Chordal structure in computer algebra: Permanents

Transcription

1 Chordal structure in computer algebra: Permanents Diego Cifuentes Laboratory for Information and Decision Systems Electrical Engineering and Computer Science Massachusetts Institute of Technology Joint work with Pablo A. Parrilo (MIT) SIAM Conference on Discrete Mathematics Cifuentes (MIT) Chordal structure and permanents SIAM-DM 16 1 / 21

2 Algebraic tools in discrete math Several problems from discrete mathematics can be approached with tools from computer algebra. Sensor network localization: Find positions, given a few known fixed anchors and pairwise distances. x i x j 2 = d 2 ij x i a k 2 = e 2 ij ij A ik B This is a system of quadratic polynomial equations. Can be solved using Gröbner bases. Cifuentes (MIT) Chordal structure and permanents SIAM-DM 16 2 / 21

3 Algebraic tools in discrete math Several problems from discrete mathematics can be approached with tools from computer algebra. Lattice walks Let A Z n consist of integer vectors. Consider a graph with vertex set N n in which α, β are adjacent if α β A. Describe the connected components. This can be solved using a primary decomposition. Cifuentes (MIT) Chordal structure and permanents SIAM-DM 16 3 / 21

4 Algebraic tools in discrete math Several problems from discrete mathematics can be approached with tools from computer algebra. Permanents (this talk) Given a n n matrix M, compute Perm(M) := π M i,π(i) i sum over permutations π. Important case: sparse matrices. Cifuentes (MIT) Chordal structure and permanents SIAM-DM 16 4 / 21

5 Algebraic tools in discrete math Many more problems can be naturally approached with algebraic tools: graph colorings, independent sets, Hamiltonian cycles, crypto problems, etc. Cifuentes (MIT) Chordal structure and permanents SIAM-DM 16 5 / 21

6 Algebraic tools in discrete math Many more problems can be naturally approached with algebraic tools: graph colorings, independent sets, Hamiltonian cycles, crypto problems, etc. However these problems are NP-hard, and these algebraic methods do not work well for relatively small problems. Cifuentes (MIT) Chordal structure and permanents SIAM-DM 16 5 / 21

7 Algebraic tools in discrete math Many more problems can be naturally approached with algebraic tools: graph colorings, independent sets, Hamiltonian cycles, crypto problems, etc. However these problems are NP-hard, and these algebraic methods do not work well for relatively small problems. Idea: exploit the graphical structure (chordality) of the problem! Cifuentes (MIT) Chordal structure and permanents SIAM-DM 16 5 / 21

8 Algebraic tools in discrete math Many more problems can be naturally approached with algebraic tools: graph colorings, independent sets, Hamiltonian cycles, crypto problems, etc. However these problems are NP-hard, and these algebraic methods do not work well for relatively small problems. Idea: exploit the graphical structure (chordality) of the problem! Complexity aspects? Identify families of tractable instances. Cifuentes (MIT) Chordal structure and permanents SIAM-DM 16 5 / 21

9 Permanent computation General facts: (Ryser 63) Best general and exact method; complexity O(n 2 n ). (Valiant 79) Computing the permanent of a matrix is #P-hard. Cifuentes (MIT) Chordal structure and permanents SIAM-DM 16 6 / 21

10 Permanent computation General facts: (Ryser 63) Best general and exact method; complexity O(n 2 n ). (Valiant 79) Computing the permanent of a matrix is #P-hard. Approximation algorithms: (Jerrum,Sinclair,Vigoda 04) FPRAS for nonnegative matrices. Cifuentes (MIT) Chordal structure and permanents SIAM-DM 16 6 / 21

11 Permanent computation General facts: (Ryser 63) Best general and exact method; complexity O(n 2 n ). (Valiant 79) Computing the permanent of a matrix is #P-hard. Approximation algorithms: (Jerrum,Sinclair,Vigoda 04) FPRAS for nonnegative matrices. Tractable under special structure: (Fisher,Kasteleyn,Temperley 67) Bipartite graph is planar. (Barvinok 96) Rank is bounded. (Courcelle et al 01) Treewidth is bounded. Cifuentes (MIT) Chordal structure and permanents SIAM-DM 16 6 / 21

12 Graph abstractions of a matrix The sparsity structure of a matrix M can be represented with a graph. Let a 1,..., a n denote the rows and x 1,..., x n denote the columns. Consider these abstractions: G, bipartite adjacency graph: (a i, x j ) E iff M i,j 0 G s, (symmetrized) adjacency graph: (i, j) E iff M i,j + M j,i 0 G X, column graph: (x j, x k ) E iff exists a i such that M i,j M i,k 0. Equivalently, the adjacency graph of M T M. Cifuentes (MIT) Chordal structure and permanents SIAM-DM 16 7 / 21

