Solving Set Cover with Pairs Problem using Quantum Annealing

Here we consider using quantum annealing to solve Set Cover with Pairs (SCP), an NP-hard combinatorial optimization problem that plays an important role in networking, computational biology, and biochemistry. We show an explicit construction of Ising Hamiltonians whose ground states encode the solution of SCP instances. We numerically simulate the time-dependent Schrödinger equation in order to test the performance of quantum annealing for random instances and compare with that of simulated annealing. We also discuss explicit embedding strategies for realizing our Hamiltonian construction on the D-wave type restricted Ising Hamiltonian based on Chimera graphs. Our embedding on the Chimera graph preserves the structure of the original SCP instance and in particular, the embedding for general complete bipartite graphs and logical disjunctions may be of broader use than that the specific problem we deal with

Parallel Metric Tree Embedding based on an Algebraic View on Moore-Bellman-Ford

A \emph{metric tree embedding} of expected \emph{stretch~$\alpha \geq 1$} maps a weighted $n$-node graph $G = (V, E, \omega)$ to a weighted tree $T = (V_T, E_T, \omega_T)$ with $V \subseteq V_T$ such that, for all $v,w \in V$, $\operatorname{dist}(v, w, G) \leq \operatorname{dist}(v, w, T)$ and $operatorname{E}[\operatorname{dist}(v, w, T)] \leq \alpha \operatorname{dist}(v, w, G)$. Such embeddings are highly useful for designing fast approximation algorithms, as many hard problems are easy to solve on tree instances. However, to date the best parallel $(\operatorname{polylog} n)$-depth algorithm that achieves an asymptotically optimal expected stretch of $\alpha \in \operatorname{O}(\log n)$ requires $\operatorname{\Omega}(n^2)$ work and a metric as input. In this paper, we show how to achieve the same guarantees using $\operatorname{polylog} n$ depth and $\operatorname{\tilde{O}}(m^{1+\epsilon})$ work, where $m = |E|$ and $\epsilon > 0$ is an arbitrarily small constant. Moreover, one may further reduce the work to $\operatorname{\tilde{O}}(m + n^{1+\epsilon})$ at the expense of increasing the expected stretch to $\operatorname{O}(\epsilon^{-1} \log n)$. Our main tool in deriving these parallel algorithms is an algebraic characterization of a generalization of the classic Moore-Bellman-Ford algorithm. We consider this framework, which subsumes a variety of previous "Moore-Bellman-Ford-like" algorithms, to be of independent interest and discuss it in depth. In our tree embedding algorithm, we leverage it for providing efficient query access to an approximate metric that allows sampling the tree using $\operatorname{polylog} n$ depth and $\operatorname{\tilde{O}}(m)$ work. We illustrate the generality and versatility of our techniques by various examples and a number of additional results

Constant-Factor FPT Approximation for Capacitated k-Median

Capacitated k-median is one of the few outstanding optimization problems for which the existence of a polynomial time constant factor approximation algorithm remains an open problem. In a series of recent papers algorithms producing solutions violating either the number of facilities or the capacity by a multiplicative factor were obtained. However, to produce solutions without violations appears to be hard and potentially requires different algorithmic techniques. Notably, if parameterized by the number of facilities k, the problem is also W[2] hard, making the existence of an exact FPT algorithm unlikely. In this work we provide an FPT-time constant factor approximation algorithm preserving both cardinality and capacity of the facilities. The algorithm runs in time 2^O(k log k) n^O(1) and achieves an approximation ratio of 7+epsilon

Fast Shortest Path Distance Estimation in Large Networks

We study the problem of preprocessing a large graph so that point-to-point shortest-path queries can be answered very fast. Computing shortest paths is a well studied problem, but exact algorithms do not scale to huge graphs encountered on the web, social networks, and other applications. In this paper we focus on approximate methods for distance estimation, in particular using landmark-based distance indexing. This approach involves selecting a subset of nodes as landmarks and computing (offline) the distances from each node in the graph to those landmarks. At runtime, when the distance between a pair of nodes is needed, we can estimate it quickly by combining the precomputed distances of the two nodes to the landmarks. We prove that selecting the optimal set of landmarks is an NP-hard problem, and thus heuristic solutions need to be employed. Given a budget of memory for the index, which translates directly into a budget of landmarks, different landmark selection strategies can yield dramatically different results in terms of accuracy. A number of simple methods that scale well to large graphs are therefore developed and experimentally compared. The simplest methods choose central nodes of the graph, while the more elaborate ones select central nodes that are also far away from one another. The efficiency of the suggested techniques is tested experimentally using five different real world graphs with millions of edges; for a given accuracy, they require as much as 250 times less space than the current approach in the literature which considers selecting landmarks at random. Finally, we study applications of our method in two problems arising naturally in large-scale networks, namely, social search and community detection.Yahoo! Research (internship

A comparison between two representatives of a set of graphs: median vs barycenter graph

In this paper we consider two existing methods to generate a representative of a given set of graphs, that satisfy the following two conditions. On the one hand, that they are applicable to graphs with any kind of labels in nodes and edges and on the other hand, that they can handle relatively large amount of data. Namely, the approximated algorithms to compute the Median Graph via graph embedding and a new method to compute the Barycenter Graph. Our contribution is to give a new algorithm for the barycenter computation and to compare it to the median Graph. To compare these two representatives, we take into account algorithmic considerations and experimental results on the quality of the representative and its robustness, on several datasets.Preprin

A comparison between two representatives of a set of graphs: median vs barycenter graph

Trabajo presentado al Joint IAPR International Workshop on Structural, Syntactic and Statistical Pattern Recognition (SSPR&SPR) celebrado en Esmirna (Turquía) del 18 al 20 de agosto de 2010.In this paper we consider two existing methods to generate a representative of a given set of graphs, that satisfy the following two conditions. On the one hand, that they are applicable to graphs with any kind of labels in nodes and edges and on the other hand, that they can handle relatively large amount of data. Namely, the approximated algorithms to compute the Median Graph via graph embedding and a new method to compute the Barycenter Graph. Our contribution is to give a new algorithm for the barycenter computation and to compare it to the median Graph. To compare these two representatives, we take into account algorithmic considerations and experimental results on the quality of the representative and its robustness, on several datasets.This work was supported by projects: 'CONSOLIDER-INGENIO 2010 Multimodal interaction in pattern recognition and computer vision' (V-00069), 'Robotica ubicua para entornos urbanos' (J-01225).Peer Reviewe