Imai, Hiroshi

Miyata, Hiroyuki

Moriyama, Sonoko

Kyoto University Research Information Repository

TitleAnalyzing automorphism groups of oriented matroids bysemidefinite programming (Computational Geometry andDiscrete Mathematics)Author(s)Miyata, Hiroyuki; Moriyama, Sonoko; Imai, HiroshiCitation数理解析研究所講究録 (2009), 1641: 8-15Issue Date2009-04URL http://hdl.handle.net/2433/140589RightType Departmental Bulletin PaperTextversionpublisherKyoto UniversityAnalyzing automorphism groups of oriented matroidsby semidefinite programmingHiroyuki Miyata* Sonoko Moriyama\dagger Hiroshi Imai*\dagger \ddaggerhmiyata@is.s.u-tokyo.ac.jp moriso@is.s.u-tokyo.ac.jp imai@is.s.u-tokyo.ac.jpAbstractMany geometric realizability problems such as the realizability problem of ori-ented matroids can be reduced to the Existential Theory of the Reals $(ETR)$ . Theauthors proposed a non-realizability certificate for a symmetric realization problemusing semidefinite programming (SDP) [9]. In this paper, we review this study.In addition, we explain that this method can detect gaps between combinatorialsymmetry and geometric symmetry by analyzing Richter-Gebert’s example [12].1 IntroductionMany geometric realizability problems can be reduced to the Enistential Theory of theReals $(ETR)$ , the problem to decide whether a given polynomial equalities and inequali-ties system with integer coefficients has a real root or not. For example, the realizabilityproblem of oriented matroids can be reduced to ETR. Therefore, one can analyze manygeometric realizability problems in terms of polynomial system, which has been investi-gated largely in computational algebra. There are two main types of approaches to ETR.One is a type of symmbolic approaches such as cylindrical algebraic decomposition [4].We obtain exact result by these approaches, but the time complexity of them are too largeand we cannot apply these approaches to large instances. The other is a type of numericalapproaches such as polynomial optimization using semidefinite programming [8, 11]. Wecan compute more fastly using these approaches, but they produce numerical errors andwe cannot obtain exact results. On the other hand, many geometric realizability problemshave the follwoing features. Firstly, their problem sizes often become large. Therefore,it is difficult to apply symbolic approaches directly to them. Sencondly, results contain-ing numerical errors do not make sense. Therefore, we cannot apply numerical approachdirectly. To overcome this difficulty, Miyata, Moriyama and Imai [9] proposed a numer-ical method to prove infeasibility of polynomial system exactly, which was based on theSDP relaxation proposed by Lasserre [8] and Parrilo [11], and analyzed the realizabilityproblem of oriented matroids using this method.*Department of Computer Science Graduate School of Information Science and Technology, The Uni-versity of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo $113arrow 0033$ , Japan.\dagger Institute of Nano Quantum Information Electronics, The University of Tokyo, 4-6-1, Komaba,Meguro-ku, Tokyo $153arrow 8505$ Japan.