Complexity of Matrix Rank and Rigidity y Jayalal Sarma M.N. z 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 n£n 2 k 3 Rank of a matrix M 2 F has the following 0 2 0 0 RM(b; n ¡ 1) doesn't exist, 6 k 7 2k equivalent de¯nitions. 6 0 0 2 0 7 unless b ¸ n . 6 k 7 ² The size of the largest submatrix with a non- 4 0 0 0 2 5 zero determinant. Rank Computation : Algebraic ! Question : Can we test this? i.e, for a given ² 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 ja j ¸ ja j ii X ij Essential Ideas.. j6=i ² From Linear Algebra : Dimension of solution Equivalent formulation : De¯ne 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 ² From Control Theory : Rank of a matrix Question : Is there a rank r matrix B 2 [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 de¯nes an interval for each entry ² 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: ² 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 de¯ned 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 ² 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. ² Can be solved in Polynomial time; Gaussian change to bring the rank below r elimination : inherently sequential. NP algorithm : Guess the \nice" singular ma- ² A natural linear algebraic optimisation ² 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 coe±cients of the characterstic polynomial are zeros. ² Interesting in a circuit complexity theory Future Work/Open Problems setting. Highly rigid linear transforma- ² Re¯ned complexity bounds by [ABO99]. tions (matrices) have some \nice" size-depth Upperbound testing is complete for C=L. ² 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 ² Is there a recursive(or better) upperbound for rigidity over in¯nite ¯elds? Classi¯cation of problems in P solved by vari- rigid(M; r; k): Given a matrix M, values r ² 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 ² 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 ² Over in¯nite ¯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 ² 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 yJoint work with Meena Mahajan (IMSc) bring the rank below r, if the change Email: fmeena,[email protected] allowed per entry is atmost b. z Full version available as ECCC technical report at http://eccc.hpi-web.de/eccc-reports/2006/TR06-100/.