A scaling-invariant algorithm for linear programming whose running time depends only on the constraint matrix

Following the breakthrough work of Tardos (Oper Res 34:250–256, 1986) in the bit-complexity model, Vavasis and Ye (Math Program 74(1):79–120, 1996) gave the first exact algorithm for linear programming in the real model of computation with running time depending only on the constraint matrix. For solving a linear program (LP) max c^⊤ x, Ax = b, x ≥ 0, A ∈ℝ^m × n , Vavasis and Ye developed a primal-dual interior point method using a ‘layered least squares’ (LLS) step, and showed that O(n^3.5log (χ̅_A+n)) iterations suffice to solve (LP) exactly, where χ̅_A is a condition measure controlling the size of solutions to linear systems related to A . Monteiro and Tsuchiya (SIAM J Optim 13(4):1054–1079, 2003), noting that the central path is invariant under rescalings of the columns of A and c , asked whether there exists an LP algorithm depending instead on the measure χ̅^*_A , defined as the minimum χ̅_AD value achievable by a column rescaling AD of A , and gave strong evidence that this should be the case. We resolve this open question affirmatively. Our first main contribution is an O(m^2 n^2 + n^3) time algorithm which works on the linear matroid of A to compute a nearly optimal diagonal rescaling D satisfying χ̅_AD≤ n(χ̅_A^*)^3 . This algorithm also allows us to approximate the value of χ̅_A up to a factor n (χ̅_A^*)^2 . This result is in surprising contrast to that of Tunçel (Math Program 86(1):219–223, 1999), who showed NP-hardness for approximating χ̅_A to within 2^poly(rank(A)) . The key insight for our algorithm is to work with ratios g_i/g_j of circuits of A —i.e., minimal linear dependencies Ag=0 —which allow us to approximate the value of χ̅_A^* by a maximum geometric mean cycle computation in what we call the ‘circuit ratio digraph’ of A . While this resolves Monteiro and Tsuchiya’s question by appropriate preprocessing, it falls short of providing either a truly scaling invariant algorithm or an improvement upon the base LLS analysis. In this vein, as our second main contribution we develop a scaling invariant LLS algorithm, which uses and dynamically maintains improving estimates of the circuit ratio digraph, together with a refined potential function based analysis for LLS algorithms in general. With this analysis, we derive an improved O(n^2.5log (n)log (χ̅^*_A+n)) iteration bound for optimally solving (LP) using our algorithm. The same argument also yields a factor n/log n improvement on the iteration complexity bound of the original Vavasis–Ye algorithm.