Unique (optimal) solutions: Complexity results for identifying and locating–dominating codes

Complexity results for identifying codes in planar graphs

Let G be a simple, undirected, connected graph with vertex set V (G) and C ⊆ V (G) be a set of vertices whose elements are called codewords. For v ∈ V (G) and r ≥ 1, let us denote by I r (v) the set of codewords c ∈ C such that d(v, c) ≤ r, where the distance d(v, c) is defined as the length of a shortest path between v and c. More generally, for A ⊆ V (G), we define I r (A) = ∪v∈AI C r (v), wh...

Optimal linear identifying codes

Complexity Results for Compressing Optimal Paths

In this work we give a first tractability analysis of Compressed Path Databases, space efficient oracles used to very quickly identify the first arc on a shortest path. We study the complexity of computing an optimal compressed path database for general directed and undirected graphs. We find that in both cases the problem is NP-complete. We also show that, for graphs which can be decomposed al...

Hardness results and approximation algorithms for identifying codes and locating-dominating codes in graphs

In a graph G = (V, E), an identifying code of G (resp. a locating-dominating code of G) is a subset of vertices C ⊆ V such that N [v]∩C 6= ∅ for all v ∈ V , and N [u] ∩C 6= N [v]∩C for all u 6= v, u, v ∈ V (resp. u, v ∈ V r C), where N [u] denotes the closed neighbourhood of v, that is N [u] = N(u) ∪ {u}. These codes model fault-detection problems in multiprocessor systems and are also used for...

Low Complexity Algorithm for Optimal Hard Decoding of Convolutional Codes

It is well known that convolutional codes can be optimally decoded by the Viterbi Algorithm (VA). We propose an optimal hard decoding technique where the VA is applied to identify the error vector rather than the information message. In this paper, we show that, with this type of decoding, the exhaustive computation of a vast majority of state to state iterations is unnecessary. Hence, under ce...