$t_{ERATO}$-SORST Quantum Computation and Information Project, Japan Science and TechnologyAgency, Daini Hongo White Bldg.201, 5-28-3, Hongo, Bunkyo-ku, Tokyo 113-0033, Japan.1641 2009 8-15 8In [10], we proposed a method to analyze automorphims groups of oriented matroidsusing semidefinite programming. Automorphism groups of vector configurations had beenstudied well, but there was little study on those of oriented matroids. We proposed toanalyze automorphism groups of oriented matroids by measuring gaps between automor-phism groups of oriented matroids and those of vector configurations. It can be formulatedas a kind of geometric realizability problems: “ Can a given oriented matroid be realizedas a vector configuration whose symmetry group coincides with that of the oriented ma-troid?”, which we call a symmetric realization problem. This problem can be reduced toETR, and we analyzed it by semidefinite programming. This problem seems difficult tobe analyzed by the existing non-realizability certificates of oriented matroids such as non-Euclidean [6], non-HK* [7] and a biquadratic final polynomial method [2], and only twoexamples with such gaps are known today [14, 12]. Actually, geometric non-realizabilitycertificates such as non-Euclidean and non-HK* seem difficult to be applied to this prob-lem, and a biquadratic final polynomial method canont be used to detect gaps betweencombinatrial symmetry and geometric symmetry by the following theorem.Theorem 1.1 ([3]) For any subgroup $G\subset Aut(M)$ we have: An oriented $M=(E, \chi)$admits a bi-quadratic final polynomial, if and only if it admits a symmetm$c$ bi-quadraticfinal polynomial with respect to $G$ .We applied our method to oriented matroids of rank 4 on up to 8 elements and thoseof rank 3 on up to 9 elements. Our method with relaxation order 1 could find orientedmatroids of rank 3 on up to 8 elements or those of rank 4 on up 9 elements with such agap, but we observed that there were gaps between the feasibility of our method wihtoutthe conditions of symmetry and those with the conditions of symmetry. In this paper, weexplain that our method with higher relaxation order can detect gaps between geometricsymmetry and combinatorial symmetry by analyzing Richter-Gebert’s example [12].2 PreliminariesIn this section, we explain symmetry of oriented matroids and that of vector configura-tions. For details about oriented matroids, see [1].First, we explain automorphism groups of vector configurations.Definition 2.1 Let $V=(v_{1}, \ldots, v_{n})\in \mathbb{R}^{nxr}$ be a vector configuration. If there exist aninversible linear map $A:\mathbb{R}^{r}arrow \mathbb{R}^{r}$ such that$A\cdot v_{i}=v_{\pi(i)}$ for all $i=1,$ $\ldots,$ $n$ ,for a permutation $\pi$ on $\{$ 1, $\ldots,$ $n\}$ , we say that $V$ has a symmetry $\pi$ and $A$ is an affineautomorphism of $V$ .Definition 2.2 The group formed by all symmetries of a vecotor configuration $V$ is calledthe automorphism group of $V$ , and we denote it by $Aut(V)$ .We immediately notice the following proposition.Proposition 2.3 Let $A$ be an affine automorphism of a vector configuration V. Then,$\det(A)=1$ or $\det(A)=-1$ .9Proof: Asume that $A\cdot v_{i}=v_{\pi_{*}}.