Number of primal and dual bases of network flow and unimodular integer programs

1. Number of primal and dual bases of network flow and unimodular integer programs Hiroki NAKAYAMA1, Takayuki ISHIZEKI2, Hiroshi IMAI1 The University of Tokyo 1. Department of Computer Science. 2. Department of Information Science.

2. (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. Our work -- Fusion of combinatorial, geometric, and algebraic approaches Integer programming; Geometric approaches Combinatorial approaches Computing Normalized volume Dual feasible bases of relaxed linear program Fusion matched edge spanning tree Algebraic Geometry transportation problem minimum cost flow problem bijection • Standard pairs • Gröbner bases Application to network problem Algebraic approaches

3. x1 Nn Nc x2 x1 3 Oc Standard pairs decomposition x2 ex. b=12 ex. Knapsack problem b varies optimal x moves variously. How large optimal x span? For each b, In fact, for any b, If , cost can be reduced by replacing is never optimal solution.

4. Nn Nc x2 x1 3 Oc Standard pairs decomposition Covering Oc by standard pairs as few as possible. standard pair indicates Standard pair decomposition: {((0,0),{x2}), ((1,0),{x2}), ((2,0),{x2})} This example has 3 standard pairs. u= (0,0) is important.

5. Lem. [Sturmfels-Thomas’94, Hosten-Thomas’01] {((0,…,0),σ): standard pairs of Oc} bijection {dual feasible bases for linear relaxation of IPA,c(b)} y 0 1/3 1 Dual polyhedron • dual polyhedron is defined by dual of integer program . vertex of polyhedron 1:1 a dual feasible basis

6. Our results • Main theoremmax#(dual feasible bases)=normalized volume of conv(A’) • Analyses of several network 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. Volume of conv(A’) For #(standard pair of O’c)

8. If A is unimodular, s.t. equality hold. 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’). Proof.

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

10. 3 1 3 3 2 2 2 2 5 1 3 Transportation problem bipartite graph K2,3 consumer cost cij supplier xij : Flow from supplier i to consumer j incidence matrix of K2,3 unimodular

11. Normalized volume for the primaltransportation problem on Km,n

12. Dual transportation problem on Km,n dependent incidence matrix of K2,3 (A I ) coefficient matrix of dual problem K2,3 (I -AT )

13. n = 3 n = 4 n = 2 Normalized volume for the dualtransportation problem on Km,n This can be shown by computing volume explicitly. 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

15. cost cij 5 1 1 3 1 1 2 4 4 7 2 1 3 2 Minimum cost flowon acyclic tournament graphs oriented complete graph tournament graph K4 incidence matrix of K4 xij : Flow from vertex i to vertex j unimodular

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

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

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