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Two-Commodity Flow is Equivalent to Linear Programming under Nearly-Linear Time Reductions [article]

Ming Ding, Rasmus Kyng, Peng Zhang
2022
We give a nearly-linear time reduction that encodes any linear program as a 2-commodity flow problem with only a small blow-up in size.  ...  We remark that our reduction does not run in strongly polynomial time and that it is open whether 2-commodity flow and linear programming are equivalent in strongly polynomial time.  ...  In nearly-linear time, we can convert this linear program to a 2-commodity flow problem which is feasible if and only if the original program is.  ... 
doi:10.48550/arxiv.2201.11587 fatcat:dgmhdo3ennc43m4zuldf44biqa

Two-Commodity Flow Is Equivalent to Linear Programming Under Nearly-Linear Time Reductions

Ming Ding, Rasmus Kyng, Peng Zhang, Mikołaj Bojańczyk, Emanuela Merelli, David P. Woodruff
2022
We give a nearly-linear time reduction that encodes any linear program as a 2-commodity flow problem with only a small blow-up in size.  ...  We remark that our reduction does not run in strongly polynomial time and that it is open whether 2-commodity flow and linear programming are equivalent in strongly polynomial time.  ...  Concretely, they showed that the linear equations that arise I C A L P 2 0 2 2 54:4 Two-Commodity Flow Is Equivalent to Linear Programming in interior point methods for multi-commodity flow are as hard  ... 
doi:10.4230/lipics.icalp.2022.54 fatcat:2d67cqhafvfz5omjy54cmzdrly

A Dual-Ascent Procedure for Large-Scale Uncapacitated Network Design

A. Balakrishnan, T. L. Magnanti, R. T. Wong
1989 Operations Research  
He shows that asymptotically it is very likely that the cost of the solution generated by this heuristic will, under certain conditions, be very close to the cost of the optimal solution.  ...  Section 2 describes dual ascent procedures for computing approximate solutions to the dual of the linear programming relaxation of the design problem formulation.  ...  We would like to thank Professor Bruce W. Lamar for providing us the data for the Less-than-Truckload Consolidation test problems.  ... 
doi:10.1287/opre.37.5.716 fatcat:6kuknncc5nbzxbnh2veqawlkjm

Multicommodity flow problems with a bounded number of paths: A flow deviation approach

Christophe Duhamel, Philippe Mahey
2006 Networks  
A heuristic control of the path generation is then embedded in the original algorithm based on the concepts of cleaning the quasi inactive paths and the reduction of the flow width for each commodity.  ...  We propose a modified version of the Flow Deviation method of Fratta, Gerla and Kleinrock to solve multicommodity problems with minimal congestion and a bounded number of active paths.  ...  This basic problem is known to be hard to solve even if it can be written as a linear program.  ... 
doi:10.1002/net.20143 fatcat:boiy5nwpczcq7b5n4djn4hqn5q

Hardness Results for Structured Linear Systems [article]

Rasmus Kyng, Peng Zhang
2017 arXiv   pre-print
We show that if the nearly-linear time solvers for Laplacian matrices and their generalizations can be extended to solve just slightly larger families of linear systems, then they can be used to quickly  ...  can solve in nearly-linear time will soon halt.  ...  commodity flows to (1+ ) accuracy in time that scales as poly( −1 ) [GK98, Mad10, Fle00, LMP + 91], even obtaining nearly-linear running times when the graph is undirected [She13, KLOS14, Pen16] .  ... 
arXiv:1705.02944v2 fatcat:stpvkn7jbrgk3p4utcfrtq34nu

On Choosing an Optimal Technology

Martin Weitzman
1967 Management science  
There is no joint production. Each hoinogeneous commodity is produced or purchased by linear processes or activities with fixed coefficients.  ...  The question of which commodities are to be considered exogenous is entirely relative to the particular production system under consideration. 2 In the case of circulating capital (intermediate materials  ...  The special structure of the problem under investigation allows us to perform such a reduction in this particular case.  ... 
doi:10.1287/mnsc.13.5.413 fatcat:fi2gvute5bgylkhbailvdd6n4a

Using life cycle assessment and techno-economic analysis in a real options framework to inform the design of algal biofuel production facilities

Jordan D. Kern, Adam M. Hise, Greg W. Characklis, Robin Gerlach, Sridhar Viamajala, Robert D. Gardner
2017 Bioresource Technology  
To do this, an existing LCA/TEA model is paired with a mixed integer linear program (MILP) and subjected to time-varying data for key operational parameters (e.g., prices for electricity, natural gas,  ...  (profits are maximized) via a mixed integer linear program (MILP).  ... 
doi:10.1016/j.biortech.2016.11.116 pmid:27965015 fatcat:anfiqyqjlfaerhou4a5u3iee7y

Stronger Bounds on Braess's Paradox and the Maximum Latency of Selfish Routing

Henry Lin, Tim Roughgarden, Éva Tardos, Asher Walkover
2011 SIAM Journal on Discrete Mathematics  
Finally, we use our family of two-commodity networks to exhibit a natural network design problem with intrinsically exponential (in)approximability.  ...  In multicommodity networks, we exhibit an infinite family of two-commodity networks, related to the Fibonacci numbers, in which both the worst-case severity of Braess's Paradox and the price of anarchy  ...  We define our notion of equivalence so that Lemma 6.1 ensures that the linear program has a bounded maximum, and then analyze the vertices of the feasible region of the linear program to derive the following  ... 
doi:10.1137/090769600 fatcat:63cvtfzcvfanhntvaeua6vvupy

