Fast Deterministic Algorithms for Matrix Completion Problems Tasuku Soma Research Institute for Mathematical Sciences, Kyoto Univ. Tasuku Soma (Kyoto Univ.) Fast Matrix Completion Algorithms 1 / 29

1 Introduction 2 Matrix Completion by Rank-One Matrices 3 Application to Network Coding 4 Mixed Skew-Symmetric Matrix Completion 5 Skew-Symmetric Matrix Completion by Rank-Two Skew-Symmetric Matrices 6 Conclusion Tasuku Soma (Kyoto Univ.) Fast Matrix Completion Algorithms 2 / 29

1 Introduction 2 Matrix Completion by Rank-One Matrices 3 Application to Network Coding 4 Mixed Skew-Symmetric Matrix Completion 5 Skew-Symmetric Matrix Completion by Rank-Two Skew-Symmetric Matrices 6 Conclusion Tasuku Soma (Kyoto Univ.) Fast Matrix Completion Algorithms 3 / 29

Matrix Completion Matrix Completion F: Field Input Matrix A(x1 , . . . , xn) over F(x1 , . . . , xn) with indeterminates x1 , . . . , xn Find α1 , . . . , αn ∈ F maximizing rank A(α1 , . . . , αn). Tasuku Soma (Kyoto Univ.) Fast Matrix Completion Algorithms 4 / 29

Matrix Completion Matrix Completion F: Field Input Matrix A(x1 , . . . , xn) over F(x1 , . . . , xn) with indeterminates x1 , . . . , xn Find α1 , . . . , αn ∈ F maximizing rank A(α1 , . . . , αn). Example F = Q, A = 1 + x1 2 + x2 x3 x4 −→ A = 2 2 1 0 (x1 := 1, x2 := 0, x3 := 1, x4 := 0) Tasuku Soma (Kyoto Univ.) Fast Matrix Completion Algorithms 4 / 29

Backgrounds A variety of combinatorial optimization problems can be formulated by matrices with indeterminates: Maximum matching, Structural rigidity, Network coding, etc. Tasuku Soma (Kyoto Univ.) Fast Matrix Completion Algorithms 5 / 29

Backgrounds A variety of combinatorial optimization problems can be formulated by matrices with indeterminates: Maximum matching, Structural rigidity, Network coding, etc. Previous Works Matrix completion for general matrices is solvable in polynomial time by a randomized algorithm if the field is sufficiently large. Deterministic algorithms are known only for special matrices (cf. polynomial identity testing) NP hard over a general field. Tasuku Soma (Kyoto Univ.) Fast Matrix Completion Algorithms 5 / 29

Our Results Our Results Deterministic polynomial time algorithms for the following matrix completion problems: Matrix completion by rank-one matrices — a faster algorithm than the previous one Mixed skew-symmetric matrix completion — the first deterministic polynomial time algorithm Skew-symmetric matrix completion by rank-two skew-symmetric matrices — the first deterministic polynomial time algorithm Tasuku Soma (Kyoto Univ.) Fast Matrix Completion Algorithms 6 / 29

Our Results Our Results Deterministic polynomial time algorithms for the following matrix completion problems: Matrix completion by rank-one matrices — a faster algorithm than the previous one Mixed skew-symmetric matrix completion — the first deterministic polynomial time algorithm Skew-symmetric matrix completion by rank-two skew-symmetric matrices — the first deterministic polynomial time algorithm They are working over an arbitrary field! Tasuku Soma (Kyoto Univ.) Fast Matrix Completion Algorithms 6 / 29

1 Introduction 2 Matrix Completion by Rank-One Matrices 3 Application to Network Coding 4 Mixed Skew-Symmetric Matrix Completion 5 Skew-Symmetric Matrix Completion by Rank-Two Skew-Symmetric Matrices 6 Conclusion Tasuku Soma (Kyoto Univ.) Fast Matrix Completion Algorithms 7 / 29

Problem Definition Matrix Completion by Rank-One Matrices Matrix completion for A = B0 + x1B1 + · · · + xnBn, where B1 , . . . , Bn are of rank one. Tasuku Soma (Kyoto Univ.) Fast Matrix Completion Algorithms 8 / 29

Problem Definition Matrix Completion by Rank-One Matrices Matrix completion for A = B0 + x1B1 + · · · + xnBn, where B1 , . . . , Bn are of rank one. Example B0 = 1 0 0 0 , B1 = 1 1 0 0 , B2 = 2 0 1 0 A = 1 + x1 + 2x2 x1 x2 0 Tasuku Soma (Kyoto Univ.) Fast Matrix Completion Algorithms 8 / 29