13 Graph abstractions of a matrix x 1 x 2 x 3 x 4 x 5 x 6 x 7 x 8 x 9 a a a a a a a a a Cifuentes (MIT) Chordal structure and permanents SIAM-DM 16 8 / 21

14 Tree decompositions Definition: A tree decomposition of a graph G = (X, E) is a pair (T, χ), where T is a tree and χ : T {0, 1} X assigns some χ(t) X to each node t of T, that satisfies the following conditions. i. The union of {χ(t)} t T is the whole vertex set X. ii. For every (x i, x j ) E, there is some node t of T with x i, x j χ(t). iii. For every x i X the set {t : x i χ(t)} forms a subtree of T. The width ω of the decomposition is the largest χ(t). Cifuentes (MIT) Chordal structure and permanents SIAM-DM 16 9 / 21

15 Chordality and treewidth The treewidth of G is the minimum width among all possible tree decompositions. The treewidth ω(g) of a graph G can be though of as a measure of complexity: the smaller ω(g), the simpler the graph (closer to a tree). Meta-theorem: NP-complete problems are easy on graphs of small treewidth. Cifuentes (MIT) Chordal structure and permanents SIAM-DM / 21

16 Chordality and treewidth The treewidth of G is the minimum width among all possible tree decompositions. The treewidth ω(g) of a graph G can be though of as a measure of complexity: the smaller ω(g), the simpler the graph (closer to a tree). Meta-theorem: NP-complete problems are easy on graphs of small treewidth. Alternatively, we can define treewidth in terms of chordal completions of the graph (chordal completions are in correspondance with tree decompositions). Cifuentes (MIT) Chordal structure and permanents SIAM-DM / 21

17 Treewidth of matrix graphs Simple fact: tw(g) 2 tw(g s ). tw(g) tw(g X ) + 1. For a fixed tw(g) both tw(g s ), tw(g X ) are unbounded. Cifuentes (MIT) Chordal structure and permanents SIAM-DM / 21

18 Treewidth of matrix graphs Simple fact: tw(g) 2 tw(g s ). tw(g) tw(g X ) + 1. For a fixed tw(g) both tw(g s ), tw(g X ) are unbounded. Previous tree decomposition methods are primarily based on the symmetrized graph G s (Courcelle et al., Flarup et al., Meer), and they do not scale well with the treewidth O(n 2 O(ω2) ). Cifuentes (MIT) Chordal structure and permanents SIAM-DM / 21

19 Our results A tree decomposition method to compute permanents, based on the bipartite graph G, with complexity Õ(n 2ω ). The algorithm naturally extends to higher dimensional problems: mixed discriminants, hyperdeterminants. Complexity Õ(n2 + n 3 ω ). Hardness results for the case of mixed volumes. Cifuentes (MIT) Chordal structure and permanents SIAM-DM / 21

20 Our results HDet tensor MVol n polytopes MDisc n matrices MVol n zonotopes (known) Hard Easy Hard (small ω) Easy (small ω) (this paper) Det matrix Perm matrix Cifuentes (MIT) Chordal structure and permanents SIAM-DM / 21

21 Basic idea For block matrices ( ) ( ) A A 0 A Perm = Perm(A) Perm(B) = Perm 0 B B B Cifuentes (MIT) Chordal structure and permanents SIAM-DM / 21

22 Permanent expansion: column graph Similarly, for the matrix A 1,1 0 A 1,3 A 1,4 0 A 2,1 0 A 2,3 A 2,4 0 M = 0 C 3,2 C 3,3 C 3,4 0 0 B 4,2 B 4,3 0 B 4,5 0 B 5,2 B 5,3 0 B 5,5 we have the following formula ( ) A1,1 A Perm(M) = Perm 1,3 A 2,1 A 2,3 ( ) A1,1 A + Perm 1,4 A 2,1 A 2,4 ( ) A1,1 A + Perm 1,4 A 2,1 A 2,4 ) Perm (C 3,4 Perm ) Perm (C 3,3 Perm ) Perm (C 3,2 Perm ( ) B4,2 B 4,5 B 5,2 B 5,5 ( B4,2 ) B 4,5 B 5,2 B 5,5 ( ) B4,3 B 4,5. B 5,3 B 5,5 To evaluate we need 14 multiplications: four 2 2 permanetns. Compare with: 4 5! = 480 multiplications using definition. Cifuentes (MIT) Chordal structure and permanents SIAM-DM / 21

23 Permanent expansion: column graph This simple formula exists because its column graph has a simple structure M 1,1 0 M 1,3 M 1,4 0 M 2,1 0 M 2,3 M 2,4 0 0 M 3,2 M 3,3 M 3,4 0 0 M 4,2 M 4,3 0 M 4,5 0 M 5,2 M 5,3 0 M 5,5 Cifuentes (MIT) Chordal structure and permanents SIAM-DM / 21

