Algebraic combinatorial optimization on the degree of determinants of noncommutative symbolic matrices Tasuku Soma (ISM) Joint work with Hiroshi Hirai (Nagoya) Yuni Iwamasa (Kyoto) Taihei Oki (Tokyo) 1 / 14

Matrix formulation of bipartite matching G = (V1 , V2 ; E): bipartite graph with |V1 | = |V2 | = n 1 2 3 1 2 3 2 / 14

Matrix formulation of bipartite matching G = (V1 , V2 ; E): bipartite graph with |V1 | = |V2 | = n Edmonds matrix: n × n symbolic matrix A s.t. aij = xij if ij ∈ E 0 otherwise 1 2 3 1 2 3 A =   x11 x12 x13 0 x22 x23 x31 0 x33   2 / 14

Matrix formulation of bipartite matching G = (V1 , V2 ; E): bipartite graph with |V1 | = |V2 | = n Edmonds matrix: n × n symbolic matrix A s.t. aij = xij if ij ∈ E 0 otherwise Then, det A = M: perfect matching ± ij∈M xij det A ̸= 0 ⇐⇒ G has a perfect matching. 1 2 3 1 2 3 A =   x11 x12 x13 0 x22 x23 x31 0 x33   det A = + x11 x22 x33 + x12 x23 x31 − x13 x22 x31 2 / 14

Edmonds’ problem [Edmonds, 1967] F: field Given: A = x1 A1 + · · · + xm Am (xi: indeterminate, Ai: n × n matrix over F) Compute: rank A (over F[x1 , . . . , xm ]) • If we can use randomness, it is easy! (Schwartz-Zippel lemma) Can we do deterministically? −→ polynomial identity testing • Includes linear matroid intersection, general matching, linear matroid parity, etc. [Lovász, 1989] 3 / 14

Various matrix formulations Linear matroid intersection ... rank-one Ai = ai b⊤ i General matching ... Tutte matrix: Ai = eu e⊤ v − ev e⊤ u for each edge uv Linear matroid parity ... rank-two skew-symmetric Ai = ai b⊤ i − bi a⊤ i 4 / 14

bipartite matching Edmonds matrix linear matroid intersection rank-one Ai general matching Tutte matrix linear matroid parity rank-two skew-symmetric Ai 5 / 14

Noncommutative Edmonds’ problem A = x1 A1 + · · · + xm Am Noncommutative rank nc-rank A = min{n − dim U + dim( i Ai U) : U subspace in Fn}. • “rank” of A where xi xj ̸= xj xi; formally defined over free skew field [Amitsur, 1966] • rank A ≤ nc-rank A ≤ 2 rank A • Optimal U is called a shrunk subspace; is dominant if dim U is minimum 6 / 14

Noncommutative Edmonds’ problem A = x1 A1 + · · · + xm Am Noncommutative rank nc-rank A = min{n − dim U + dim( i Ai U) : U subspace in Fn}. • “rank” of A where xi xj ̸= xj xi; formally defined over free skew field [Amitsur, 1966] • rank A ≤ nc-rank A ≤ 2 rank A • Optimal U is called a shrunk subspace; is dominant if dim U is minimum Noncommutative Edmonds’ problem Compute: nc-rank A Recent Breakthrough: Computing nc-rank of A is in P [Garg et al., 2020; Ivanyos, Qiao, and Subrahmanyam, 2018; Hamada and Hirai, 2021] 6 / 14

bipartite matching Edmonds matrix linear matroid intersection rank-one Ai general matching Tutte matrix linear matroid parity rank-two skew-symmetric Ai fractional matching Tutte matrix frac linear matroid parity rank-two skew-symmetric Ai rank nc-rank rank = nc-rank 7 / 14

Weighted Edmonds’ problem Given: A[w] = tw1 x1 A1 + · · · + twm xm Am (xi: indeterminate, Ai ∈ Fn×n, t: new indeterminate, weight wi ∈ Z) Compute: degt det A[w] 8 / 14