Previous Works In the case of B0 = 0: Lov´ asz ’89 This can be reduced to linear matroid intersection. solvable in O(mn1.62) time using the algorithm of Gabow & Xu ’96 m: the larger of row and column sizes, n: # of indeterminates Tasuku Soma (Kyoto Univ.) Fast Matrix Completion Algorithms 9 / 29

Previous Works In the case of B0 = 0: Lov´ asz ’89 This can be reduced to linear matroid intersection. solvable in O(mn1.62) time using the algorithm of Gabow & Xu ’96 For the general case: Ivanyos, Karpinski & Saxena ’10 An optimal solution can be found in O(m4.37n) time. m: the larger of row and column sizes, n: # of indeterminates Tasuku Soma (Kyoto Univ.) Fast Matrix Completion Algorithms 9 / 29

Previous Works In the case of B0 = 0: Lov´ asz ’89 This can be reduced to linear matroid intersection. solvable in O(mn1.62) time using the algorithm of Gabow & Xu ’96 For the general case: Ivanyos, Karpinski & Saxena ’10 An optimal solution can be found in O(m4.37n) time. Our Result An optimal solution can be found in O((m + n)2.77) time. m: the larger of row and column sizes, n: # of indeterminates Tasuku Soma (Kyoto Univ.) Fast Matrix Completion Algorithms 9 / 29

Idea For A = B0 + x1B1 + · · · + xnBn (Bi = uiv i (i = 1, . . . , n)) ˜ A :=                                               1 ... 1 0 v 1 . . . vn x1 ... xn 1 ... 1 0 0 u1 · · · un B0                                               . Tasuku Soma (Kyoto Univ.) Fast Matrix Completion Algorithms 10 / 29

Idea For A = B0 + x1B1 + · · · + xnBn (Bi = uiv i (i = 1, . . . , n)) ˜ A :=                                               1 ... 1 0 v 1 . . . vn x1 ... xn 1 ... 1 0 0 u1 · · · un B0                                               . Lemma rank ˜ A = 2n + rank A Tasuku Soma (Kyoto Univ.) Fast Matrix Completion Algorithms 10 / 29

Algorithm Each indeterminate appears only once in ˜ A! (˜ A is a mixed matrix) Tasuku Soma (Kyoto Univ.) Fast Matrix Completion Algorithms 11 / 29

Algorithm Each indeterminate appears only once in ˜ A! (˜ A is a mixed matrix) Harvey, Karger & Murota ’05 Matrix completion for a mixed matrix can be done in O(m2.77) time. m: the larger of row and column sizes, n: # of indeterminates Tasuku Soma (Kyoto Univ.) Fast Matrix Completion Algorithms 11 / 29

Algorithm Each indeterminate appears only once in ˜ A! (˜ A is a mixed matrix) Harvey, Karger & Murota ’05 Matrix completion for a mixed matrix can be done in O(m2.77) time. ↓ Apply to ˜ A Theorem Matrix completion by rank-one matrices can be done in O((m + n)2.77) time. m: the larger of row and column sizes, n: # of indeterminates Tasuku Soma (Kyoto Univ.) Fast Matrix Completion Algorithms 11 / 29

Min-Max Theorem Theorem For A = B0 + x1B1 + · · · + xnBn, max{rank A : x1 , . . . , xn} = min rank 0 [vj : j J] [uj : j ∈ J] B0 : J ⊆ {1, . . . , n} . Tasuku Soma (Kyoto Univ.) Fast Matrix Completion Algorithms 12 / 29

Min-Max Theorem Theorem For A = B0 + x1B1 + · · · + xnBn, max{rank A : x1 , . . . , xn} = min rank 0 [vj : j J] [uj : j ∈ J] B0 : J ⊆ {1, . . . , n} . Corollary (Lov´ asz ’89) If B0 = 0, then max{rank A : x1 , . . . , xn} = min{dim uj : j ∈ J + dim vj : j J : J ⊆ {1, . . . , n}} Tasuku Soma (Kyoto Univ.) Fast Matrix Completion Algorithms 12 / 29

Simultaneous Matrix Completion by Rank-One Matrices Simultaneous Matrix Completion by Rank-One Matrices F: Field Input Collection A of matrices in the form of B0 + x1B1 + · · · + xnBn Find Value assignments αi ∈ F for each indeterminate xi maximizing the rank of every matrix in A Tasuku Soma (Kyoto Univ.) Fast Matrix Completion Algorithms 13 / 29

