Stateless Distributed Gradient Descent for Positive Linear Programs

We develop a framework of distributed and stateless solutions for packing and covering linear programs (LPs), which are solved by multiple agents operating in a cooperative but uncoordinated manner. Our model has a separate “agent” controlling each variable, and an agent is allowed to read off the current values only of those constraints in which it has nonzero coefficients. This is a natural model for many distributed applications like flow control, maximum bipartite matching, and dominating sets. The most appealing features of our algorithms are their simplicity and polylogarithmic convergence. For the packing LP $\max\{c\cdot x\mid Ax\leq b,$ $x\geq0\}$, the algorithm associates a dual variable $y_i=\exp[\frac{1}{\epsilon}(\frac{A_ix}{b_i}-1)]$ for each constraint $i$, and each agent $j$ iteratively increases (resp., decreases) $x_j$ multiplicatively if $A_j^\top y$ is too small (resp., large) as compared to $c_j$. Our algorithm, starting from a feasible solution, always maintains feasibility and computes a $(1+\epsilon)$ approximation in $\mathrm{poly}(\frac{\ln(mn\cdot A_{\max})}{\epsilon})$ rounds. Here $m$ and $n$ are number of rows and columns of $A$, and $A_{\max}$, also known as the “width” of the LP, is the ratio of the maximum and the minimum nonzero values taken by the expression $A_{ij}/(b_ic_j)$ as the pair $i,j$ varies over the matrix. A similar algorithm works for the covering LP $\min\{b\cdot y\mid A^\top y\geq c,$ $y\geq0\}$ as well. While exponential dual variables have been used in several packing/covering linear programming (LP) algorithms before [S. Plotkin, D. Shmoys, and E. Tardos, Math. Oper. Res., 20 (1995), pp. 257–301; Y. Bartal, J. W. Byers, and D. Raz, Proceedings of the IEEE Symposium on Foundations of Computer Science, 1997; N. Garg and J. Könemann, SIAM J. Comput., 37 (2007), pp. 630–652; L. K. Fleischer, SIAM J. Discrete Math., 13 (2000), pp. 505–520; N. E. Young, Proceedings of the IEEE Symposium on Foundations of Computer Science, 2001; C. Koufogiannakis and N. E. Young, Proceedings of the IEEE Symposium on Foundations of Computer Science, 2007], this is the first algorithm which is both stateless and has polylogarithmic convergence. Our algorithms can be thought of as applying distributed gradient descent/ascent on a carefully chosen potential. Our analysis differs from those of previous multiplicative update based algorithms and argues that while the current solution is far away from optimality, the potential function decreases/increases by a significant factor.