## Solitaire Cones (1996)

Venue: | Discrete Applied Mathematics |

Citations: | 5 - 0 self |

### BibTeX

@TECHREPORT{Avis96solitairecones,

author = {David Avis and Antoine Deza},

title = {Solitaire Cones},

institution = {Discrete Applied Mathematics},

year = {1996}

}

### OpenURL

### 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 NP-completeness result; 3. a method of generating large classes of facets; 4. a complete characterization of 0-1 facets; 5. exponential upper and lower bounds (in the dimension) on the number of facets; 6. results on the number of facets, incidence, adjacency and ...

### Citations

277 | Convex Polytopes - GRÜNBAUM - 1967 |

192 | A pivoting algorithm for convex hulls and vertex enumeration of arrangements and polyhedra, Discrete and Computational Geometry (8 - Avis, Fukuda - 1992 |

192 | Lectures on Polytopes, Graduate Texts in Mathematics 152 - ZIEGLER - 1995 |

58 | lrs: A revised implementation of the reverse search vertex enumeration algorithm - Avis |

46 |
On an extension of the maximum-flow minimum cut theorem to multicommodity flow
- Iri
- 1971
(Show Context)
Citation Context ...+ z st with 1sr ! s ! tsng: The Japanese theorem of Iri and Onaga and Kakusho gives a necessary and sufficient condition for the feasibility of the fractional multicommodity flow problem. Theorem 2.2 =-=[17, 20]-=- A multiflow problem c; d is fractionally feasible if and only if d\Gammac 2F n . Facets of the flow cone are useful to show the infeasibility of multiflow problems. In the previous example, if we cha... |

40 |
On Feasibility Conditions of Multicommodity Flows in Networks
- Onaga, Kakusho
- 1971
(Show Context)
Citation Context ...+ z st with 1sr ! s ! tsng: The Japanese theorem of Iri and Onaga and Kakusho gives a necessary and sufficient condition for the feasibility of the fractional multicommodity flow problem. Theorem 2.2 =-=[17, 20]-=- A multiflow problem c; d is fractionally feasible if and only if d\Gammac 2F n . Facets of the flow cone are useful to show the infeasibility of multiflow problems. In the previous example, if we cha... |

30 | Combinatorial aspects of convex polytopes, in - Bayer, Lee - 1993 |

30 |
Combinatorial approaches to multiflow problems
- Lomonosov
- 1985
(Show Context)
Citation Context ...t with y rsts0 and 1sr ! s ! tsng: (2.3) F 3 is illustrated in Fig. 2.6. The dual of the flow cone is the much studied metric cone (see for example, Avis [1], Deza, Deza and Fukuda [11] and Lomonosov =-=[19]-=-): M n = fz : z rssz rt + z st with 1sr ! s ! tsng: The Japanese theorem of Iri and Onaga and Kakusho gives a necessary and sufficient condition for the feasibility of the fractional multicommodity fl... |

27 |
The Ins & Outs of Peg Solitaire
- Beasley
- 1985
(Show Context)
Citation Context ...has uncertain origins, and different legends attest to its discovery by various cultures. An authoritative account with a long annotated bibliography can be found in the comprehensive book of Beasley =-=[7]-=-. The book mentions an engraving of Berey, dated 1697, of a lady with a Solitaire board. The book also contains a quotation of Leibniz [18] which was written for the Berlin Academy in 1710. Although s... |

26 | On the extreme rays of the metric cone
- Avis
- 1980
(Show Context)
Citation Context ...the corresponding mn-vectors. Lemma 1.3 For nB ? 3, the solitaire cone is full dimensional, that is, the dimension of SB equals hB the number of holes of the board B. Proof. Its dual S B contains the =-=[1; 2]-=- hB cube. 2 The following result obtained in 1961 is credited to Boardman (who apparently has not published anything on the subject) by Beasley [7], page 87. Theorem 1.4 Equation 1.2, that is, F \Gamm... |

