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37
Approximate graph coloring by semidefinite programming
 Proc. 35 th IEEE FOCS, IEEE
, 1994
"... a coloring is called the chromatic number of�, and is usually denoted by��.Determining the chromatic number of a graph is known to be NPhard (cf. [19]). Besides its theoretical significance as a canonical NPhard problem, graph coloring arises naturally in a variety of applications such as register ..."
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Cited by 180 (7 self)
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a coloring is called the chromatic number of�, and is usually denoted by��.Determining the chromatic number of a graph is known to be NPhard (cf. [19]). Besides its theoretical significance as a canonical NPhard problem, graph coloring arises naturally in a variety of applications such as register allocation [11, 12, 13] is the maximum degree of any vertex. Beand timetable/examination scheduling [8, 40]. In many We consider the problem of coloring�colorable graphs with the fewest possible colors. We give a randomized polynomial time algorithm which colors a 3colorable graph on vertices with� � ���� colors where sides giving the best known approximation ratio in terms of, this marks the first nontrivial approximation result as a function of the maximum degree. This result can be generalized to�colorable graphs to obtain a coloring using�� � ��� � � � �colors. Our results are inspired by the recent work of Goemans and Williamson who used an algorithm for semidefinite optimization problems, which generalize linear programs, to obtain improved approximations for the MAX CUT and MAX 2SAT problems. An intriguing outcome of our work is a duality relationship established between the value of the optimum solution to our semidefinite program and the Lovász�function. We show lower bounds on the gap between the optimum solution of our semidefinite program and the actual chromatic number; by duality this also demonstrates interesting new facts about the�function. 1
Quantum vs. classical communication and computation
 Proc. 30th Ann. ACM Symp. on Theory of Computing (STOC ’98
, 1998
"... We present a simple and general simulation technique that transforms any blackbox quantum algorithm (à la Grover’s database search algorithm) to a quantum communication protocol for a related problem, in a way that fully exploits the quantum parallelism. This allows us to obtain new positive and ne ..."
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Cited by 126 (15 self)
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We present a simple and general simulation technique that transforms any blackbox quantum algorithm (à la Grover’s database search algorithm) to a quantum communication protocol for a related problem, in a way that fully exploits the quantum parallelism. This allows us to obtain new positive and negative results. The positive results are novel quantum communication protocols that are built from nontrivial quantum algorithms via this simulation. These protocols, combined with (old and new) classical lower bounds, are shown to provide the first asymptotic separation results between the quantum and classical (probabilistic) twoparty communication complexity models. In particular, we obtain a quadratic separation for the boundederror model, and an exponential separation for the zeroerror model. The negative results transform known quantum communication lower bounds to computational lower bounds in the blackbox model. In particular, we show that the quadratic speedup achieved by Grover for the OR function is impossible for the PARITY function or the MAJORITY function in the boundederror model, nor is it possible for the OR function itself in the exact case. This dichotomy naturally suggests a study of boundeddepth predicates (i.e. those in the polynomial hierarchy) between OR and MAJORITY. We present blackbox algorithms that achieve near quadratic speed up for all such predicates.
Consequences and Limits of Nonlocal Strategies
, 2010
"... Thispaperinvestigatesthepowersandlimitationsofquantum entanglementinthecontext of cooperative games of incomplete information. We give several examples of such nonlocal games where strategies that make use of entanglement outperform all possible classical strategies. One implication ofthese examples ..."
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Cited by 72 (17 self)
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Thispaperinvestigatesthepowersandlimitationsofquantum entanglementinthecontext of cooperative games of incomplete information. We give several examples of such nonlocal games where strategies that make use of entanglement outperform all possible classical strategies. One implication ofthese examplesis that entanglement canprofoundly affectthesoundness property of twoprover interactive proof systems. We then establish limits on the probability with which strategies making use of entanglement can win restricted types of nonlocal games. These upperbounds mayberegardedasgeneralizationsof Tsirelsontypeinequalities, which place bounds on the extent to which quantum information can allow for the violation of Bell inequalities. We also investigate the amount of entanglement required by optimal and nearly optimal quantum strategies forsome games.
Semidefinite Programming and Integer Programming
"... We survey how semidefinite programming can be used for finding good approximative solutions to hard combinatorial optimization problems. ..."
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Cited by 48 (7 self)
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We survey how semidefinite programming can be used for finding good approximative solutions to hard combinatorial optimization problems.
The Cost of Exactly Simulating Quantum Entanglement With Classical Communication
, 1999
"... We investigate the amount of communication that must augment classical local hidden variable models in order to simulate the behaviour of entangled quantum systems. We consider the scenario where a bipartite measurement is given from a set of possibilities and the goal is to obtain exactly the same ..."
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Cited by 39 (11 self)
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We investigate the amount of communication that must augment classical local hidden variable models in order to simulate the behaviour of entangled quantum systems. We consider the scenario where a bipartite measurement is given from a set of possibilities and the goal is to obtain exactly the same correlations that arise when the actual quantum system is measured. We show that, in the case of a single pair of qubits in a Bell state, a constant number of bits of communication is always sufficientregardless of the number of measurements under consideration. We also show that, in the case of a system of n Bell states, a constant times 2 n bits of communication are necessary. 1 Introduction Bell's celebrated theorem [1] shows that certain scenarios involving bipartite quantum measurements result in correlations that are impossible to simulate with a classical system if the measurement events are spacelike separated. If the measurement events are timelike separated then classical s...
