Quantum information processing is at the crossroads of physics, mathematics and computer science. It is concerned with what we can and cannot do with quantum information that goes beyond the abilities of classical information processing devices. Communication complexity is an area of classical computer science that aims at quantifying the amount of communication necessary to solve distributed computational problems. Quantum communication complexity uses quantum mechanics to reduce the amount of communication that would be classically required.
Also, Aravind has generalized his own magic square idea @cite_14 to a two-player pseudo-telepathy game in which the players share @math Bell states, @math being an arbitrary odd number larger than 1.
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Bell inequality violations correspond to behavior of entangled quantum systems that cannot be simulated classically. We give two new two-player games with Bell inequality violations that are stronger, fully explicit, and arguably simpler than earlier work.The first game is based on the Hidden Matching problem of quantum communication complexity, introduced by Bar-Yossef, Jayram, and Kerenidis. This game can be won with probability 1 by a quantum strategy using a maximally entangled state with local dimension n (e.g., log n EPR-pairs), while we show that the winning probability of any classical strategy differs from 1 2 by at most O(log(n) sqrt(n)).The second game is based on the integrality gap for Unique Games by Khot and Vishnoi and the quantum rounding procedure of Kempe, Regev, and Toner. Here n-dimensional entanglement allows to win the game with probability 1 (log n)^2, while the best winning probability without entanglement is 1 n. This near-linear ratio ( Bell inequality violation'') is near-optimal, both in terms of the local dimension of the entangled state, and in terms of the number of possible outputs of the two players. Consider the following probabilistic one-player game: The board is a graph with @math vertices, which initially contains no edges. In each step, a new edge is drawn uniformly at random from all nonedges and is presented to the player, henceforth called Painter. Painter must assign one of @math available colors to each edge immediately, where @math is a fixed integer. The game is over as soon as a monochromatic copy of some fixed graph @math has been created, and Painter's goal is to “survive” for as many steps as possible before this happens. We present a new technique for deriving upper bounds on the threshold of this game, i.e., on the typical number of steps Painter will survive with an optimal strategy. More specifically, we consider a deterministic two-player variant of the game where the edges are chosen not randomly, but by a second player Builder. However, Builder has to adhere to the restriction that, for some real number @math , the ratio of edges to vertices in all subgraphs of the evolving board ne... This paper investigates a zero-sum game played on a weighted connected graph @math between two players, the tree player and the edge player. At each play, the tree player chooses a spanning tree @math and the edge player chooses an edge @math . The payoff to the edge player is @math , defined as follows: If @math lies in the tree @math then @math ; if @math does not lie in the tree then @math , where @math is the weight of edge @math and @math is the weight of the unique cycle formed when edge @math is added to the tree @math . The main result is that the value of the game on any @math -vertex graph is bounded above by @math . It is conjectured that the value of the game is @math . The game arises in connection with the @math -server problem on a road network; i.e., a metric space that can be represented as a multigraph @math in which each edge @math represents a road of length @math . It is shown that, if the value of the game on @math is @math , then there is a randomized strategy that achieves a competitive ratio of @math against any oblivious adversary. Thus, on any @math -vertex road network, there is a randomized algorithm for the @math -server problem that is @math competitive against oblivious adversaries. At the heart of the analysis of the game is an algorithm that provides an approximate solution for the simple network design problem. Specifically, for any @math -vertex weighted, connected multigraph, the algorithm constructs a spanning tree @math such that the average, over all edges @math , of @math is less than or equal to @math . This result has potential application to the design of communication networks. It also improves substantially known estimates concerning the existence of a sparse basis for the cycle space of a graph. We study the following one-person game against a random graph process: the Player's goal is to @math -colour a random sequence of edges @math of a complete graph on @math vertices, avoiding a monochromatic triangle for as long as possible. The game is over when a monochromatic triangle is created. The online version of the game requires that the Player should colour each edge as it comes, before looking at the next edge.While it is not hard to prove that the expected length of this game is about @math , the proof of the upper bound suggests the following relaxation: instead of colouring online, the random graph is generated in only two rounds, and the Player colours the edges of the first round before the edges of the second round are thrown in. Given the size of the first round, how many edges can there be in the second round for the Player to be likely to win? In the extreme case, when the first round consists of a random graph with @math edges, where @math is a positive constant, we show that the Player can win with high probability only if constantly many edges are generated in the second round.