$ : Because $Aut(V)$ is a finite group, there exists $m\in N$such that $\pi^{m}=id$, where $id$ is the identity. Thus, $A^{m}(v_{1}, \ldots., v_{n})=(v_{1}, \ldots, v_{n})$ . Then,we choose a maximal linear independent set $\{v_{i_{1}}, \ldots, v_{i_{r}}\}$ of $V$ and make a inversiblematrix $M_{V}=(v_{i_{1}}, \ldots, v_{i_{r}})$ . Then, it satisfies $A^{m}M_{V}=M_{V}$ and we can conclude that$A^{m}=I$ , where $I$ is the identity matrix. Therefore, $\det(A)^{m}=1$ . It implies $\det(A)=1$ or$\det(A)=-1$ .An automorphism whose determinant is $-1$ is called a reflection, and that whose deter-minant is 1 is called a rotation. Considering the above discussion, symmetry of orientedmatroids is defined as follows.Definition 2.4 Let $M=(E, \chi)$ be an n-element rank-r oriented matroid.$Aut(M):=\{\sigma\in S_{n}|\sigma\cdot\chi=\chi or \sigma\cdot\chi=-\chi\}$is called the automorphism group of $M$ .An automorphism $\sigma$ of $M$ such that $\sigma\cdot\chi=\chi$ is called a rotation of $M$ and we denotethe set of all rotations of $M$ by Rot$(M)$ . Similarly, an automorphism $\sigma$ of $M$ such that$\sigma\cdot\chi=-\chi$ is called a reflection of $M$ and we denote the set of all reflections of $M$ byRef$(M)$ .3 Analyzing gaps between combinatorial symmetryand geometric symmetry by SDPIn this section, we review a formulation of a non-realizability certificate of the followingproblem by semidefinite programming [10] in order to analyze automorphism groups oforiented matroids.Problem 1 Let $M=(E, \chi)$ be an oriented matroid. Then, is there a vector configuration$V$ such that $V$ is a realization of $M$ and $Aut(V)=Aut(M)^{Q}$Let $M=(E, \chi)$ be an oriented matroid of rank $r$ where $|E|=n$ . Suppose that $M$ canbe realized as a vector configuration whose automorphism group$\cdot$ is $Aut(M)$ . Then thereexists a vector configuration $V=(v_{i})_{1=1}^{n}\in \mathbb{R}^{rxn}$ such that for $1\leq i_{1}<\ldots<i_{r}\leq n$ ,$1\leq j_{1}<\ldots<j_{r}\leq n,$ $\sigma\in Rot(M)$ and $\tau\in Ref(M)$ ,$\{\begin{array}{l}[i_{1}\ldots i_{r}][i_{1}\cdots j_{r}]-\sum_{k=1}^{r}[i_{k}i_{2}\ldots i_{r}]b_{1}\cdots j_{k-1}i_{1}j_{k+1}\ldots j_{r}]=0,(Grassmann- Pl\text{\"{u}} cker relations)sign ([i_{1}\ldots i_{f}]) = \chi(i_{1}, -, i_{r}),(Conditions of chirotope)[i_{1}\ldots i_{r}]=[\sigma(i_{1})\ldots\sigma(i_{r})],[i_{1}\ldots i_{r}]=-[\tau(i_{1})\ldots\tau(i_{r})],(Conditions of symmetry)\end{array}$10where $[i_{1}\ldots i_{r}];=\det(v_{i_{1}}, \ldots, v_{i_{r}})$ . Then, we rewrite the above system as follows. For$1\leq i_{1}<\ldots<i_{r}\leq n$ and $1\leq j_{1}<\ldots<j_{r}\leq n$ ,$\{\begin{array}{l}x(i_{1},\ldots,i,)_{t}(j_{1},\ldots,j_{r})-\sum_{k=1}^{r}x_{Ci_{2}\ldots i_{r})_{1}Cj_{k-\iota}i_{1}j_{k+1}\ldots j_{r})}=0k1\cdots,sign (x_{(i_{1},\ldots,i_{r})\rangle 0_{1},\ldots,j_{r})}) =\chi(i_{1}, \ldots, i_{r})\cdot\chi(j_{1}, \ldots,j_{r}),sign (x_{(i_{1}\ldots.,i_{f})}) =\chi(i_{1}, \ldots, i_{r}),x_{(i_{1},\ldots,i_{r})} =x_{(\sigma(i_{1}))’\sigma(|_{r}))},x_{(i_{1},\ldots,i_{r})} = -x_{(\tau(i_{1}),\ldots,\tau(i_{r}))},x_{(i_{1},\ldots,i_{r})_{\gamma}(J_{1},\ldots ir)} = [i_{1}\ldots i_{r}]b_{1}\cdots