Towards Effective Exact Algorithms for the Maximum Balanced Biclique Problem

The Maximum Balanced Biclique Problem (MBBP) is a prominent model with numerous applications. Yet, the problem is NP-hard and thus computationally challenging. We propose novel ideas for designing effective exact algorithms for MBBP. Firstly, we introduce an Upper Bound Propagation procedure to pre-compute an upper bound involving each vertex. Then we extend an existing branch-and-bound algorithm by integrating the pre-computed upper bounds. We also present a set of new valid inequalities induced from the upper bounds to tighten an existing mathematical formulation for MBBP. Lastly, we investigate another exact algorithm scheme which enumerates a subset of balanced bicliques based on our upper bounds. Experiments show that compared to existing approaches, the proposed algorithms and formulations are more efficient in solving a set of random graphs and large real-life instances

Multiparticle entanglement purification for two-colorable graph states

We investigate multiparticle entanglement purification schemes which allow one to purify all two colorable graph states, a class of states which includes e.g. cluster states, GHZ states and codewords of various error correction codes. The schemes include both recurrence protocols and hashing protocols. We analyze these schemes under realistic conditions and observe for a generic error model that the threshold value for imperfect local operations depends on the structure of the corresponding interaction graph, but is otherwise independent of the number of parties. The qualitative behavior can be understood from an analytically solvable model which deals only with a restricted class of errors. We compare direct multiparticle entanglement purification protocols with schemes based on bipartite entanglement purification and show that the direct multiparticle entanglement purification is more efficient and the achievable fidelity of the purified states is larger. We also show that the purification protocol allows one to produce private entanglement, an important aspect when using the produced entangled states for secure applications. Finally we discuss an experimental realization of a multiparty purification protocol in optical lattices which is issued to improve the fidelity of cluster states created in such systems.Comment: 22 pages, 8 figures; replaced with published versio

Simpler, faster and shorter labels for distances in graphs

We consider how to assign labels to any undirected graph with n nodes such that, given the labels of two nodes and no other information regarding the graph, it is possible to determine the distance between the two nodes. The challenge in such a distance labeling scheme is primarily to minimize the maximum label lenght and secondarily to minimize the time needed to answer distance queries (decoding). Previous schemes have offered different trade-offs between label lengths and query time. This paper presents a simple algorithm with shorter labels and shorter query time than any previous solution, thereby improving the state-of-the-art with respect to both label length and query time in one single algorithm. Our solution addresses several open problems concerning label length and decoding time and is the first improvement of label length for more than three decades. More specifically, we present a distance labeling scheme with label size (log 3)/2 + o(n) (logarithms are in base 2) and O(1) decoding time. This outperforms all existing results with respect to both size and decoding time, including Winkler's (Combinatorica 1983) decade-old result, which uses labels of size (log 3)n and O(n/log n) decoding time, and Gavoille et al. (SODA'01), which uses labels of size 11n + o(n) and O(loglog n) decoding time. In addition, our algorithm is simpler than the previous ones. In the case of integral edge weights of size at most W, we present almost matching upper and lower bounds for label sizes. For r-additive approximation schemes, where distances can be off by an additive constant r, we give both upper and lower bounds. In particular, we present an upper bound for 1-additive approximation schemes which, in the unweighted case, has the same size (ignoring second order terms) as an adjacency scheme: n/2. We also give results for bipartite graphs and for exact and 1-additive distance oracles

Perfectly Secure Communication, based on Graph-Topological Addressing in Unique-Neighborhood Networks

We consider network graphs $G=(V,E)$ in which adjacent nodes share common secrets. In this setting, certain techniques for perfect end-to-end security (in the sense of confidentiality, authenticity (implying integrity) and availability, i.e., CIA+) can be made applicable without end-to-end shared secrets and without computational intractability assumptions. To this end, we introduce and study the concept of a unique-neighborhood network, in which nodes are uniquely identifiable upon their graph-topological neighborhood. While the concept is motivated by authentication, it may enjoy wider applicability as being a technology-agnostic (yet topology aware) form of addressing nodes in a network

A Graph-based Framework for Transmission of Correlated Sources over Broadcast Channels

In this paper we consider the communication problem that involves transmission of correlated sources over broadcast channels. We consider a graph-based framework for this information transmission problem. The system involves a source coding module and a channel coding module. In the source coding module, the sources are efficiently mapped into a nearly semi-regular bipartite graph, and in the channel coding module, the edges of this graph are reliably transmitted over a broadcast channel. We consider nearly semi-regular bipartite graphs as discrete interface between source coding and channel coding in this multiterminal setting. We provide an information-theoretic characterization of (1) the rate of exponential growth (as a function of the number of channel uses) of the size of the bipartite graphs whose edges can be reliably transmitted over a broadcast channel and (2) the rate of exponential growth (as a function of the number of source samples) of the size of the bipartite graphs which can reliably represent a pair of correlated sources to be transmitted over a broadcast channel.Comment: 36 pages, 9 figure