It is shown that the problem of computing the minimum distance of a binary linear code is NP-hard, and the corresponding decision problem is NP-complete. This result constitutes a proof of the conjecture of Berlekamp, McEliece, and van Tilborg (1978). Extensions and applications of this result to other problems in coding theory are discussed.

The intractability of computing the minimum distance of a code

Wireless localization techniques are an area that has attracted interest from both industry and academia, with self-localization capability providing a highly desirable characteristic of wireless sensor networks. Localization Algorithms and Strategies for Wireless Sensor Networks encompasses the significant and fast growing area of wireless localization techniques. This book provides comprehensive and up-to-date coverage of topics and fundamental theories underpinning measurement techniques and localization algorithms. A useful compilation for academicians, researchers, and practitioners, this Premier Reference Source contains relevant references and the latest studies emerging out of the wireless sensor network field.

Localization Algorithms and Strategies for Wireless Sensor Networks: Monitoring and Surveillance Techniques for Target Tracking

Minimizing the size of an identifying or locating-dominating code in a graph is NP-hard

We propose a new framework for providing robust location detection in emergency response systems, based on the theory of identifying codes. The key idea of this approach is to allow sensor coverage areas to overlap in such a way that each resolvable position is covered by a unique set of sensors. In this setting, determining a sensor-placement with a minimum number of sensors is equivalent to constructing an optimal identifying code, an NP-complete problem in general. We thus propose and analyze a new polynomial-time algorithm for generating irreducible codes for arbitrary topologies. We also generalize the concept of identifying codes to incorporate robustness properties that are critically needed in emergency networks and provide a polynomial-time algorithm to compute irreducible robust identifying codes. Through analysis and simulation, we show that our approach typically requires significantly fewer sensors than existing proximity-based schemes. Alternatively, for a fixed number of sensors, our scheme can provide robustness in the face of sensor failures or physical damage to the system.

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Robust location detection in emergency sensor networks

We start withan overviewof algorithmiccomplexity problemsin coding theory We then show that the problemof com- puting the minimumdiktanceof a binaryIinwr code is NP-hard,and the corresponding deci~"on problemis W-complete. This constitutes a proof of the conjecture Bedekamp, McEliece,vanTilborg, dating back to 1978. Extensionsand applications of this result to other problemsin codingtheqv are discussed. 1. Introduction. This paper is organized in a manner contrapositive to the talk. In the talk, we give a detailed overview of algorithmic complexity issueain codhg theory. Herein, westart with an overviewof the prominentrole that computational complexity plays in coding theory: we briefly surveythe numerousalgorithmic complexity issuesthat arise in code construction and decoding. Subsequently,we focus on one specific problem. Namely, we consider the problem of computing the minimum distance of a linear code, which is also equivalentto the problem of finding the shortest cycle in a linear matroid over a finite field (90). A long-standing conjecture (12) says that this problem is NP-hard, and we will settle this conjecture in the ai%rmative. We provide a detailed proof. We hope that our proof illustrates some of the elegant techniques used in coding theory today, such as the construction of MDS codes via Vandermonde matri- ces (15, 59, 72) and concatenated coding (23, 31), for instance. It is interesting that algebraic techniques of this kind cart be employed to answer an essentially combinatorial ques- tion. The complexity of computing the minimum distance and the proof of the conjecture of Berlekamp,McEliece, and vanTilborg (12) will be only briefly mentioned in the talk.

/pdf/algorithmic-complexity-in-coding-theory-and-the-minimum-31m8l6zpxu.pdf

Algorithmic complexity in coding theory and the minimum distance problem

Consider a connected undirected graph G = (V, E), a subset of vertices C ⊆ V, and an integer r ≥ 1; for any vertex v ∈ V, let B r (v) denote the ball of radius r centered at v, i.e., the set of all vertices within distance r from v. If for all vertices v ∈ V (respectively, v ∈ V\C), the sets Br(v) ∩ C are all nonempty and different, then we call C an r-identifying code (respectively, an r-locating-dominating code). We study the smallest cardinalities or densities of these codes in chains (finite or infinite) and cycles.

Identifying and locating-dominating codes on chains and cycles

Let G =( V;E) be a connected undirected graph and S a subset of vertices. If for all vertices v 2 V , the sets Br(v) \ S are all nonempty and dierent, where Br(v) denotes the set of all points within distance r from v ,t hen we callS an r-identifying code. We give constructive upper bounds on the best possible density of r-identifying codes in four innite regular graphs, for small values of r.

/pdf/identifying-codes-with-small-radius-in-some-infinite-regular-uoojij8ood.pdf

Antoine Lobstein

Papers

Minimizing the size of an identifying or locating-dominating code in a graph is NP-hard

Identifying and locating-dominating codes on chains and cycles

Identifying Codes with Small Radius in Some Infinite Regular Graphs

On identifying codes.

Extremal cardinalities for identifying and locating-dominating codes in graphs