Complexity of Matrix Rank and Rigidity †

Jayalal Sarma M.N. ‡ The Institute of Mathematical Sciences, C.I.T. Campus, Chennai, India.

Turing/Circuit Model : Combinatorial ! RM(b, r) need not always exist ! Consider, Matrix Rank Seperation of small classes : Unknown 2k 0 0 0 nn  k  Rank of a matrix M ∈ F has the following 0 2 0 0 RM(b, n 1) doesn’t exist,  k  2k equivalent denitions.  0 0 2 0  unless b n .  k  The size of the largest submatrix with a non-  0 0 0 2  zero determinant. Rank Computation : Algebraic ! Question : Can we test this? i.e, for a given The number of linearly independent Characterising computation might help. matrix M, bound b, target rank r, can we ef- rows/columns of a matrix. Several applications do have inherent structure ciently test, whether RM(b, r) exists ? rank bound: Given a matrix M and a value for the matrices. It is NP-hard for arbitrary r and NP-complete r, is rank(M) < r?. M = [ai,j] is diagonally dominant if for the case of singularity.

Motivation |a | |a | ii X ij Essential Ideas.. j6=i From Linear Algebra : Dimension of solution Equivalent formulation : Dene an interval of Fun fact : If for all i, the dominance is strict space of a system of linear equations. matrices [A] where m b a m + b then M in non-singular. ij ij ij From Control Theory : Rank of a matrix Question : Is there a rank r matrix B ∈ [A] Matrix type rank bound singular can be used to classify a linear system as such that M B has atmost k non-zero en- C L C L controllable, or observable. General = -complete = -complete tries? [ABO99] [ABO99] The bound b denes an interval for each entry From Algorithmics : Some natural algorith- Sym.Non-neg. C=L-complete C=L-complete of the matrix. The determinant is a multlilin- mic problems can be expressed in terms of [ABO99] [ABO99] ear polynomial on the entries of the matrix. rank and determinant computation. Sym.Non-neg. Now use the following lemma: From Complexity Theory : Characterising Diag. Dom. L-complete L-complete 0 0 complexity classes might facilitate applica- Diagonal TC -complete in AC Zero-on-an-edge Lemma tion of the well studied algebraic techniques. For a multilinear polynomial p on t vari- How close is M to a rank r matrix? ables, consider the t-dimensional hypercube Computing the Rank Given a matrix M and r < n, rigidity dened by the interval of each of the vari- of the matrix M (RM(r)) is the number ables. If there is a zero of the polynomial The naive approach : EXPonential time. of entries of the matrix that we need to in the hypercube then there is a zero on an edge of the hypercube. Can be solved in Polynomial time; Gaussian change to bring the rank below r elimination : inherently sequential. NP algorithm : Guess the “nice” singular ma- A natural linear algebraic optimisation Rank can be computed in NC. Elegant trix and verify. Hardness: A reduction from problem, with important applications in MAX-CUT problem. parallel algorithm ([Mul87]) by relating the control theory. problem to testing if some coecients of the characterstic polynomial are zeros. Interesting in a circuit complexity theory Future Work/Open Problems setting. Highly rigid linear transforma- Rened complexity bounds by [ABO99]. tions (matrices) have some “nice” size-depth Upperbound testing is complete for C=L. Are there characterisations of other small tradeo in circuits computing them [Val77]. complexity classes (like NL) using the rank/determinant computation? Complexity Theory Preliminaries Computing Rigidity Is there a recursive(or better) upperbound for rigidity over innite elds? Classication of problems in P solved by vari- rigid(M, r, k): Given a matrix M, values r Can bounded rigidity be decided in NP? - is ous resource bounded models of computation. and k, is RM(r) k?. there a generalisation of the zero-on-an-edge Class Resource Bound Complete Problem Over any nite eld F, rigid is in NP. Over lemma to arbitrary rank? L log space Reachability in F , rigid is NP-hard too [Des] : reduction deterministic TM undirected forests 2 References from Nearest Neighbour Decoding problem. NL log space Reachability in nondeterminsitic TM directed graphs Over innite elds we don’t even know if it [ABO99] E. Allender, R. Beals, and M. Ogi- C=L log space Singularity of is decidable. hara. The complexity of matrix rank “balanced” NTM boolean matrices If k is constant, restriced to boolean matri- and feasible systems of linear equa- 0 Computational Complex- AC poly size, constant Reachability in ces, rigid is C=L-complete. tions. In depth circuits const. width maze ity, 8, 99-126, 1999, 1999. In many applications, the amount of change TC1 AC0 + “majority” Testing Majority of the matrix entries dictates the cost. So we [Mul87] K. Mulmuley. A fast parallel algo- would like the changes to be small. rithm to compute the rank of a ma- trix over an arbitrary eld. Combi- natorica, 7:101–104, 1987. Bounded Rigidity [Val77] L. G. Valiant. Graph theoretic ar- M r < n Given a matrix and and guments in low-level complexity. In b bounded rigidity M , of the matrix MFCS, 1977. (RM(b, r)) is the number of entries of the matrix that we need to change to †Joint work with Meena Mahajan (IMSc) bring the rank below r, if the change Email: {meena,jayalal}@imsc.res.in allowed per entry is atmost b. ‡ Full version available as ECCC technical report at http://eccc.hpi-web.de/eccc-reports/2006/TR06-100/