Optimal Auctions vs. Anonymous Pricing

For selling a single item to agents with independent but non-identically distributed values, the revenue optimal auction is complex. With respect to it, Hartline and Roughgarden (2009) showed that the approximation factor of the second-price auction with an anonymous reserve is between two and four. We consider the more demanding problem of approximating the revenue of the ex ante relaxation of the auction problem by posting an anonymous price (while supplies last) and prove that their worst-case ratio is e. As a corollary, the upper-bound of anonymous pricing or anonymous reserves versus the optimal auction improves from four to $e$. We conclude that, up to an $e$ factor, discrimination and simultaneity are unimportant for driving revenue in single-item auctions.Comment: 19 pages, 6 figures, To appear in 56th Annual IEEE Symposium on Foundations of Computer Science (FOCS 2015

Resource-Aware Cost-Sharing Mechanisms with Priors

In a decentralized system with $m$ machines, we study the selfish scheduling problem where each user strategically chooses which machine to use. Each machine incurs a cost, which is a function of the total load assigned to it, and some cost-sharing mechanism distributes this cost among the machine's users. The users choose a machine aiming to minimize their own share of the cost, so the cost-sharing mechanism induces a game among them. We approach this problem from the perspective of a designer who can select which cost-sharing mechanism to use, aiming to minimize the price of anarchy (PoA) of the induced games. Recent work introduced the class of \emph{resource-aware} cost-sharing mechanisms, whose decisions can depend on the set of machines in the system, but are oblivious to the total number of users. These mechanisms can guarantee low PoA bounds for instances where the cost functions of the machines are all convex or concave, but can suffer from very high PoA for cost functions that deviate from these families. In this paper we show that if we enhance the class of resource-aware mechanisms with some prior information regarding the users, then they can achieve low PoA for a much more general family of cost functions. We first show that, as long as the mechanism knows just two of the participating users, then it can assign special roles to them and ensure a constant PoA. We then extend this idea to settings where the mechanism has access to the probability with which each user is present in the system. For all these instances, we provide a mechanism that achieves an expected PoA that is logarithmic in the expected number of users

A Submodular Approach for Electricity Distribution Network Reconfiguration

Distribution network reconfiguration (DNR) is a tool used by operators to balance line load flows and mitigate losses. As distributed generation and flexible load adoption increases, the impact of DNR on the security, efficiency, and reliability of the grid will increase as well. Today, heuristic-based actions like branch exchange are routinely taken, with no theoretical guarantee of their optimality. This paper considers loss minimization via DNR, which changes the on/off status of switches in the network. The goal is to ensure a radial final configuration (called a spanning tree in the algorithms literature) that spans all network buses and connects them to the substation (called the root of the tree) through a single path. We prove that the associated combinatorial optimization problem is strongly NP-hard and thus likely cannot be solved efficiently. We formulate the loss minimization problem as a supermodular function minimization under a single matroid basis constraint, and use existing algorithms to propose a polynomial time local search algorithm for the DNR problem at hand and derive performance bounds. We show that our algorithm is equivalent to the extensively used branch exchange algorithm, for which, to the best of our knowledge, we pioneer in proposing a theoretical performance bound. Finally, we use a 33-bus network to compare our algorithm\u27s performance to several algorithms published in the literature

Clustering on k-Edge-Colored Graphs

International audienceWe study the Max k-colored clustering problem, where, given an edge-colored graph with k colors, we seek to color the vertices of the graph so as to find a clustering of the vertices maximizing the number (or the weight) of matched edges, i.e. the edges having the same color as their extremities. We show that the cardinality problem is NP-hard even for edge-colored bipartite graphs with a chromatic degree equal to two and k ≥ 3. Our main result is a constant approximation algorithm for the weighted version of the Max k-colored clustering problem which is based on a rounding of a natural linear programming relaxation. For graphs with chromatic degree equal to two, we improve this ratio by exploiting the relation of our problem with the Max 2-and problem. We also present a reduction to the maximum-weight independent set (IS) problem in bipartite graphs which leads to a polynomial time algorithm for the case of two colors