Weighted Edmonds’ problem Given: A[w] = tw1 x1 A1 + · · · + twm xm Am (xi: indeterminate, Ai ∈ Fn×n, t: new indeterminate, weight wi ∈ Z) Compute: degt det A[w] Example A: Edmonds matrix of a bipartite graph, w: edge weight det A[w] = M: perfect matching ±tw(M) ij∈M xij degt det A[w] = max M: perfect matching w(M) Maximum weight perfect matching! 8 / 14

Noncommutative weighted Edmonds’ problem Given: A[w] = tw1x1 A1 + · · · + twpxp Ap (xi: nc-indeterminate, Ai ∈ Fn×n, t: commutative indeterminate, weight wi ∈ Z) Compute: degt Det A[w] Here, Det is the Dieudonné determinant, an nc-version of ordinary determinant. • Known result: strongly polynomial deterministic algorithm [Hirai and Ikeda, 2022] • Caveat: not combinatorial; relies on Frank–Tardos framework (so time complexity is merely poly) 9 / 14

bipartite matching Edmonds matrix linear matroid intersection rank-one Ai general matching Tutte matrix linear matroid parity rank-two skew-symmetric Ai fractional matching Tutte matrix frac linear matroid parity rank-two skew-symmetric Ai det Det det = Det 10 / 14

Our results • Combinatorial algorithm for weighted noncommutative Edmonds’ problem without Frank–Tardos framework. • Time-complexity: O(n3NCRK), where NCRK = time complexity to find dominant shrunk subspace of n × n matrix. • Application: first poly-time membership algorithm for rank-2 Brascamp-Lieb polytope. (previous result: NP ∩ co-NP [Franks, Soma, and Goemans, 2023]) • Can be extended to compute degrees of all k × k minors: ∆k (A[w]) = max degt Det(A[w]S,T ) : S, T ∈ [n] k for k = 0, 1, . . . , n. 11 / 14

Min-max theorem of degDet Theorem (Hirai (2019)) degt Det A[w] = min P,Q∈GLn,α,β∈Zn i αi + βi : degt ((t−α)PAk Q(t−β))ij ≤ −wk (i, j ∈ [n], k ∈ [m]) , where (t−α) :=   t−α1 ... t−αn  . • Can recover min-max theorems for bipartite matching, linear matroid intersection, fractional matching, and fractional linear matroid parity. • Our algorithm maintains feasible P, Q, α, β. 12 / 14

Algorithm (at very high level) “Algebraic Hungarian method” feasible solution P, Q ∈ GLn, α, β ∈ Zn nc-rank algorithm 1. modified unweighted instance 2. optimal shrunk subspace 3. update dual • Nc-rank algorithm needs to find optimal shrunk subspace • Strongly poly-time if nc-rank alg finds the dominant shrunk subspace; possible by several (but not all) nc-rank algorithms [Ivanyos, Qiao, and Subrahmanyam, 2018; Franks, Soma, and Goemans, 2023] • Inspired by weighted fractional matroid parity algorithm by Gijswijt–Pap [Gijswijt and Pap, 2013] 13 / 14

Conclusion Our results • First combinatorial strongly polynomial algorithm for weighted noncommutative Edmonds’ problem • Application: first poly-time membership algorithm for rank-2 Brascamp–Lieb polytope Future direction • More applications in combinatorial optimization? 14 / 14

Conclusion Our results • First combinatorial strongly polynomial algorithm for weighted noncommutative Edmonds’ problem • Application: first poly-time membership algorithm for rank-2 Brascamp–Lieb polytope Future direction • More applications in combinatorial optimization? Thanks! Questions? 14 / 14

