Computing the Integer Points of a Polyhedron

Transcription

1 Computing the Integer Points of a Polyhedron Complexity Estimates Rui-Juan Jing 1,2 and Marc Moreno Maza 2,3 1 Key Laboratoty of Mathematics Mechnization, Academy of Mathematics and Systems Science, Chinese of Academy of Sciences 2 University of Western Ontario, London, Ontario 3 IBM Center for Advanced Studies, Markham, Ontario CASC 2017, September 19

2 Plan Recall the algorithm Complexity Experiments Application Summary

3 Motivations 1. Data dependence analysis and scheduling of for-loop nests of computer programs, 2. support for decision problems in Presburger arithmetic, 3. manipulation of Z-polyhedra.

4 Algorithm IntegerSolve(K) relies on three sub-procedures. Procedure 1: IntegerNormalize(Ax b): Solve the equation systems and remove the redundant inequalities;

5 Algorithm IntegerSolve(K) relies on three sub-procedures. Procedure 1: IntegerNormalize(Ax b): Solve the equation systems and remove the redundant inequalities; Procedure 2: DarkShadow(Mt v) Any integer point in the dark shadow can be lifted to an integer point of the original polyhedron (represented by (Mt v))

6 Algorithm IntegerSolve(K) relies on three sub-procedures. Procedure 1: IntegerNormalize(Ax b): Solve the equation systems and remove the redundant inequalities; Procedure 2: DarkShadow(Mt v) Any integer point in the dark shadow can be lifted to an integer point of the original polyhedron (represented by (Mt v)) Procedure 3: GreyShadow(Mt v) Output the grey shadow parts of polyhedron represented by Mt v, each integer point in every grey shadow part corresponding to one integer point satisfying Mt v and the number of variables to be dealt with is less than the length of t.

7 Plan Recall the algorithm Complexity Experiments Application Summary

8 Complexity-FM elimination K polyhedron in R d, defined by m inequalities

9 Complexity-FM elimination K polyhedron in R d, defined by m inequalities K is full-dimentional Ax b (Ax b has no implicit equations.)

10 Complexity-FM elimination K polyhedron in R d, defined by m inequalities K is full-dimentional Ax b (Ax b has no implicit equations.) k-dimentional face of K { A I k x = b Ik, A I Ik b I Ik (I = {1,..., m}, I k I with d k elements.)

11 Complexity-FM elimination K polyhedron in R d, defined by m inequalities K is full-dimentional Ax b (Ax b has no implicit equations.) k-dimentional face of K { A I k x = b Ik, A I Ik b I Ik (I = {1,..., m}, I k I with d k elements.) Lemma Let f d,m,k be the number of k-dimensional faces of K. Then, we have f d,m,k ( m d k ). Therefore, we have f d,m,0 + f d,m,1 + + f d,m,d 1 m d.

12 Complexity-FM elimination Proposition { A I k x = b Ik, A I Ik x b I Ik IntegerNormalize Mt v M, v (k + 1) k+1 2 L k+1.

13 Complexity-FM elimination Proposition { A I k x = b Ik, A I Ik x b I Ik IntegerNormalize Mt v M, v (k + 1) k+1 2 L k+1. Notation Given a linear program with total bit size H and with d variables LP(d, H): the number of bit operations required for solving it. Karmarkar s algorithm: LP(d, H) O(d 3.5 H 2 log H log log H).

14 Complexity-FM elimination Proposition { A I k x = b Ik, A I Ik x b I Ik IntegerNormalize Mt v M, v (k + 1) k+1 2 L k+1. Notation Given a linear program with total bit size H and with d variables LP(d, H): the number of bit operations required for solving it. Karmarkar s algorithm: LP(d, H) O(d 3.5 H 2 log H log log H). Proposition Given a polyhedron K in R d, which is defined by m inequalities and with coefficient maximum bit size h, one can perform Fourier-Motzkin elimination within O(d 2 m 2d LP(d, 2 d hd 2 m d )) bit operations.

15 Complexity of our algorithm Hypothesis During the execution of the function call IntegerSolve(K), for any polyhedral set K, input of a recursive call, each facet of the dark shadow of K is parallel to some facet of the real shadow of K.

