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26
All quantum adversary methods are equivalent
 THEORY OF COMPUTING
, 2006
"... The quantum adversary method is one of the most versatile lowerbound methods for quantum algorithms. We show that all known variants of this method are equivalent: spectral adversary (Barnum, Saks, and Szegedy, 2003), weighted adversary (Ambainis, 2003), strong weighted adversary (Zhang, 2005), an ..."
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Cited by 29 (5 self)
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The quantum adversary method is one of the most versatile lowerbound methods for quantum algorithms. We show that all known variants of this method are equivalent: spectral adversary (Barnum, Saks, and Szegedy, 2003), weighted adversary (Ambainis, 2003), strong weighted adversary (Zhang, 2005), and the Kolmogorov complexity adversary (Laplante and Magniez, 2004). We also present a few new equivalent formulations of the method. This shows that there is essentially one quantum adversary method. From our approach, all known limitations of these versions of the quantum adversary method easily follow.
Boundederror quantum state identification and exponential separations in communication complexity
 In Proc. of the 38th Symposium on Theory of Computing (STOC
, 2006
"... We consider the problem of boundederror quantum state identification: given either state α0 or state α1, we are required to output ‘0’, ‘1 ’ or ‘? ’ (“don’t know”), such that conditioned on outputting ‘0 ’ or ‘1’, our guess is correct with high probability. The goal is to maximize the probability o ..."
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Cited by 25 (14 self)
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We consider the problem of boundederror quantum state identification: given either state α0 or state α1, we are required to output ‘0’, ‘1 ’ or ‘? ’ (“don’t know”), such that conditioned on outputting ‘0 ’ or ‘1’, our guess is correct with high probability. The goal is to maximize the probability of not outputting ‘?’. We prove a direct product theorem: if we are given two such problems, with optimal probabilities a and b, respectively, and the states in the first problem are pure, then the optimal probability for the joint boundederror state identification problem is O(ab). Our proof is based on semidefinite programming duality. Using this result, we present two exponential separations in the simultaneous message passing model of communication complexity. First, we describe a relation that can be computed with O(log n) classical bits of communication in the presence of shared randomness, but needs Ω(n 1/3) communication if the parties don’t share randomness, even if communication is quantum. This shows the optimality of Yao’s recent exponential simulation of sharedrandomness protocols by quantum protocols without shared randomness. Combined with an earlier separation in the other direction due to BarYossef et al., this shows that the quantum SMP model is incomparable with the classical sharedrandomness SMP model. Second, we describe a relation that can be computed with O(log n) classical bits of communication in the presence of shared entanglement, but needs Ω((n / log n) 1/3) communication if the parties share randomness but no entanglement, even if communication is quantum. This is the first example in communication complexity of a situation where entanglement buys you much more than quantum communication.
Theta Bodies for Polynomial Ideals
, 2008
"... Abstract. Inspired by a question of Lovász, we introduce a hierarchy of nested semidefinite relaxations of the convex hull of real solutions to an arbitrary polynomial ideal, called theta bodies of the ideal. For the stable set problem in a graph, the first theta body in this hierarchy is exactly Lo ..."
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Cited by 17 (3 self)
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Abstract. Inspired by a question of Lovász, we introduce a hierarchy of nested semidefinite relaxations of the convex hull of real solutions to an arbitrary polynomial ideal, called theta bodies of the ideal. For the stable set problem in a graph, the first theta body in this hierarchy is exactly Lovász’s theta body of the graph. We prove that theta bodies are, up to closure, a version of Lasserre’s relaxations for real solutions to ideals, and that they can be computed explicitly using combinatorial moment matrices. Theta bodies provide a new canonical set of semidefinite relaxations for the max cut problem. For vanishing ideals of finite point sets, we give several equivalent characterizations of when the first theta body equals the convex hull of the points. We also determine the structure of the first theta body for all ideals. 1.
Online Oblivious Routing
 In Proceedings of ACM Symposium in Parallelism in Algorithms and Architectures (SPAA
, 2003
"... We consider an online version of the oblivious routing problem. Oblivious routing is the problem of picking a routing between each pair of nodes (or a set of flows), without knowledge of the traffic or demand between each pair, with the goal of minimizing the maximum congestion on any edge in the gr ..."