References I Amitsur, S. A. (1966). “Rational identities and applications to algebra and geometry”. In: Journal of Algebra 3, pp. 304–359. Barthe, F. (1998). “On a Reverse Form of the Brascamp-Lieb Inequality”. In: Inventiones Mathematicae 134.2, pp. 335–361. doi: 10.1007/s002220050267. Bennett, J., A. Carbery, M. Christ, and T. Tao (2008). “The Brascamp-Lieb inequalities: Finiteness, structure and extremals”. In: Geometric and Functional Analysis 17, pp. 1343–1415. Brascamp, H. and E. Lieb (1976). “Best constants in Young’s inequality, its converse, and its generalization to more than three functions”. In: Advances in Mathematics 20, pp. 151–173. Chang, S., D. C. Llewellyn, and J. H. Vande Vate (2001). “Two-lattice polyhedra: duality and extreme points”. In: Discrete Mathematics 237, pp. 63–95. Edmonds, J. (1967). “Systems of distinct representatives and linear algebra”. In: Journal of Research of the National Bureau of Standards 71B, pp. 241–245. Franks, C., T. Soma, and M. X. Goemans (2023). “Shrunk subspaces via operator Sinkhorn iteration”. In: Proceedings of the 2023 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA). SIAM, pp. 1655–1668. Garg, A., L. Gurvits, R. Oliveira, and A. Wigderson (2020). “Operator scaling: theory and applications”. In: Foundations of Computational Mathematics 20, pp. 223–290. Gijswijt, D. and G. Pap (2013). “An algorithm for weighted fractional matroid matching”. In: Journal of Combinatorial Theory, Series B 103, pp. 509–520. Hamada, M. and H. Hirai (2021). “Computing the nc-rank via discrete convex optimization on CAT(0) spaces”. In: SIAM Journal on Applied Geometry and Algebra 5, pp. 455–478. Hirai, H. (2019). “Computing the degree of determinants via discrete convex optimization on Euclidean buildings”. In: SIAM Journal on Applied Geometry and Algebra 3, pp. 523–557. 1 / 5

References II Hirai, H. and M. Ikeda (2022). “A cost-scaling algorithm for computing the degree of determinants”. In: Computational Complexity 31. Ivanyos, G., Y. Qiao, and K. V. Subrahmanyam (2018). “Constructive noncommutative rank computation in deterministic polynomial time over fields of arbitrary characteristics”. In: Computational Complexity 27, pp. 561–593. Lovász, L. (1989). “Singular spaces of matrices and their application in combinatorics”. In: Boletim da Sociedade Brasileira de Matemática 20, pp. 87–99. Vande Vate, J. H. (1992). “Fractional matroid matchings”. In: Journal of Combinatorial Theory, Series B 55, pp. 133–145. 2 / 5

Dieudonné determinant Bruhat decomposition Any square matrix A over a skew field K can be decomposed as A = LDPU where L is lower unitriangular, D diagonal, P permutation, U upper unitriangular over K. Dieudonné determinant Det A := sgn P · i dii mod [K×, K×] • Det(AB) = Det A Det B • Det A = 0 ⇐⇒ nc-rank A < n 3 / 5

Brascamp–Lieb Polyotope [Bennett et al., 2008] Bi: ni × n real matrix, row full-rank (i = 1, . . . , m) BLP =    x ∈ Rm : m i=1 xi dim(Bi V ) ≥ dim(V ) (V : subspace in Rn) m i=1 xi ni = n xi ≥ 0 (i = 1, . . . , m)    rank-r BL polytope △ ⇐⇒ ni = r (i = 1, . . . , m) • Characterization of optimal constant of Brascamp–Lieb inequality [Brascamp and Lieb, 1976] • rank-1 BL = base polytope of linear matroid [Barthe, 1998] • rank-2 BL = fractional linear matroid parity polytope ⊋ common base polytope of two linear matroids [Franks, Soma, and Goemans, 2023] 4 / 5

Frac linear matroid parity polytope [Vande Vate, 1992] Lines ℓi = span{ai , bi } ⊆ Fn (i = 1, . . . , m) FM := x ∈ Rm : m i=1 dim(ℓi ∩ V )xi ≤ dim V (V : subspace in Fn) xi ≥ 0 (i = 1, . . . , m) • Totally dual half-integrality [Vande Vate, 1992] • integral vertex of FM = parity base • For F = R, FM coincides with (down-closure of) rank-2 BL polytope [Franks, Soma, and Goemans, 2023] • Combinatorial alg for unweighted [Chang, Llewellyn, and Vande Vate, 2001] and weighted problems [Gijswijt and Pap, 2013]; polynomial-time in arithmetic model 5 / 5