Water use and salinity in the Murray?Darling Basin: A state-contingent model

David Adamson, Thilak Mallawaarachchi, John Quiggin
2007 Australian Journal of Agricultural and Resource Economics  
The object of the present paper is to show how the linear and non-linear programming models commonly used in modelling problems such as those arising in the Murray-Darling Basin may be adapted to incorporate  ...  The supply of water for irrigation is subject to climatic and policy uncertainty.  ...  By the time a limit, know as the Cap, was imposed on diversions in 1995, prohibiting further growth in average annual allocations, nearly 100 per cent of normally available flows had been allocated, and  ... 
doi:10.1111/j.1467-8489.2007.00380.x fatcat:oz6xguppjfbxfekx7673rwgco4

Multi-hop Routing and Scheduling in Wireless Networks in the SINR Model [chapter]

Guy Even, Yakov Matsri, Moti Medina
2012 Lecture Notes in Computer Science  
For linear powers, Wan et al. [WFJ + 11] obtain a O(log n)-approximation for THROUGHPUT-ROUTE. The algorithm is based on a reduction to the single-slot problem using the ellipsoid method.  ...  Every bucket consists of a set of links that have nearly equivalent reception powers. We denote the number of nonempty buckets by σ.  ...  The linear program formulation that we use for computing the multi-commodity flow f is as follows.  ... 
doi:10.1007/978-3-642-28209-6_16 fatcat:ir2fv5pabvcdja6n76zvjswdee

Achieving Target Equilibria in Network Routing Games without Knowing the Latency Functions [article]

Umang Bhaskar, Katrina Ligett, Leonard J. Schulman, Chaitanya Swamy
2014 arXiv   pre-print
We show that, under fairly general settings, one can efficiently compute edge tolls that induce a given target multicommodity flow in a nonatomic routing game using a polynomial number of queries to an  ...  We obtain tighter bounds on the query complexity for series-parallel networks, and single-commodity routing games with linear latency functions, and complement these with a query-complexity lower bound  ...  Each query is a Stackelberg routing, which is a flow of volume at most α times the total volume, and returns the equilibrium flow under this Stackelberg routing.  ... 
arXiv:1408.1429v1 fatcat:rhap5mhmfnbzjc7hxcadekwmx4

Achieving Target Equilibria in Network Routing Games without Knowing the Latency Functions

Umang Bhaskar, Katrina Ligett, Leonard J. Schulman, Chaitanya Swamy
2014 2014 IEEE 55th Annual Symposium on Foundations of Computer Science  
We show that, under fairly general settings, one can efficiently compute edge tolls that induce a given target multicommodity flow in a nonatomic routing game using a polynomial number of queries to an  ...  We obtain tighter bounds on the query complexity for series-parallel networks, and single-commodity routing games with linear latency functions, and complement these with a query-complexity lower bound  ...  Each query is a Stackelberg routing, which is a flow of volume at most α times the total volume, and returns the equilibrium flow under this Stackelberg routing.  ... 
doi:10.1109/focs.2014.12 dblp:conf/focs/BhaskarLSS14 fatcat:gdqccw6bpngrvcibok2zyrevpm

Outdoor Downward-Facing Optical Flow Odometry with Commodity Sensors [chapter]

Michael Dille, Ben Grocholsky, Sanjiv Singh
2010 Springer Tracts in Advanced Robotics  
A common tactic in such cases is to position visually, and we present a visual odometry implementation with the unusual reliance on optical mouse sensors to report vehicle velocity.  ...  at times more accurate than that achievable by wheel odometry and visibly more desirable for perception purposes than that provided by a high-end GPS-INS system.  ...  Acknowledgements This research was supported in part by NASA under grants, NNX07AE30G, John Caruso, Project Manager, and NNX08AJ99G, Robert Ambrose, Program Scientist.  ... 
doi:10.1007/978-3-642-13408-1_17 fatcat:fcn36ghky5d7tbat3pqsi35b34

Hardness Results for Laplacians of Simplicial Complexes via Sparse-Linear Equation Complete Gadgets [article]

Ming Ding, Rasmus Kyng, Maximilian Probst Gutenberg, Peng Zhang
2022 arXiv   pre-print
We prove this by a nearly-linear time reduction from general linear equations to combinatorial Laplacians of 2-complexes.  ...  It is known that nearly-linear time solvers exist for graph Laplacians. However, nearly-linear time solvers for combinatorial Laplacians are only known for restricted classes of complexes.  ...  Therefore, the two-step reduction is a nearly-linear time reduction while the one-step reduction is not. • Why encode into a 2-complex rather than a 1-complex?  ... 
arXiv:2202.05011v1 fatcat:uywzjonex5bdxh4noylvuplyl4

Optimisation in Forest Management

Abdulğaffar Kaya, Pete Bettinger, Kevin Boston, Ramazan Akbulut, Zennure Ucar, Jacek Siry, Krista Merry, Chris Cieszewski
2016 Current Forestry Reports  
These approaches continue to evolve, and in the last 5 years, the research has expanded to demonstrate how complex, non-linear relationships can be recognised and incorporated into planning processes at  ...  Through this review, we found that landscape-level optimisation is a relatively new and expanding area of research, most often performed by one large public landowner in regions where the resulting plan  ...  [81] developed a linear programming model to minimise harvest reductions due to losses from an insect infestation while imposing a wood flow constraint.  ... 
doi:10.1007/s40725-016-0027-y fatcat:4hwmuvxuyfhxzekdqkn5sexnre
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