16 Complexity-our algorithm S D G D G D G D G D G

17 Complexity-our algorithm S D G D G D G D G D G number of pathes T : T m d 2 d 3d 3 L 3d 3 coefficient bound M in any node in a path: M d 3d 2 d 4d 3 L 6d 3

18 Complexity-our algorithm S D G D G D G D G D G number of pathes T : T m d 2 d 3d 3 L 3d 3 coefficient bound M in any node in a path: M d 3d 2 d 4d 3 L 6d 3 Theorem Under our Hypothesis, the function call IntegerSolve(K) runs within O(m 2d 2 d 4d 3 L 4d 3 LP(d, m d d 4 (log d + log L))) bit operations.

19 Plan Recall the algorithm Complexity Experiments Application Summary

20 Experiments IntegerSolve is implemented in the Polyhedra library and available from Example m d L m o L o?hyp t H t P Tetrahedron yes TruncatedTetrahedron yes Presburger yes Presburger yes Bounded no Bounded yes Bounded no Unbounded no Unbounded no Unbounded no P no Sys yes Sys yes Automatic yes Automatic yes Table: Implementation

21 Plan Recall the algorithm Complexity Experiments Application Summary

22 Application Solve integer programming: min lex (x 1,..., x d ) Ax b, x Z d Example Problem: min lex (x 3, x 2, x 1 ) 3x 1 2x 2 + x 3 7 2x 1 + 2x 2 x x 1 + x 2 + 3x 3 15 x 2 25 x 1, x 2, x 3 Z

23 Application Example 3x 1 2x 2 + x 3 7 2x 1 + 2x 2 x 3 12 Input: K 1, assume x 1 > x 2 > x 3. 4x 1 + x 2 + 3x 3 15 x 2 25 Output: K 1 1, K 2 1, K 3 1, K 4 1, K 5 1 given by: 3x 1 2x 2 + x 3 7 2x 1 + 2x 2 x x 1 + x 2 + 3x x 2 x x x 3 67 x x 3 17, x 1 = 15 x 2 = 27, x 3 = 16 x 1 = 18 x 2 = 33, x 3 = 18 x 1 = 19 x 1 = 14 x 2 = 50 + t x 2 = 25, x 3 = t x 3 = t 16.

24 Application min(x 3, x 2, x 1 ) K 1 Z 3 min(x 3, x 2, x 1 ) min(x 3, x 2, x 1 ) min(x 3, x 2, x 1 ) min(x 3, x 2, x 1 ) min(x 3, x 2, x 1 ) K1 1 Z3 K1 2 Z3 K1 3 Z3 K1 4 Z3 K1 5 Z3 (2, 8, 4) (16, 27, 15) (18, 33, 18) (15, 25, 14) (0, 25, 19) (0, 25, 19)

25 Plan Recall the algorithm Complexity Experiments Application Summary

26 Summary Assuming that each facet of the dark shadow of a polyhedron is parallel to some facet of its real shadow, we prove that our algorithm runs in time single exponential in the dimension d of the ambient space.

27 Summary Assuming that each facet of the dark shadow of a polyhedron is parallel to some facet of its real shadow, we prove that our algorithm runs in time single exponential in the dimension d of the ambient space. This assumption is almost always verified in practice, in particular for problems coming from computer program analysis.

28 Summary Assuming that each facet of the dark shadow of a polyhedron is parallel to some facet of its real shadow, we prove that our algorithm runs in time single exponential in the dimension d of the ambient space. This assumption is almost always verified in practice, in particular for problems coming from computer program analysis. Taking advantage of the good structure of the simpler polyhedra, we give an application to solve the lexicographic minimum of some variable orders.

29 Summary Assuming that each facet of the dark shadow of a polyhedron is parallel to some facet of its real shadow, we prove that our algorithm runs in time single exponential in the dimension d of the ambient space. This assumption is almost always verified in practice, in particular for problems coming from computer program analysis. Taking advantage of the good structure of the simpler polyhedra, we give an application to solve the lexicographic minimum of some variable orders. Works in progress A CilkPlus version of the Polyhedra library Parametric integer programming (PIP) in support of automatic parallelization.