The Lovász theta function and a semidefinite programming relaxation of vertex cover
 SIAM JOURNAL ON DISCRETE MATHEMATICS
, 1998
"... Let vc(G) denote the minimum size of a vertex cover of a graph G =(V,E). It is well known that one can approximate vc(G) to within a factor of 2 in polynomial time; and despite considerable investigation, no (2−ε)approximation algorithm has been found for any ε>0. Because of the many connections ..."
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Cited by 38 (1 self)
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Let vc(G) denote the minimum size of a vertex cover of a graph G =(V,E). It is well known that one can approximate vc(G) to within a factor of 2 in polynomial time; and despite considerable investigation, no (2−ε)approximation algorithm has been found for any ε>0. Because of the many connections between the independence number α(G) and the Lovász theta function ϑ(G), and because vc(G) =V −α(G), it is natural to ask how well V −ϑ(G) approximates vc(G). It is not difficult to show that these quantities are within a factor of 2 of each other (V −ϑ(G) is never less than the value of the canonical linear programming relaxation of vc(G)); our main result is that vc(G) can be more than (2 − ε) times V −ϑ(G) for any ε>0. We also investigate a stronger lower bound than V −ϑ(G) for vc(G).
Quantum Communication and Complexity
 Theoretical Computer Science
, 2000
"... In the setting of communication complexity, two distributed parties want to compute a function depending on both their inputs, using as little communication as possible. The required communication can sometimes be significantly lowered if we allow the parties the use of quantum communication. We sur ..."
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Cited by 32 (13 self)
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In the setting of communication complexity, two distributed parties want to compute a function depending on both their inputs, using as little communication as possible. The required communication can sometimes be significantly lowered if we allow the parties the use of quantum communication. We survey the main results of the young area of quantum communication complexity: its relation to teleportation and dense coding, the main examples of fast quantum communication protocols, lower bounds, and some applications. 1 Introduction The area of communication complexity deals with the following type of problem. There are two separated parties, called Alice and Bob. Alice receives some input x 2 X, Bob receives some y 2 Y , and together they want to compute some function f(x; y). As the value f(x; y) will generally depend on both x and y, neither Alice nor Bob will have sufficient information to do the computation by themselves, so they will have to communicate in order to achieve their go...
Approximating the independence number via the ϑfunction
, 1994
"... We study the relationship between the independence number of a graph and its semidefinite relaxation, the Lov'asz `function. We deduce an improved approximation algorithm for the independence number. If a graph on n vertices has an independence number n=k + m, for some fixed integer k 3 and some ..."
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Cited by 31 (5 self)
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We study the relationship between the independence number of a graph and its semidefinite relaxation, the Lov'asz `function. We deduce an improved approximation algorithm for the independence number. If a graph on n vertices has an independence number n=k + m, for some fixed integer k 3 and some m ? 0, the algorithm finds, in random polynomial time, an independent set of size ~ \Omega\Gamma m 3=(k+1) ). This is the first improvement upon the Ramsey Theory based algorithm of Boppana and Halldorsson that finds an independent set of size\Omega\Gamma m 1=(k\Gamma1) ) in such a graph. The algorithm is based on semidefinite programming, some properties of the `function, and the recent algorithm of Karger, Motwani and Sudan for approximating the chromatic number of a graph. If the `function of an n vertex graph is at least Mn 1\Gamma2=h , for some absolute constant M , we describe another, related algorithm that finds an independent set of size h. Finally, while it is e...
Quantum Computing and Communication Complexity
 EATCS Bulletin
, 2000
"... Quantum computing combines the framework of quantum mechanics with that of computer science. In this paper we give a short introduction to quantum computing and survey the results in the area of quantum communication complexity. 1 ..."
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Cited by 14 (0 self)
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Quantum computing combines the framework of quantum mechanics with that of computer science. In this paper we give a short introduction to quantum computing and survey the results in the area of quantum communication complexity. 1
Quantum communication complexity
 Foundations of Physics
"... Can quantum communication be more efficient than its classical counterpart? Holevo’s theorem rules out the possibility of communicating more than n bits of classical information by the transmission of n quantum bits—unless the two parties are entangled, in which case twice as many classical bits can ..."
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Cited by 12 (6 self)
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Can quantum communication be more efficient than its classical counterpart? Holevo’s theorem rules out the possibility of communicating more than n bits of classical information by the transmission of n quantum bits—unless the two parties are entangled, in which case twice as many classical bits can be communicated but no more. In apparent contradiction, there are distributed computational tasks for which quantum communication cannot be simulated efficiently by classical means. In some cases, the effect of transmitting quantum bits cannot be achieved classically short of transmitting an exponentially larger number of bits. In a similar vein, can entanglement be used to save on classical communication? It is well known that entanglement on its own is useless for the transmission of information. Yet, there are distributed tasks that cannot be accomplished at all in a classical world when communication is not allowed, but that become possible if the noncommunicating parties share prior entanglement. This leads to the question of how expensive it is, in terms of classical communication, to provide an exact simulation of the spooky power of entanglement. KEY WORDS: Bell’s theorem; communication complexity; distributed computation; entanglement simulation; pseudotelepathy; spooky communication.