Compression with wildcards: All induced metric subgraphs
Marcel Wild
TL;DR
The paper tackles the problem of enumerating metric and geodesically-convex vertex subsets in graphs, introducing two algorithms: AllMetricSets for compressed enumeration of metric subsets $X\subseteq V$ and AllGeConvSets for output-polynomial enumeration of geodesically-convex subsets. It establishes theoretical results showing that GeConv$(G)$ can be enumerated in $O(Nn^6)$ time and AccMet$(G)$ in $O(N^2n^3)$ time, with special graph classes like distance-hereditary and Ptolemaic graphs yielding further structure (e.g., Met$(G)=Conn(G)$). The work extends from trees, where all three notions coincide, to general graphs by reducing metric-set enumeration to Boolean superclauses and using 012-row representations to compress the model sets, complemented by detailed numerical experiments. The findings demonstrate the viability and limitations of compressed and polynomial-time enumeration for metric and geodesically-convex sets, with practical implications for network analysis and related applications.
Abstract
Driven by applications in the natural, social and computer sciences several algorithms have been proposed to enumerate all sets $X\s V$ of vertices of a graph $G=(V,E)$ that induce a {\it connected} subgraph. We offer two algorithms for enumerating all $X$'s that induce (more exquisite) {\it metric} subgraphs. Specifically, the first algorithm, called {\tt AllMetricSets}, generates these $X$'s in a compressed format. The second algorithm generates all (accessible) metric sets one-by-one but is provably output-polynomial. Mutatis mutandis the same holds for the geodesically-convex sets $X\s V$, this being a natural strengthening of "metric". The Mathematica command {\tt BooleanConvert} features prominently.