26 |
Geometry of Cuts and Metrics. Algorithms and Combinatorics
- Deza, Laurent
- 1997
(Show Context)
Citation Context ...orem 2.3. 7. The 0-1 facets of the flow cone are the incidence vectors of cuts in the complete graph. The cone generated by these facets is the dual of the well studied cut cone, see Deza and Laurent =-=[12]-=-. Papernov [21] gave a complete characterization of multiflow problems for which the metric cone F n in Theorem 2.2 can be replaced by the dual of the cut cone. For example, single commodity flow prob... |

23 |
A C Implementation of the Reverse Search Vertex Enumeration Algorithm
- Avis
- 1994
(Show Context)
Citation Context ...ious computer programs are available. The results in this paper were obtained using the double description method cdd implemented by Fukuda [13], and the reverse search method lrs implemented by Avis =-=[3]-=-. We made use of these codes to completely generate all facets for some small boards as reported in later sections (such as the 930 048 facets for the toric closure of the 4 by 4 board). For realistic... |

23 | Information bounds are weak in the shortest distance problem
- Graham, Yao, et al.
- 1980
(Show Context)
Citation Context ... 4.4, 3. Their numbers of facets are bounded above and below by an exponential in the dimension, see Lemma 3.7 and Theorem 3.8 and, for the metric cone, Avis [2] (lower bound) and Graham, Yao and Yao =-=[15]-=- (upper bound). 4. Their extremes rays are also extremes rays of their relaxations by their f0; 1g-valued facets, see Theorem 5.1. 5. While the extremes rays of the solitaire cone are conjectured to b... |

22 |
Generalized Hi-Q is NP-complete
- Uehara, Iwata
(Show Context)
Citation Context ...ng configuration Final Figure 1.1: A feasible English solitaire peg game with possible first and last moves The complexity of the feasibility problem for the n by n game was shown by Uehara and Iwata =-=[23]-=- to be NP-complete, so easily checked necessary and sufficient conditions 3 for feasibility are unlikely to exist. One of the earliest tools used to show the infeasibility of certain starting and fini... |

18 |
The cut cone, L 1 embeddability, complexity, and multicommodity flows
- Avis, Deza
- 1991
(Show Context)
Citation Context ...ows), and we will show similarities between this problem and the peg game. For more details on the relationship between the metric cone and multicommodity flows, see the survey paper of Avis and Deza =-=[4]-=-. We give a brief sketch here. 2.1 Metric cone and multicommodity flows Let K n denote the complete graph on n vertices. For each edge ij we define a non-negative integer capacity c ij and demand d ij... |

16 | On skeletons, diameters and volumes of metric polyhedra - Deza, Deza, et al. - 1996 |

15 |
The ridge graph of the metric polytope and some relatives
- Deza, Deza
- 1994
(Show Context)
Citation Context ... solitaire cone Sm\Thetan are adjacent if and only if they are not strongly conflicting, for ns4 two extreme rays of the dual metric cone M n are adjacent if and only if they are not conflicting, see =-=[9]-=-. The following conjectures are based on Remark 4.5 and other similarities between the solitaire cone and the dual metric cone investigated in Section 2. 27 Conjecture 4.6 1. For ms3, the f0; 1g-value... |

15 |
Critical graphs of a given diameter
- Plesńık
- 1975
(Show Context)
Citation Context ...Proof. (1), (2) The adjacency and the diameter are straightforward. (3) The number of 2-faces of a cone is half of the total adjacency of its skeleton. (4) We recall the following result of Plesn' ik =-=[22]-=-: the edge connectivity of a graph of diameter 2 equals its minimum degree. Then, since for ns5 the cone S 1\Thetan has diameter 2 and since c e (S 1\Thetan ) = 2; 3 for n = 3; 4, we have c e (S 1\The... |

13 |
D.S.: Computers and Intractability
- Garey, Johnson
- 1979
(Show Context)
Citation Context ...d \Gamma cs3f 123 + 2f 341 + f 342 = z. The problem is integer feasible and the multiflow is unsaturated as there is a residual capacity of 1 on arc 1; 3. Even, Itai and Shamir (see Garey and Johnson =-=[14]-=- p. 217) proved the following result showing that integer feasibility is an intractable problem in general. Theorem 2.1 It is an NP-complete to decide if c; d are integer feasible for the multiflow pr... |