j_{r}],x_{(i_{1},\ldots,i_{r})} = [i_{1}\ldots i_{r}]\end{array}$where $x_{(i_{\sigma\langle 1)},\ldots,i_{\sigma(r)})_{I}(j_{\tau(1)’\cdots\dot{O}_{\tau(r)}},)}$ denotes sgn$(\sigma’)$ . sgn$(\tau’)\cdot x_{(i_{1},\ldots,t_{r}),(j_{1,\ldots\dot{O}r})}$for $1\leq i_{1}<\ldots<i_{r}\leq n$ and $1\leq j_{1}<\ldots<j_{f}\leq n$ , and $\sigma’$ and $\tau$‘ are permutations on$\{$ 1, $\ldots,$ $r\}$ .Here we consider a vector $u_{1}$ given by listing 1 and elements of $\Lambda(n, r)$ in lexicographicorder:$u_{1}:=(1, [1,2, \ldots,r], [1,2, \ldots, r-1, r+1], \ldots, [n-r,n-r+1, \ldots,n])^{T}$ .Then, under the constraints $x_{(i_{1},\ldots,i_{r})_{2}(j_{1},:\cdot\cdot ir)}=[i_{1}\ldots i_{f}]$la $1\cdots j_{f}]$ and $x_{(i_{1},\ldots,i_{r})}=[i_{1}\ldots i_{r}]$ for$1\leq i_{1}<\ldots<i_{r}\leq n$ and $1\leq j_{1}<\ldots<Jr\leq n$ ,$(=u_{1}u_{1}^{T})$ should be a positive semidefinite symmetric matrix. Therefore if $M$ is realizable,the following system (SDP A) is feasible: for $1\leq i_{1}<\ldots<i_{r}\leq n$ and $1\leq j_{1}<\ldots<j_{r}\leq$$n$ ,$\{\begin{array}{l}x_{(i_{1},\ldots,:_{r})_{i}(j_{1},\ldots,j_{r})}-\sum_{k=1}^{r}x_{(j_{k}i_{2}\ldots i_{r}),(j_{1}\ldots j_{karrow 1}t_{1}j_{k+1}\ldots j_{r})}=0,sign (x_{(i_{1},\ldots,i_{r}),(j_{1},\ldots ir)})=\chi(i_{1}, \ldots,i_{r})\cdot\chi(j_{1}, \ldots,j_{r}),sign (x_{(i_{1},\ldots,i,)}) =\chi(i_{1}, \ldots, i_{r}),x_{(i_{1},\ldots,i_{r})} =x_{(\sigma(:_{1}),\ldots,\sigma(i_{r}))},x_{(i_{1},\ldots,i_{r})} = -x_{(\tau(i_{1}),\ldots,\tau(i_{r}))},X is a positive semidefinite symmetric matrix,\end{array}$ (SDP A)where $x_{(i_{\sigma(1)},\ldots,i_{\sigma(r)})_{1}(j_{d\langle 1)},\ldots,j_{r(r)})}$ denotes sgn$(\sigma’)$ . sgn $(\tau’)\cdot x_{(i_{1},\ldots,i_{r})_{2}C_{1}\cdots,j_{r})}$for all $i_{1},$ $\ldots,$ $i_{r},j_{1},$ $\ldots,j_{r}$ for $1\leq i_{1}<\ldots<i_{r}\leq n$ and $1\leq j_{1}<\ldots<j_{r}\leq n$ , and $\sigma’$ and $\tau’$are permutations on $\{1_{1}\ldots,r\}$ .Remarkl. The relaxation explained above corresponds with the SDP relaxation of apolynomial system with relaxation order 1 which was introduced in [8]. If we use thehigher order SDP relaxation, we would obtain a more powerful certificate. However, theproblem size will become very large. For detail about the higher order SDP relaxation,11see [8].Remark2. We considered all Grassmann-Pl\"ucker relations as constraints in the abovediscussion, but Grassmann-Pl\"ucker relations have redundancy. One way to deal with thisissue is considering a minmal generating set of Grassmann-Pl\"ucker ideal. For a minimalgenerating set of Grassmann-Pl\"ucker ideal, see [9].4 Experimental resultsIn [10], we applied our method to oriented matroids on up to 8 elements using Finschi andFukuda’s database of oriented matroids [5] to look up oriented matroids and SeDuMi [13,15] to solve SDPs.Table 1: The number of oriented matroids [5] (reorientation class)We decided non-realizability of 797 oriented matroids of rank 4 on 8 elements and 9oriented matroids of rank 3 on 9 elements but all of them were non-realizable even withoutthe conditions of symmetry. On the other hand, we observed that feasibility of SDPswithout the