About strongly polynomial time algorithms for quadratic optimization over submodular constraints

1995 ◽  
Vol 69 (1-3) ◽  
pp. 269-309 ◽  
Author(s):  
Dorit S. Hochbaum ◽  
Sung-Pil Hong
Keyword(s):  
Polynomial Time ◽  
Quadratic Optimization ◽  
Strongly Polynomial Time ◽  
Polynomial Time Algorithms ◽  
Strongly Polynomial

scholarly journals On the Proximity of Markets with Integral Equilibria

2019 ◽  
Vol 33 ◽  
pp. 1748-1755 ◽  
Author(s):  
Siddharth Barman ◽  
Sanath Kumar Krishnamurthy
Keyword(s):  
Polynomial Time ◽  
Fair Division ◽  
Prior Work ◽  
Indivisible Goods ◽  
Polynomial Time Algorithms ◽  
Fairness And Efficiency ◽  
Pareto Efficient ◽  
Efficient Allocations ◽  
Strong Existence ◽  
Strongly Polynomial

We study Fisher markets that admit equilibria wherein each good is integrally assigned to some agent. While strong existence and computational guarantees are known for equilibria of Fisher markets with additive valuations (Eisenberg and Gale 1959; Orlin 2010), such equilibria, in general, assign goods fractionally to agents. Hence, Fisher markets are not directly applicable in the context of indivisible goods. In this work we show that one can always bypass this hurdle and, up to a bounded change in agents’ budgets, obtain markets that admit an integral equilibrium. We refer to such markets as pure markets and show that, for any given Fisher market (with additive valuations), one can efficiently compute a “near-by,” pure market with an accompanying integral equilibrium.Our work on pure markets leads to novel algorithmic results for fair division of indivisible goods. Prior work in discrete fair division has shown that, under additive valuations, there always exist allocations that simultaneously achieve the seemingly incompatible properties of fairness and efficiency (Caragiannis et al. 2016); here fairness refers to envyfreeness up to one good (EF1) and efficiency corresponds to Pareto efficiency. However, polynomial-time algorithms are not known for finding such allocations. Considering relaxations of proportionality and EF1, respectively, as our notions of fairness, we show that fair and Pareto efficient allocations can be computed in strongly polynomial time.

scholarly journals Efficient algorithm for Minimum cost flow problem with partial lane reversals

2019 ◽  
Vol 36 (1-2) ◽  
pp. 51-59
Author(s):  
Urmila Pyakurel
Keyword(s):  
Polynomial Time ◽  
Flow Problem ◽  
Minimum Cost ◽  
Minimum Cost Flow ◽  
Strongly Polynomial Time ◽  
Parallel Networks ◽  
Minimum Cost Flow Problem ◽  
Parallel Network ◽  
Cost Flow ◽  
Strongly Polynomial

In this paper, we investigate the minimum cost flow problem in two terminal series parallel network. We present modified minimum cost flow algorithm that computes the maximum dynamic and the earliest arrival flows in strongly polynomial time and also preserves all unused arc capacities. We also present strongly polynomial time minimum cost partial contraflow algorithm that solves both problems with partial reversals of arc capacities on two terminal series parallel networks.

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