A strongly polynomial algorithm for generalized flow maximization

A strongly polynomial algorithm is given for the generalized flow maximization problem. It uses a new variant of the scaling technique, called continuous scaling. The main measure of progress is that within a strongly polynomial number of steps, an arc can be identified that must be tight in every dual optimal solution, and thus can be contracted. As a consequence of the result, we also obtain a strongly polynomial algorithm for the linear feasibility problem with at most two nonzero entries per column in the constraint matrix.Comment: minor correction

Truthful Assignment without Money

We study the design of truthful mechanisms that do not use payments for the generalized assignment problem (GAP) and its variants. An instance of the GAP consists of a bipartite graph with jobs on one side and machines on the other. Machines have capacities and edges have values and sizes; the goal is to construct a welfare maximizing feasible assignment. In our model of private valuations, motivated by impossibility results, the value and sizes on all job-machine pairs are public information; however, whether an edge exists or not in the bipartite graph is a job's private information. We study several variants of the GAP starting with matching. For the unweighted version, we give an optimal strategyproof mechanism; for maximum weight bipartite matching, however, we show give a 2-approximate strategyproof mechanism and show by a matching lowerbound that this is optimal. Next we study knapsack-like problems, which are APX-hard. For these problems, we develop a general LP-based technique that extends the ideas of Lavi and Swamy to reduce designing a truthful mechanism without money to designing such a mechanism for the fractional version of the problem, at a loss of a factor equal to the integrality gap in the approximation ratio. We use this technique to obtain strategyproof mechanisms with constant approximation ratios for these problems. We then design an O(log n)-approximate strategyproof mechanism for the GAP by reducing, with logarithmic loss in the approximation, to our solution for the value-invariant GAP. Our technique may be of independent interest for designing truthful mechanisms without money for other LP-based problems.Comment: Extended abstract appears in the 11th ACM Conference on Electronic Commerce (EC), 201

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

Shortest path and maximum flow problems in planar flow networks with additive gains and losses

In contrast to traditional flow networks, in additive flow networks, to every edge e is assigned a gain factor g(e) which represents the loss or gain of the flow while using edge e. Hence, if a flow f(e) enters the edge e and f(e) is less than the designated capacity of e, then f(e) + g(e) = 0 units of flow reach the end point of e, provided e is used, i.e., provided f(e) != 0. In this report we study the maximum flow problem in additive flow networks, which we prove to be NP-hard even when the underlying graphs of additive flow networks are planar. We also investigate the shortest path problem, when to every edge e is assigned a cost value for every unit flow entering edge e, which we show to be NP-hard in the strong sense even when the additive flow networks are planar