Light Spanners

A $t$-spanner of a weighted undirected graph $G=(V,E)$, is a subgraph $H$ such that $d_H(u,v)\le t\cdot d_G(u,v)$ for all $u,v\in V$. The sparseness of the spanner can be measured by its size (the number of edges) and weight (the sum of all edge weights), both being important measures of the spanner's quality -- in this work we focus on the latter. Specifically, it is shown that for any parameters $k\ge 1$ and $\epsilon>0$, any weighted graph $G$ on $n$ vertices admits a $(2k-1)\cdot(1+\epsilon)$-stretch spanner of weight at most $w(MST(G))\cdot O_\epsilon(kn^{1/k}/\log k)$, where $w(MST(G))$ is the weight of a minimum spanning tree of $G$. Our result is obtained via a novel analysis of the classic greedy algorithm, and improves previous work by a factor of $O(\log k)$.Comment: 10 pages, 1 figure, to appear in ICALP 201

Weak Fourier-Schur sampling, the hidden subgroup problem, and the quantum collision problem

Schur duality decomposes many copies of a quantum state into subspaces labeled by partitions, a decomposition with applications throughout quantum information theory. Here we consider applying Schur duality to the problem of distinguishing coset states in the standard approach to the hidden subgroup problem. We observe that simply measuring the partition (a procedure we call weak Schur sampling) provides very little information about the hidden subgroup. Furthermore, we show that under quite general assumptions, even a combination of weak Fourier sampling and weak Schur sampling fails to identify the hidden subgroup. We also prove tight bounds on how many coset states are required to solve the hidden subgroup problem by weak Schur sampling, and we relate this question to a quantum version of the collision problem.Comment: 21 page

An efficient quantum algorithm for the hidden subgroup problem in extraspecial groups

Extraspecial groups form a remarkable subclass of p-groups. They are also present in quantum information theory, in particular in quantum error correction. We give here a polynomial time quantum algorithm for finding hidden subgroups in extraspecial groups. Our approach is quite different from the recent algorithms presented in [17] and [2] for the Heisenberg group, the extraspecial p-group of size p3 and exponent p. Exploiting certain nice automorphisms of the extraspecial groups we define specific group actions which are used to reduce the problem to hidden subgroup instances in abelian groups that can be dealt with directly.Comment: 10 page

The Optimal Single Copy Measurement for the Hidden Subgroup Problem

The optimization of measurements for the state distinction problem has recently been applied to the theory of quantum algorithms with considerable successes, including efficient new quantum algorithms for the non-abelian hidden subgroup problem. Previous work has identified the optimal single copy measurement for the hidden subgroup problem over abelian groups as well as for the non-abelian problem in the setting where the subgroups are restricted to be all conjugate to each other. Here we describe the optimal single copy measurement for the hidden subgroup problem when all of the subgroups of the group are given with equal a priori probability. The optimal measurement is seen to be a hybrid of the two previously discovered single copy optimal measurements for the hidden subgroup problem.Comment: 8 pages. Error in main proof fixe

Ontology-based data access with databases: a short course

Ontology-based data access (OBDA) is regarded as a key ingredient of the new generation of information systems. In the OBDA paradigm, an ontology defines a high-level global schema of (already existing) data sources and provides a vocabulary for user queries. An OBDA system rewrites such queries and ontologies into the vocabulary of the data sources and then delegates the actual query evaluation to a suitable query answering system such as a relational database management system or a datalog engine. In this chapter, we mainly focus on OBDA with the ontology language OWL 2QL, one of the three profiles of the W3C standard Web Ontology Language OWL 2, and relational databases, although other possible languages will also be discussed. We consider different types of conjunctive query rewriting and their succinctness, different architectures of OBDA systems, and give an overview of the OBDA system Ontop

Jamming Model for the Extremal Optimization Heuristic

Extremal Optimization, a recently introduced meta-heuristic for hard optimization problems, is analyzed on a simple model of jamming. The model is motivated first by the problem of finding lowest energy configurations for a disordered spin system on a fixed-valence graph. The numerical results for the spin system exhibit the same phenomena found in all earlier studies of extremal optimization, and our analytical results for the model reproduce many of these features.Comment: 9 pages, RevTex4, 7 ps-figures included, as to appear in J. Phys. A, related papers available at http://www.physics.emory.edu/faculty/boettcher

Computational Indistinguishability between Quantum States and Its Cryptographic Application

We introduce a computational problem of distinguishing between two specific quantum states as a new cryptographic problem to design a quantum cryptographic scheme that is "secure" against any polynomial-time quantum adversary. Our problem, QSCDff, is to distinguish between two types of random coset states with a hidden permutation over the symmetric group of finite degree. This naturally generalizes the commonly-used distinction problem between two probability distributions in computational cryptography. As our major contribution, we show that QSCDff has three properties of cryptographic interest: (i) QSCDff has a trapdoor; (ii) the average-case hardness of QSCDff coincides with its worst-case hardness; and (iii) QSCDff is computationally at least as hard as the graph automorphism problem in the worst case. These cryptographic properties enable us to construct a quantum public-key cryptosystem, which is likely to withstand any chosen plaintext attack of a polynomial-time quantum adversary. We further discuss a generalization of QSCDff, called QSCDcyc, and introduce a multi-bit encryption scheme that relies on similar cryptographic properties of QSCDcyc.Comment: 24 pages, 2 figures. We improved presentation, and added more detail proofs and follow-up of recent wor

On the complexity of 2d discrete fixed point problem

Abstract. We study a computational complexity version of the two dimensional Sperner problem, which states that any three coloring of vertices of a triangulated triangle, satisfying some boundary conditions, will have a trichromatic triangle. In introducing a complexity class PPAD, Papadimitriou [9] proved a three dimensional analogue is PPAD-complete about fifteen years ago. The complexity of 2D-SPERNER itself has remained open since then. We settle this open problem with a PPAD-complete proof. The result also allows us to derive the computational complexity characterization of a discrete version of the 2-dimensional Brouwer fixed point problem, improving a recent result of Daskalakis, Goldberg and Papadimitriou [4]. Those hardness results for the simplest version of those problems provide very useful tools to the study of other important problems in the PPAD class.