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's performance to several algorithms published in the literature.
Abstract. The inefficiency of the Wardrop equilibrium of nonatomic routing games can be eliminated by placing tolls on the edges of a network so that the socially optimal flow is induced as an equilibrium flow. A solution where the minimum number of edges are tolled may be preferable over others due to its ease of implementation in real networks. In this paper we consider the minimum tollbooth (M IN T B) problem, which seeks social optimum inducing tolls with minimum support. We prove for single commodity networks with linear latencies that the problem is NP-hard to approximate within a factor of 1.1377 through a reduction from the minimum vertex cover problem. Insights from network design motivate us to formulate a new variation of the problem where, in addition to placing tolls, it is allowed to remove unused edges by the social optimum. We prove that this new problem remains NP-hard even for single commodity networks with linear latencies, using a reduction from the partition problem. On the positive side, we give the first exact polynomial solution to the M IN T B problem in an important class of graphs-series-parallel graphs. Our algorithm solves M IN T B by first tabulating the candidate solutions for subgraphs of the series-parallel network and then combining them optimally.
Understanding the behavior of no-regret dynamics in general N -player games is a fundamental question in online learning and game theory. A folk result in the field states that, in finite games, the empirical frequency of play under no-regret learning converges to the game's set of coarse correlated equilibria. By contrast, our understanding of how the day-to-day behavior of the dynamics correlates to the game's Nash equilibria is much more limited, and only partial results are known for certain classes of games (such as zero-sum or congestion games). In this paper, we study the dynamics of follow the regularized leader (FTRL), arguably the most well-studied class of no-regret dynamics, and we establish a sweeping negative result showing that the notion of mixed Nash equilibrium is antithetical to no-regret learning. Specifically, we show that any Nash equilibrium which is not strict (in that every player has a unique best response) cannot be stable and attracting under the dynamics of FTRL. This result has significant implications for predicting the outcome of a learning process as it shows unequivocally that only strict (and hence, pure) Nash equilibria can emerge as stable limit points thereof.
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