The quantum adversary method is one of the most versatile lower-bound 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.

We consider the problem of bounded-error 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 bounded-error 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 shared-randomness protocols by quantum protocols without shared randomness. Combined with an earlier separation in the other direction due to Bar-Yossef et al., this shows that the quantum SMP model is incomparable with the classical shared-randomness 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.

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.

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.

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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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 max-cut problem studied by Goemans and Williamson [6]. Since the problem is NP-hard 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 worst-case performance ratio at least alpha = 0.87856... . This is a generalization...

In communication, we usually encode our messages using the letters in a particular alphabet for transmission in some kind of channels. However, with the presence of noise in the channel, it might be the case when some letters can be confused with others. Assume some pairs of letters are similar in representation, and the noise may flip a letter into another if their representation is similar, but can’t flip those that are “different”. In this case, we want to find a set of messages which are pairwise “different”, such that we can distinguish them even if noise is present. We can model the similarity between the letters with a graph G, whose vertices are the letter and in which adjacency means that the letters can be confused. Then the maximum number of one-letter messages that can be sent without danger of confusion is obviously α(G), which is the maximum independent set number introduced in Lecture 23. For longer messages, we define the graph product to model the graph representing the danger of confusion between messages: G × G = {(V × V, E 2)} where E2 is the set of vertices with at least one coordinate being similar, i.e.

Abstract. The theta bodies of a polynomial ideal are a series of semidefinite programming relaxations of the convex hull of the real variety of the ideal. In this paper we construct the theta bodies of the vanishing ideal of cycles in a binary matroid. Applied to cuts in graphs, this yields a new hierarchy of semidefinite programming relaxations of the cut polytope of the graph. If the binary matroid avoids certain minors we can characterize when the first theta body in the hierarchy equals the cycle polytope of the matroid. Specialized to cuts in graphs, this result solves a problem posed by Lovász. 1.

This survey paper is intended for the graduate students and researchers who are interested in Operations Research, have solid understanding of linear optimization but are not familiar with Semidefinite Programming (SDP). Here, I provide a very gentle introduction to SDP, some entry points for further look into the SDP literature, and brief introductions to some selected well-known applications which may be attractive to such audience and in turn motivate them to learn more about semidefinite optimization.