24 Results Lemma: Let (T, χ) be a tree decomposition of G X. Let c 1,..., c k be the children of a node t T. Then perm(a Tt, Y ) = k perm(a t, Y t ) perm(a Tcj, Y cj ) Y j=1 where the sum is over all Y = (Y t, Y c1,..., Y ck ) such that: Y = Y t (Y c1 Y ck ) χ(t cj ) \ χ(t) Y cj χ(t cj ) Y t χ(t). Lemma: The above formula can be evaluated in Õ(k 2ω ). Cifuentes (MIT) Chordal structure and permanents SIAM-DM / 21

25 Results Lemma: Let (T, χ) be a tree decomposition of G X. Let c 1,..., c k be the children of a node t T. Then perm(a Tt, Y ) = k perm(a t, Y t ) perm(a Tcj, Y cj ) Y j=1 where the sum is over all Y = (Y t, Y c1,..., Y ck ) such that: Y = Y t (Y c1 Y ck ) χ(t cj ) \ χ(t) Y cj χ(t cj ) Y t χ(t). Lemma: The above formula can be evaluated in Õ(k 2ω ). Proof: Rewrite the equation as P t (Ȳ ) = k Q t (Ȳ t ) Q cj (Ȳ cj ) Ȳ t Ȳ c1 Ȳ ck =Ȳ j=1 and use the fast subset convolution (Björklund et al. 07). Cifuentes (MIT) Chordal structure and permanents SIAM-DM / 21

26 Permanent expansion: bipartite graph Perm(M) = perm({a 1, a 2}, {x 1, x 4}) perm({a 3, a 4, a 5}, {x 2, x 3, x 5}) + perm({a 1, a 2, a 3}, {x 1, x 3, x 4}) perm({a 4, a 5}, {x 2, x 5}) M 3,3 perm({a 1, a 2}, {x 1, x 4}) perm({a 4, a 5}, {x 2, x 5}) perm({a 1, a 2, a 3}, {x 1, x 3, x 4}) = M 3,3 perm({a 1, a 2}, {x 1, x 4}) + M 3,4 perm({a 1, a 2}, {x 1, x 3}) perm({a 3, a 4, a 5}, {x 2, x 3, x 5}) = M 3,3 perm({a 4, a 5}, {x 2, x 5}) + M 3,2 perm({a 4, a 5}, {x 3, x 5}) To evaluate we need 16 multiplications. Cifuentes (MIT) Chordal structure and permanents SIAM-DM / 21

27 Results Theorem: Let M be a matrix with associated bipartite graph G. Then we can compute Perm(M) in Õ(n 2ω ) where ω = tw(g). Cifuentes (MIT) Chordal structure and permanents SIAM-DM / 21

28 Results Theorem: Let M be a matrix with associated bipartite graph G. Then we can compute Perm(M) in Õ(n 2ω ) where ω = tw(g). The mixed discriminant of n matrices of size n n is Disc(A 1,..., A n ) := sgn(π)sgn(ρ) π,ρ i A i π(i),ρ(i) where the sum is over pairs of permutations π, ρ. Theorem: Let M be a list of matrices with associated tripartite graph G. Then we can compute Disc(M) in Õ(n2 + n 3 ω ), where ω = tw(g). Cifuentes (MIT) Chordal structure and permanents SIAM-DM / 21

29 Further generalizations The first Cayley hyperdeterminant is the simplest generalization of the determinant to multidimensional arrays. Theorem: Let M be a square d-dimensional tensor of length n, with d-partite graph G. We can compute its hyperdeterminant in Õ(n2 + n 3 ω ). Cifuentes (MIT) Chordal structure and permanents SIAM-DM / 21

30 Further generalizations The first Cayley hyperdeterminant is the simplest generalization of the determinant to multidimensional arrays. Theorem: Let M be a square d-dimensional tensor of length n, with d-partite graph G. We can compute its hyperdeterminant in Õ(n2 + n 3 ω ). The mixed volume is a geometric generalization of determinants and permanents to arbitrary convex bodies on R n. Theorem: Computing mixed volumes of n zonotopes remains #P-hard when their associated graph has bounded treewidth. Cifuentes (MIT) Chordal structure and permanents SIAM-DM / 21

31 Summary HDet tensor MVol n polytopes MDisc n matrices MVol n zonotopes (known) Hard Easy Hard (small ω) Easy (small ω) (this paper) Det matrix Perm matrix D. Cifuentes, P.A. Parrilo, An efficient tree decomposition method for permanents and mixed discriminants. arxiv: D. Cifuentes, P.A. Parrilo, Exploiting chordal structure in polynomial ideals: a Gröbner basis approach. arxiv: D. Cifuentes, P.A. Parrilo, Chordal networks of polynomial ideals. arxiv: Thanks for your attention! Cifuentes (MIT) Chordal structure and permanents SIAM-DM / 21