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.

翻译：具有充分矩阵的线性互补问题构成了LCPs的一个重要子类，而该子类问题是否能在多项式时间内求解仍是一个重要的开放问题。Kojima、Megiddo、Noma和Yoshise于1991年提出了一种内点算法，该算法能在输入规模和系数矩阵M的所谓障碍数$\hatκ(M)$的多项式有界时间内求解具有充分矩阵的线性互补问题。然而，该数值在位编码长度上可能是指数级的。事实上，此前对$\hatκ(M)$的上界一无所知。针对de Klerk和E.-Nagy（《数学规划》，2011）提出的一个开放问题，我们给出了$\hatκ(M)$在M的位复杂度下的指数级上界。该结果基于对充分矩阵的一种新刻画。这一新刻画还导致了对Väliaho关于充分矩阵与$\mathcal{P}^*$矩阵等价性定理（《线性代数及其应用》，1996）的一个简洁新证明。注意到可以通过正对角矩阵对行和列进行重新缩放得到等价LCP，我们将$\hatκ^\star(M)$定义为在这种重新缩放下可实现的最佳障碍数。我们的第二个主要结果是针对具有充分矩阵的LCPs的一种算法，其运行时间在输入规模和优化值$\hatκ^\star(M)$上是多项式有界的。该算法基于以下观察：近似最优的行重新缩放集合构成一个凸集。我们的算法结合了行重新缩放集上的椭球法，以及一种运行时间依赖于矩阵障碍数的内点算法。如果内点算法未能在期望时间内求解LCP，它将为椭球法提供分离超平面以寻找更优的重新缩放。