symmetry constriants and those of SDPs with the symmetry constraints weredifferent. Actually, 357 oriented matroids of rank 4 on 8 elements and 9 oriented matroidsof rank 3 on 9 elements cannot be decided to be non-realizable by our method withoutconstraints of symmetry, but decided to be non-realizable with symmetry constraints.It does not occur for a biquadratic final polynomial method, and implies possibility ofdetecting gaps between geometric symmetry and combinatorial symmetry. We actuallyexplain that our method can detect such gaps in the next section.5 Analyzing Richter-Gebert’s exampleIn [12]. Richter-Gebert constructed an example of 14-element rank-3 oriented matroidwith combinatorial symmetry which cannot be realized geometrically. In this section,we analyze this example using our method. Richter’s example is an oriented matroidassociated with the vector configuration $X=(x_{1}, \ldots, x_{14})$ defined by $x_{1}=(1,0,0),$ $x_{2}=$$(0,1,0),$ $x_{3}=(1,0,1),$ $x_{4}=(0,1,1),$ $x_{6}=(1-t)x_{3}+(1+t)x_{4},$ $x_{6}=x_{5}x_{2}\wedge x_{1}x_{4}$ ,$x_{7}=x_{5}x_{1}\wedge x_{2}x_{3},$ $x_{8}=x_{6}x_{3}\wedge x_{5}x_{1},$ $x_{9}=x_{7}x_{4}\wedge x_{5}x_{2},$ $x_{10}=x_{3}x_{4}\wedge x_{8}x_{2},$ $x_{11}=x_{3}x_{4}\wedge x_{9}x_{1}$ ,12$x_{12}=x_{7}x_{10}\wedge x_{11}x_{2_{t}}x_{13}=x_{6}x_{11}\wedge x_{10}x_{1},$ $x_{14}=x_{1}x_{3}\wedge x_{2}x_{4}$ for $t\in(-3+\sqrt{8},0)\cup(0,3-\sqrt{8})$ .Note that it has a combinatorial reflection symmetry $\pi$ :$\pi=(\begin{array}{l}1234567891011121314214357698l110l31214\end{array})$ ,and [12, 13, 14] has a positive value. We prove the oriented matroid associated with $X$cannot be realized as a vector configuration with a geometric symmetry $\pi$ .First, because of homogeneousity of Grassmann-Pl\"ucker system, we can assume that$[123]=1,$ $[134]=-1$ .By the conditions of symmetry, we obtain$[124]=1,$ $[234]=-1$ .By applying the conditions of symmetry to the Grassmann-Pl\"ucker relation “ $[312][345]arrow$$[314][325]+[315][324]=0$”, we obtain the following relation:$[145]+[135]=0$ .Similarly, we obtain the following relation by focusing on the Grassman-Pl\"ucker relation“ $[123][145]-[124][135]+[125][134]=0$” :$[125]=2[145]$ .Using the above conditions, we compute $x_{i}$ for $i=6,7,$ $\ldots,$ $14$ .$x_{6}=x_{5}x_{2}\wedge x_{1}x_{4}$ $=[145]x_{2}+x_{5}$ ,$x_{7}=x_{5}x_{1}\wedge x_{2}x_{3}$ $=-[145]x_{1}-x_{5}$ ,$xs=x_{6}x_{3}\wedge x_{5}x_{1}$ $=[145]^{2}x_{2}+2[145]^{2}x_{3}+[145]x_{5}$ ,$x_{9}=x_{7}x_{4}\wedge x_{5}x_{2}$ $=[145]^{2}x_{1}+2[145]^{2}x_{4}+[145]x_{5}$ ,$x_{10}=x_{3}x_{4}\wedge x_{8}x_{2}$ $=-[145]^{2}x_{4}-3[145]^{2}x_{3}$ ,$x_{11}=x_{3}x_{4}\wedge x_{9}x_{1}$ $=-3[145]^{2}x_{4}-[145]^{2_{X_{3}}}$ ,$x_{12}=x_{7}x_{10}\wedge x_{11}x_{2}$ $=-8[145]^{5}x_{1}+18[145]^{5}x_{3}+6[145]^{5}x_{4}-S[145]^{4}x_{5}$,$x_{13}=x_{6}x_{11}\wedge x_{10}x_{1}$ $=-8[145]^{5}x_{2}-6[145]^{5}r_{3}-1S[145]^{5}x_{4}-S[145]^{4}x_{5}$ ,$x_{14}=x_{1}x_{3}\wedge x_{2}x_{4}$ $=-x_{1}+x_{3}$ .Then, we evaluate [12, 13, 14] and obtain$[$ 12, 13, $14]=-[12\}13,1]+[12,13,3]=-96[145]^{10}$ .Therefore, [12, 13, 14] should be non-positive, which contradicts the assumption. Clearly,it