10 |
On existence of multicommodity flows
- Papernov
(Show Context)
Citation Context ...e 0-1 facets of the flow cone are the incidence vectors of cuts in the complete graph. The cone generated by these facets is the dual of the well studied cut cone, see Deza and Laurent [12]. Papernov =-=[21]-=- gave a complete characterization of multiflow problems for which the metric cone F n in Theorem 2.2 can be replaced by the dual of the cut cone. For example, single commodity flow problems are in thi... |

7 |
Fukuda: On skeletons, diameters and volumes of metric polyhedra
- Deza, Deza, et al.
- 1996
(Show Context)
Citation Context ... X r;s;t y rst f rst with y rsts0 and 1sr ! s ! tsng: (2.3) F 3 is illustrated in Fig. 2.6. The dual of the flow cone is the much studied metric cone (see for example, Avis [1], Deza, Deza and Fukuda =-=[11]-=- and Lomonosov [19]): M n = fz : z rssz rt + z st with 1sr ! s ! tsng: The Japanese theorem of Iri and Onaga and Kakusho gives a necessary and sufficient condition for the feasibility of the fractiona... |

7 | On the solitaire cone and its relationship to multicommodity flows - Avis, Deza - 2001 |

6 |
Extremal Metrics Induced by Graphs
- Avis
- 1980
(Show Context)
Citation Context ...the corresponding mn-vectors. Lemma 1.3 For nB ? 3, the solitaire cone is full dimensional, that is, the dimension of SB equals hB the number of holes of the board B. Proof. Its dual S B contains the =-=[1; 2]-=- hB cube. 2 The following result obtained in 1961 is credited to Boardman (who apparently has not published anything on the subject) by Beasley [7], page 87. Theorem 1.4 Equation 1.2, that is, F \Gamm... |

6 | On the skeleton of the dual cut polytope
- Deza, Deza
- 1994
(Show Context)
Citation Context ...e are conjectured to be of maximum incidence in its relaxation by its f0; 1g-valued facets, the corresponding result is proved for the dual metric cone, see item 2 of Conjecture 5.3 and Deza and Deza =-=[10]-=-, 6. We have M n = CS Ln where L n is the line graph of the complete graph on n nodes, see Theorem 2.3. 7. The 0-1 facets of the flow cone are the incidence vectors of cuts in the complete graph. The ... |

5 |
K.: Purging Pegs Properly. Winning Ways for your mathematical plays. 2
- R, Guy
- 1982
(Show Context)
Citation Context ...ious study of the game is reported in the 19th century, the modern mathematical study of the game dates to the 1960s at Cambridge University. The group was lead by Conway who has written a chapter in =-=[8]-=- on various mathematical aspects of the subject. One of the problems studied by the Cambridge group is the following basic feasibility problem of peg solitaire: For a given board B, starting configura... |

5 |
cdd reference manual, version 0.56
- Fukuda
- 1995
(Show Context)
Citation Context ...f a convex hull or vertex enumeration problem, for which various computer programs are available. The results in this paper were obtained using the double description method cdd implemented by Fukuda =-=[13]-=-, and the reverse search method lrs implemented by Avis [3]. We made use of these codes to completely generate all facets for some small boards as reported in later sections (such as the 930 048 facet... |

5 |
Miscellenea Berolinensia ad incrementum scientarium 1
- Leibniz
- 1710
(Show Context)
Citation Context ...ibliography can be found in the comprehensive book of Beasley [7]. The book mentions an engraving of Berey, dated 1697, of a lady with a Solitaire board. The book also contains a quotation of Leibniz =-=[18]-=- which was written for the Berlin Academy in 1710. Although some serious study of the game is reported in the 19th century, the modern mathematical study of the game dates to the 1960s at Cambridge Un... |

1 | Purging Pegs Prop13 erly, Winning Ways for your mathematical plays - Berlekamp, Guy - 1982 |