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Summary: On the solitaire cone and its relationship to
multicommodity flows
David AVIS Antoine DEZA
January 18, 1999
Abstract
The classical game of Peg Solitaire has uncertain origins, but was certainly popular by the
time of Louis XIV, and was described by Leibniz in 1710. The modern mathematical study
of the game dates to the 1960s, when the solitaire cone was first described by Boardman and
Conway. Valid inequalities over this cone, known as pagoda functions, were used to show the
infeasibility of various peg games. In this paper we study the extremal structure of solitaire
cones for a variety of boards, and relate their structure to the well studied metric cone. In
particular we give:
1. an equivalence between the multicommodity flow problem with associated dual metric
cone and a generalized peg game with associated solitaire cone;
2. a related NPcompleteness result;
3. a method of generating large classes of facets;
4. a complete characterization of 01 facets;
5. exponential upper and lower bounds (in the dimension) on the number of facets;
6. results on the number of facets, incidence and adjacency relationships and diameter for
small rectangular, toric and triangular boards;
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