1. Number of primal and dual bases of network flow and unimodular integer programs Hiroki NAKAYAMA 1, Takayuki ISHIZEKI 2, Hiroshi IMAI 1 The University of Tokyo 1. Department of Computer Science. 2. Department of Information Science.

2. Our work -- Fusion of combinatorial, geometric, and algebraic approaches Combinatorial approaches Algebraic approaches Geometric approaches Computing Normalized volume Application to network problem Standard pairs Gröbner bases Integer programming; Dual feasible bases of relaxed linear program transportation problem minimum cost flow problem matched edge spanning tree Fusion bijection Algebraic Geometry (1) We give a theorem and a proof that max #(dual feasible bases) = normalized volume. (2) We analyzed several network problems by computing normalized volume.

3. Standard pairs decomposition x1x1 x2x2 NcNc NnNn OcOc 3 In fact, for any b, If, cost can be reduced by replacing x1x1 x2x2 is never optimal solution. ex. b=12 ex. Knapsack problem b variesoptimal x moves variously. For each b, How large optimal x span?

4. Standard pairs decomposition x1x1 x2x2 NcNc NnNn OcOc 3 Covering Oc by standard pairs as few as possible. Standard pair decomposition: {((0,0),{x 2 }), ((1,0),{x 2 }), ((2,0),{x 2 })} standard pair indicates This example has 3 standard pairs. u= (0,0) is important.

5. Dual polyhedron dual polyhedron is defined by dual of integer program. 0 1/31 y vertex of polyhedron 1:1 a dual feasible basis Lem. [Sturmfels-Thomas’94, Lem. [Sturmfels-Thomas’94, Hosten-Thomas’01] Hosten-Thomas’01] ((0,…,0),σ) {((0,…,0),σ): standard pairs of O c } bijection dual feasible bases {dual feasible bases for linear relaxation of IP A,c (b)}

6. Our results Main theorem max#(dual feasible bases)=normalized volume of conv(A’) Analyses of several network problem dual feasible basesprimal feasible bases transportation problem minimum cost flow problem ▲ Normalized Volume

7. Normalized volume When vertices of conv(A) are in the lattice, normalized volume of conv(A) is calculated by normalization such that ex. normalized volume of red polytope = 6 Lem. For Volume of conv(A’) #(standard pair of O’ c )

8. Sketch of proof Main Theorem. If A is unimodular, then there exists a cost vector c s.t. #(dual feasible bases) = normalized volume of conv(A’). If A is unimodular, s.t. equality hold. Proof.

9. Our results Main theorem -- max#(dual feasible bases)=normalized volume of conv(A’) Analyses of several network problem ▲ Normalized Volume dual feasible basesprimal feasible bases transportation problem minimum cost flow problem To compute dual (resp. primal) feasible bases, we think of primal (resp. dual) problem.

10. Transportation problem 3 3 5 3 2 1 1 3 2 2 2 supplier consumer cost c ij incidence matrix of K 2,3 x ij : Flow from supplier i to consumer j bipartite graph K 2,3 unimodular

11. Normalized volume for the primal transportation problem on K m,n

12. Dual transportation problem on K m,n incidence matrix of K 2,3 dependent coefficient matrix of dual problem K 2,3 (A I ) (I -A T )

13. Normalized volume for the dual transportation problem on K m,n This can be shown by computing volume explicitly. n = 2 n = 3 n = 4 example: Normalized Vol. = 2 Normalized Vol. = 6 Normalized Vol. = 12

14. Our results Main theorem -- max#(dual feasible bases)=normalized volume of conv(A’) Analyses of several network problem ▲ Normalized Volume dual feasible basesprimal feasible bases transportation problem minimum cost flow problem

15. Minimum cost flow on acyclic tournament graphs tournament graph K 4 x ij : Flow from vertex i to vertex j unimodular incidence matrix of K 4 1 3 2 4 cost c ij 1 1 1 1 3 2 5 4 2 7 oriented complete graph

16. k j l i k j l i ○ ✕ k j l i ✕ Normalized volume for the primal minimum cost flow When the cost vector satisfies #(feasible bases) becomes maximum. O(4d)O(4d) Normalized volume of conv(A’) = #(spanning trees) s.t. [Gelfand-Graev-Postnikov ’96]

17. Both can be shown by using Gröbner bases. (Please see proceedings for the proof.) By considering a dual problem of min-cost flow, Min-case of #(feasible bases) for (primal/dual) min-cost flow When the cost vector satisfies #(standard pairs) becomes minimum 1. 1 3 2 4 1 2 5 4 10 1 Normalized volume (= max #(primal feasible bases) is unknown.

18. Summary We showed the maximum number of feasible bases of dual polyhedron for unimodular IP in terms of volume of polyhedron. We applied to show the maximum number of vertices of the feasible polyhedron for (primal and dual) transportation problems (primal and dual) min-cost flow problems

19. Future works (Open problems) Prove Main theorem by purely combinatorial approach Compute exact volume of primal polyhedron for min-cost flow problem. Apply this method to other combinatorial problems knapsack problem problems on general graph