Faster strongly polynomial algorithms for the unbalanced transportation problem and assignment problem with monge costs

Networks ◽  
2013 ◽  
Vol 62 (2) ◽  
pp. 136-148 ◽  
Author(s):  
Balachandran Vaidyanathan
Keyword(s):  
Assignment Problem ◽  
Transportation Problem ◽  
Polynomial Algorithms ◽  
Strongly Polynomial Algorithms ◽  
Strongly Polynomial

scholarly journals Weakly and strongly polynomial algorithms for computing the maximum decrease in uniform arc capacities

2016 ◽  
Vol 26 (2) ◽  
pp. 159-171
Author(s):  
Mehdi Ghiyasvand
Keyword(s):  
Polynomial Time ◽  
Maximum Flow ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Polynomial Algorithms ◽  
Directed Network ◽  
Strongly Polynomial Algorithms ◽  
Maximum Decrease ◽  
Strongly Polynomial

In this paper, a new problem on a directed network is presented. Let D be a feasible network such that all arc capacities are equal to U. Given a t > 0, the network D with arc capacities U - t is called the t-network. The goal of the problem is to compute the largest t such that the t-network is feasible. First, we present a weakly polynomial time algorithm to solve this problem, which runs in O(log(nU)) maximum flow computations, where n is the number of nodes. Then, an O(m2n) time approach is presented, where m is the number of arcs. Both the weakly and strongly polynomial algorithms are inspired by McCormick and Ervolina (1994).

scholarly journals Lipschitz Continuity and Approximate Equilibria

Algorithmica ◽  
2020 ◽  
Vol 82 (10) ◽  
pp. 2927-2954
Author(s):  
Argyrios Deligkas ◽  
John Fearnley ◽  
Paul Spirakis
Keyword(s):  
Polynomial Time ◽  
Lipschitz Continuity ◽  
Payoff Function ◽  
Polynomial Algorithms ◽  
Continuous Action ◽  
Lipschitz Continuous ◽  
Payoff Functions ◽  
Action Spaces ◽  
Best Responses ◽  
Strongly Polynomial

Abstract In this paper, we study games with continuous action spaces and non-linear payoff functions. Our key insight is that Lipschitz continuity of the payoff function allows us to provide algorithms for finding approximate equilibria in these games. We begin by studying Lipschitz games, which encompass, for example, all concave games with Lipschitz continuous payoff functions. We provide an efficient algorithm for computing approximate equilibria in these games. Then we turn our attention to penalty games, which encompass biased games and games in which players take risk into account. Here we show that if the penalty function is Lipschitz continuous, then we can provide a quasi-polynomial time approximation scheme. Finally, we study distance biased games, where we present simple strongly polynomial time algorithms for finding best responses in $$L_1$$ L 1 and $$L_2^2$$ L 2 2 biased games, and then use these algorithms to provide strongly polynomial algorithms that find 2/3 and 5/7 approximate equilibria for these norms, respectively.

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