Simultaneous Matrix Completion by Rank-One Matrices Simultaneous Matrix Completion by Rank-One Matrices F: Field Input Collection A of matrices in the form of B0 + x1B1 + · · · + xnBn Find Value assignments αi ∈ F for each indeterminate xi maximizing the rank of every matrix in A Theorem A solution of simultaneous matrix completion by rank-one matrices can be found in polynomial time, if |F| > |A|. Tasuku Soma (Kyoto Univ.) Fast Matrix Completion Algorithms 13 / 29

1 Introduction 2 Matrix Completion by Rank-One Matrices 3 Application to Network Coding 4 Mixed Skew-Symmetric Matrix Completion 5 Skew-Symmetric Matrix Completion by Rank-Two Skew-Symmetric Matrices 6 Conclusion Tasuku Soma (Kyoto Univ.) Fast Matrix Completion Algorithms 14 / 29

Network Coding Network communication model s.t. intermediate nodes can perform coding Classical model Network coding Tasuku Soma (Kyoto Univ.) Fast Matrix Completion Algorithms 15 / 29

Multicast Problem with Linearly Correlated Sources Messages in source nodes are linearly correlated Each sink node demands the original messages x1 & x2 Theorem A solution of this multicast can be found in polynomial time. Approach: simultaneous matrix completion by rank-one matrices. Tasuku Soma (Kyoto Univ.) Fast Matrix Completion Algorithms 16 / 29

1 Introduction 2 Matrix Completion by Rank-One Matrices 3 Application to Network Coding 4 Mixed Skew-Symmetric Matrix Completion 5 Skew-Symmetric Matrix Completion by Rank-Two Skew-Symmetric Matrices 6 Conclusion Tasuku Soma (Kyoto Univ.) Fast Matrix Completion Algorithms 17 / 29

Problem Definition Mixed Skew-Symmetric Matrix Completion Matrix completion for a skew-symmetric matrix s.t. each indeterminate appears twice (mixed skew-symmetric matrix). Example A =           0 −1 1 1 0 0 −1 0 0           +           0 x 0 −x 0 y 0 −y 0           =           0 −1 + x 1 1 − x 0 y −1 −y 0           Tasuku Soma (Kyoto Univ.) Fast Matrix Completion Algorithms 18 / 29

Our Result There were no algorithms for this problem, but we can compute the rank. Murota ’03 (←Geelen, Iwata & Murota ’03) The rank of an m × m mixed skew-symmetric matrix can be computed in O(m4) time. Tasuku Soma (Kyoto Univ.) Fast Matrix Completion Algorithms 19 / 29

Our Result There were no algorithms for this problem, but we can compute the rank. Murota ’03 (←Geelen, Iwata & Murota ’03) The rank of an m × m mixed skew-symmetric matrix can be computed in O(m4) time. Our Result Matrix completion for an m × m mixed skew-symmetric matrix can be done in O(m4) time. Tasuku Soma (Kyoto Univ.) Fast Matrix Completion Algorithms 19 / 29

Rank of Mixed Skew-Symmetric Matrix Lemma (Murota ’03) For an m × m mixed skew-symmetric matrix A = Q + T (Q: constant part, T: indeterminates part), rank A = max |FQ FT | : both Q[FQ], T[FT ] are nonsingular RHS is linear delta-covering. Optimal FQ and FT can be found in O(m4) time (Geelen, Iwata & Murota ’03). Tasuku Soma (Kyoto Univ.) Fast Matrix Completion Algorithms 20 / 29

Support Graph and Pfaffian Support graph: A =                0 −2 1 1 2 0 0 3 −1 0 0 2 1 −3 −2 0                Tasuku Soma (Kyoto Univ.) Fast Matrix Completion Algorithms 21 / 29

Support Graph and Pfaffian Support graph: A =                0 −2 1 1 2 0 0 3 −1 0 0 2 1 −3 −2 0                Pfaffian: pf A := M:perfect matching in G ± ij∈M Aij = A12A34 − A13A24 Lemma det A = (pf A)2 Tasuku Soma (Kyoto Univ.) Fast Matrix Completion Algorithms 21 / 29

Sketch of Algorithm Algorithm 1: Find an optimal solution FQ and FT for linear delta-covering. 2: Find a perfect matching M in the support graph of T[FT ]. Tasuku Soma (Kyoto Univ.) Fast Matrix Completion Algorithms 22 / 29

Sketch of Algorithm Algorithm 1: Find an optimal solution FQ and FT for linear delta-covering. 2: Find a perfect matching M in the support graph of T[FT ]. 3: for each ij ∈ M do 4: Substitute α to Tij so that Q[FQ ∪ {i, j}] will be nonsingular after substitution. 5: FQ := FQ ∪ {i, j} 6: end for Tasuku Soma (Kyoto Univ.) Fast Matrix Completion Algorithms 22 / 29

