A note on polylinking flow networks

This is a supplementary note on M. X. Goemans, S. Iwata, and R. Zenklusen’s paper that proposes a flow model based on polylinking systems. Their flow model is a series (or tandem) connection of polylinking systems. We can consider an apparently more general model of a polylinking flow network which consists of an ordinary arc-capacitated network endowed with polylinking systems on the vertex set, one for each vertex of the network. This is a natural, apparent generalization of polymatroidal flow model of E. L. Lawler and C. U. Martel and of generalized-polymatroidal flow model of R. Hassin. We give a max-flow min-cut formula for the polylinking network flow problem and discuss some acyclic flow property of polylinking flows

Curvature and Optimal Algorithms for Learning and Minimizing Submodular Functions

We investigate three related and important problems connected to machine learning: approximating a submodular function everywhere, learning a submodular function (in a PAC-like setting [53]), and constrained minimization of submodular functions. We show that the complexity of all three problems depends on the 'curvature' of the submodular function, and provide lower and upper bounds that refine and improve previous results [3, 16, 18, 52]. Our proof techniques are fairly generic. We either use a black-box transformation of the function (for approximation and learning), or a transformation of algorithms to use an appropriate surrogate function (for minimization). Curiously, curvature has been known to influence approximations for submodular maximization [7, 55], but its effect on minimization, approximation and learning has hitherto been open. We complete this picture, and also support our theoretical claims by empirical results.Comment: 21 pages. A shorter version appeared in Advances of NIPS-201

Notes on Graph Cuts with Submodular Edge Weights

Generalizing the cost in the standard min-cut problem to a submodular cost function immediately makes the problem harder. Not only do we prove NP hardness even for nonnegative submodular costs, but also show a lower bound of (|V |1/3) on the approximation factor for the (s, t) cut version of the problem. On the positive side, we propose and compare three approximation algorithms with an overall approximation factor of O(min|V |,p|E| log |V |) that appear to do well in practice