Talk)

Quantum states generally require exponential sized classical descriptions, but the long conjectured area law provides hope that a large class of natural quantum states can be described succinctly. Recent progress in formally proving the area law is described

1. In t reduction Efficiency Considerations in Using Semi-random Sources.

Randomness is an important computational resource, and has found application in such diverse computational tasks as combinatorial algorithms, syn-chronization and deadlock resolution protocols, encrypting data and cryptographic protocols. Blum [Bl] pointed out the fundamental fact that whereas all these applications of randomness assume a source of independent, unbiased bits, the available physical sources of randomness (such as zener diodes) suffer seriously from problems of correlation. A general Permission to copy without fee all or port of this material is granted provided that the copies are not made or distributed for direct commercial advantage, the ACM coPyright notice and the title of the publication and its date appear. and no&e is given that copy-ing is by Permission of the Association for Compting Machinery. To copy otherwise, or to republish, requires a fee and/or specific permis-sion

NC Algorithms for Comparability Graphs, Interval Graphs, and Unique Perfect Matchings

Laszlo Lovasz recently posed the following problem: "Is there an NC algorithm for testing if a given graph has a unique perfect matching ?" We present such an algorithm for bipartite graphs. We also give NC algorithms for obtaining a transitive orientation of a comparability graph, and an interval representation of an interval graph. These enable us to obtain an NC algorithm for finding a maximum matching in an incomparability graph. 1 Introduction Karp, Upfal and Wigderson [9] have recently shown that the maximum matching problem is in Random NC 3 (RNC 3 ). This result has since been improved to RNC 2 by Mulmuley, Vazirani, and Vazirani [16]. It remains open whether there is a deterministic NC algorithm for this problem. A first step might be to obtain an NC algorithm for testing if a graph has a perfect matching. An RNC algorithm for this problem exists, based on a method of Lovasz [13] (see [1]). Rabin and Vazirani [18] give an NC algorithm for obtaining perfect matchings in..