can also be checked by semidefinite programming and thus we conclude that our methodcan detect a gap between combinatorial symmetry and geometric symmetry.136 Conclusion and Future worksIn this paper, we reviewed a non-realizability certificate for a symmetric realization prob-lem by semidefinite programming [10]. We saw that our method with relaxation order 1had not be able to find gaps between combinatrial symmetry and geometric symmetry fororiented matroids of rank 4 on up to 8 elements and those of rank 3 on up to 9 elements,but we explained that our method with higher relaxation order can detect such gaps byanalyzing an example in [12].References[1] A. Bj\"omer, M. Las Vergnas B. Sturmfels, N. White and G. Ziegler, $Oi\dot{v}ented$ Matroids,Cambridge University Press, 1999.[2] J. Bokowski and J. Richter, On the finding of final polynomials. European Journalof Combinatorics, 11:21-34, 1990.[3] J. Bokowski, B. Sturmfels, Computational Synthetic Geometry, No 1355, Springer,Heidelberg, 1989.[4] G. Collins, Quantifier Elimination for Real Closed Fields by Cylindrical AlgebraicDecomposition, LNCS 32. Springer Verlag, 1975.[5] L. Finschi, Catalog of oriented matroids,http: //www.om.math.ethz.ch/?$p=catom$ .[6] K. Fukuda, $Or\dot{v}ented$ Matroid Programming, PhD thesis, University of Waterloo,1982.[7] K. FXtkuda, S. Moriyama, and Y. Okamoto, The Holt-Klee condition for orientedmatroids, to appear.http: //arxiv.org/pdf/math.CO/0612073.[8] J. B. Lasserre, Global optimization with polynomials and the problems of moments,SIAM Joumal on optimization, 11:796-817, 2001.[9] H. Miyata, S. Moriyama and H. Imai, Deciding non-realizability of oriented matroidsby semidefinite programming, Pacific Journal of optimization, accepted.[10] H. Miyata, S. Moriyama and H. Imai, On the symmetry of oriented matroids, inpreparation.[11] P. A. Parrilo, Semidefinite programming relaxation for semialgebraic problems, Math-ematical Programming. 96:293-320, 2003.[12] J. Richter-Gebert, Two interesting oriented matroids, Documenta Mathematica1:137-148, 1996.[13] SeDuMi, http:$//sedumi$ .mcmaster.$ca/$ .14[14] P. Shor, Stretchability of pseudolines is NP-hard, In Applied Geometry and DiscreteMathemaics The Victor Klee Festschrift (P. Gritzmann, B. Sturmfels, eds.), DIMACSSeries in Discrete Mathematics and Theoretical Computer Science, Amer. Math. Soc.,Providence, RI, 4 (1991), 531-554.[15] J. F. Sturm, Sedumi 1.02, a matlab toolbox for optimization over symmetric cones,Optimization Methods and Sofiware, 11,12:625-653, 1999.[16] B. Sturmfels, Algomthms in Invariant Theory, Texts and Monographs in SymbolicComputation. Springer-Verlag, Vienna, 1993.[17] B. Sturmfels, Grobner Bases and Convex Polytopes, volume 8 of University LectureSeries, American Mathematical Society, 1996.15

Analyzing automorphism groups of oriented matroids by semidefinite programming (Computational Geometry and Discrete Mathematics)

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Analyzing automorphism groups of oriented matroids by semidefinite programming (Computational Geometry and Discrete Mathematics)

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