Handicap reduction for linear complementarity problems
Linear Complementarity Problems (LCPs) with sufficient matrices form an important subclass of LCPs, and it remains a significant open question whether problems in this class can be solved in polynomial time. Kojima, Megiddo, Noma, and Yoshise gave an Interior Point Algorithm (IPA) in 1991, that can solve LCPs with sufficient matrices in time bounded polynomially in the input size and the so-called handicap number $\hatκ(M)$ of the coefficient matrix $M$. However, this value can be exponentially large in the bit encoding length. In fact, no upper bounds were previously known on $\hatκ(M)$. Settling an open question raised in de Klerk and E.-Nagy (Math Programming, 2011), we give an exponential upper bound on $\hatκ(M)$ in the bit-complexity of $M$. This is based on a new characterization of sufficient matrices. The new characterization also leads to a simple new proof of Väliaho's theorem on the equivalence of sufficient and $\mathcal{P}^*$-matrices (Linear Algebra and its Applications, 1996). Noting that one can obtain an equivalent LCP by rescaling the rows and columns by a positive diagonal matrix, we define $\hatκ^\star(M)$ as the best possible handicap number achievable under such rescalings. Our second main result is an algorithm for LCPs with sufficient matrices, where the running time is polynomially bounded in the input size and in the optimized value $\hatκ^\star(M)$. This algorithm is based on the observation that the set of near-optimal row-rescalings forms a convex set. Our algorithm combines the Ellipsoid Method over the set of row rescalings, and an IPA with running time dependent on the handicap number of the matrix. If the IPA fails to solve the LCP in the desired running time, it provides a separation oracle to the Ellipsoid Method to find a better rescaling.