Sketch of Algorithm Algorithm 1: Find an optimal solution FQ and FT for linear delta-covering. 2: Find a perfect matching M in the support graph of T[FT ]. 3: for each ij ∈ M do 4: Substitute α to Tij so that Q[FQ ∪ {i, j}] will be nonsingular after substitution. 5: FQ := FQ ∪ {i, j} 6: end for 7: Substitute 0 to the rest of indeterminates 8: return the resulting matrix Tasuku Soma (Kyoto Univ.) Fast Matrix Completion Algorithms 22 / 29

Sketch of Algorithm How can we find α s.t. Q[FQ ∪ {i, j}] will be nonsingular? A = Q + T Tasuku Soma (Kyoto Univ.) Fast Matrix Completion Algorithms 23 / 29

Sketch of Algorithm How can we find α s.t. Q[FQ ∪ {i, j}] will be nonsingular? A = Q + T A = Q + T Tasuku Soma (Kyoto Univ.) Fast Matrix Completion Algorithms 23 / 29

Sketch of Algorithm How can we find α s.t. Q[FQ ∪ {i, j}] will be nonsingular? A = Q + T A = Q + T Lemma Q : modified matrix of Q as Q ij := Qij + α, Q ji := Qji − α pf Q [FQ ∪ {i, j}] = pf Q[FQ ∪ {i, j}] ± α · pf Q[FQ] Tasuku Soma (Kyoto Univ.) Fast Matrix Completion Algorithms 23 / 29

Sketch of Algorithm Finally, we obtain Q s.t. rank Q = rank A. Theorem Matrix completion for an m × m mixed skew-symmetric matrix can be done in O(m4) time. Using delta-covering algortihm of Geelen, Iwata & Murota ’03 Tasuku Soma (Kyoto Univ.) Fast Matrix Completion Algorithms 24 / 29

1 Introduction 2 Matrix Completion by Rank-One Matrices 3 Application to Network Coding 4 Mixed Skew-Symmetric Matrix Completion 5 Skew-Symmetric Matrix Completion by Rank-Two Skew-Symmetric Matrices 6 Conclusion Tasuku Soma (Kyoto Univ.) Fast Matrix Completion Algorithms 25 / 29

Problem Definition Skew-Symmetric Matrix Completion by Rank-Two Skew-Symmetric Matrices Matrix completion for A = B0 + x1B1 + · · · + xnBn, where B0 is skew-symmetric and B1 , . . . , Bn are rank-two skew-symmteric Tasuku Soma (Kyoto Univ.) Fast Matrix Completion Algorithms 26 / 29

Our Result In the case of B0 = 0: Lov´ asz ’89 This can be reduced to linear matroid parity. solvable in O(m3n) time using the algorithm of Gabow & Stallman ’86. Tasuku Soma (Kyoto Univ.) Fast Matrix Completion Algorithms 27 / 29

Our Result In the case of B0 = 0: Lov´ asz ’89 This can be reduced to linear matroid parity. solvable in O(m3n) time using the algorithm of Gabow & Stallman ’86. For the general case: Our Result An optimal solution can be found in O((m + n)4) time. Idea: Reduction to mixed skew-symmetric matrix completion (similar to matrix completion by rank-one matrices) Tasuku Soma (Kyoto Univ.) Fast Matrix Completion Algorithms 27 / 29

1 Introduction 2 Matrix Completion by Rank-One Matrices 3 Application to Network Coding 4 Mixed Skew-Symmetric Matrix Completion 5 Skew-Symmetric Matrix Completion by Rank-Two Skew-Symmetric Matrices 6 Conclusion Tasuku Soma (Kyoto Univ.) Fast Matrix Completion Algorithms 28 / 29

Conclusion Our Results Faster algorithm and Min-Max theorem for matrix completion by rank-one matrices. Application for multicast problem with linearly correlated sources. First deterministic polynomial time algorithm for mixed skew-symmetric matrix completion. First deterministic polynomial time algorithm for skew-symmetric matrix completion by rank-two skew-symmetric matrices. Tasuku Soma (Kyoto Univ.) Fast Matrix Completion Algorithms 29 / 29

Conclusion Our Results Faster algorithm and Min-Max theorem for matrix completion by rank-one matrices. Application for multicast problem with linearly correlated sources. First deterministic polynomial time algorithm for mixed skew-symmetric matrix completion. First deterministic polynomial time algorithm for skew-symmetric matrix completion by rank-two skew-symmetric matrices. Future Works Application of skew-symmetric matrix completion Matrix completion for other types of matrices Tasuku Soma (Kyoto Univ.) Fast Matrix Completion Algorithms 29 / 29