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Cited by 13 (1 self)
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We consider an online version of the oblivious routing problem. Oblivious routing is the problem of picking a routing between each pair of nodes (or a set of flows), without knowledge of the traffic or demand between each pair, with the goal of minimizing the maximum congestion on any edge in the graph. In the online version of the problem, we consider a "repeatedgame" setting, in which the algorithm is allowed to choose a new routing each night, but is still oblivious to the demands that will occur the next day. The cost of the algorithm at every time step is its competitive ratio, or the ratio of its congestion to the minimum possible congestion for the demands at that time step.
Maximum algebraic connectivity augmentation is NPhard
 Oper. Res. Lett
"... The algebraic connectivity of a graph, which is the secondsmallest eigenvalue of the Laplacian of the graph, is a measure of connectivity. We show that the problem of adding a specified number of edges to an input graph to maximize the algebraic connectivity of the augmented graph is NPhard. ..."
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Cited by 7 (0 self)
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The algebraic connectivity of a graph, which is the secondsmallest eigenvalue of the Laplacian of the graph, is a measure of connectivity. We show that the problem of adding a specified number of edges to an input graph to maximize the algebraic connectivity of the augmented graph is NPhard.
Semidefinite programs for completely bounded norms
, 2009
"... The completely bounded trace and spectral norms in finite dimensions are shown to be expressible by semidefinite programs. This provides an efficient method by which these norms may be both calculated and verified, and gives alternate proofs of some known facts about them. ..."
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Cited by 5 (2 self)
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The completely bounded trace and spectral norms in finite dimensions are shown to be expressible by semidefinite programs. This provides an efficient method by which these norms may be both calculated and verified, and gives alternate proofs of some known facts about them.
Linear vs. Semidefinite Extended Formulations: Exponential Separation and Strong Lower Bounds
, 2011
"... We solve a 20year old problem posed by M. Yannakakis and prove that there exists no polynomialsize linear program (LP) whose associated polytope projects to the traveling salesman polytope, even if the LP is not required to be symmetric. Moreover, we prove that this holds also for the maximum cut ..."
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Cited by 5 (1 self)
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We solve a 20year old problem posed by M. Yannakakis and prove that there exists no polynomialsize linear program (LP) whose associated polytope projects to the traveling salesman polytope, even if the LP is not required to be symmetric. Moreover, we prove that this holds also for the maximum cut problem and the stable set problem. These results follow from a new connection that we make between oneway quantum communication protocols and semidefinite programming reformulations of LPs.
Undecidability of linear inequalities in graph homomorphism densities
 Journal of the American Mathematical Society
"... Many fundamental theorems in extremal graph theory can be expressed as algebraic inequalities between subgraph densities. As is explained below, for dense graphs, it is possible to replace subgraph densities with homomorphism densities. An easy observation shows that one can convert any algebraic in ..."
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Cited by 4 (0 self)
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Many fundamental theorems in extremal graph theory can be expressed as algebraic inequalities between subgraph densities. As is explained below, for dense graphs, it is possible to replace subgraph densities with homomorphism densities. An easy observation shows that one can convert any algebraic inequality between
Approximation Bounds for Quadratic Maximization with Semidefinite Programming Relaxation
, 2003
"... In this paper, we consider a class of quadratic maximization problems. One important instance in that class is the famous quadratic maximization formulation of the maxcut problem studied by Goemans and Williamson [6]. Since the problem is NPhard in general, following Goemans and Williamson, we app ..."
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Cited by 2 (0 self)
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In this paper, we consider a class of quadratic maximization problems. One important instance in that class is the famous quadratic maximization formulation of the maxcut problem studied by Goemans and Williamson [6]. Since the problem is NPhard in general, following Goemans and Williamson, we apply the approximation method based on the semidefinite programming (SDP) relaxation. For a subclass of the problems, including the ones studied by Helmberg [9] and Zhang [23], we show that the SDP relaxation approach yields an approximation solution with the worstcase performance ratio at least alpha = 0.87856... . This is a generalization...