Abstract
Given an edge-weighted graph G and ε > 0, a (1 + ε)-spanner is a spanning subgraph G′ whose shortest path distances approximate those of G within a factor of 1 + ε. For G from certain graph families (such as bounded genus graphs and apex graphs), we know that light spanners exist. That is, we can compute a (1 + ε)-spanner G′ with total edge weight at most a constant times the weight of a minimum spanning tree. This constant may depend on ε and the graph family, but not on the particular graph G nor on the edge weighting. The existence of light spanners is essential in the design of approximation schemes for the metric TSP (the traveling salesman problem) and similar graph-metric problems.
In this paper we make some progress towards the conjecture that light spanners exist for every minor-closed graph family: we show that light spanners exist for graphs with bounded pathwidth, and they are computed by a greedy algorithm. We do this via the intermediate construction of light monotone spanning trees in such graphs.
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Grigni, M., Hung, HH. (2012). Light Spanners in Bounded Pathwidth Graphs. In: Rovan, B., Sassone, V., Widmayer, P. (eds) Mathematical Foundations of Computer Science 2012. MFCS 2012. Lecture Notes in Computer Science, vol 7464. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-32589-2_42
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DOI: https://doi.org/10.1007/978-3-642-32589-2_42
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