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Max-min Fair Rate Allocation and Routing in Energy Harvesting Networks: Algorithmic Analysis

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This paper considers max-min fair rate allocation and routing in energy harvesting networks where fairness is required among both the nodes and the time slots. Unlike most previous work on fairness, we focus on multihop topologies and consider different routing methods. We assume a predictable energy profile and focus on the design of efficient and optimal algorithms that can serve as benchmarks for distributed and approximate algorithms. We first develop an algorithm that obtains a max-min fair rate assignment for any routing that is specified at the input. We then turn to the problem of determining a “good” routing. For time-invariable unsplittable routing, we develop an algorithm that finds routes that maximize the minimum rate assigned to any node in any slot. For fractional routing, we derive a combinatorial algorithm for the time-invariable case with constant rates. We show that the time-variable case is at least as hard as the 2-commodity feasible flow problem and design an FPTAS to combat the high running time. Finally, we show that finding an unsplittable routing or a routing tree that provides lexicographically maximum rate assignment (i.e., the best in the max-min fairness terms) is NP-hard, even for a time horizon of a single slot. Our analysis provides insights into the problem structure and can be applied to other related fairness problems.

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  1. 1.

    Note that node i’s sensing rate (generated flow) can change over time, even though the routing does not change.

  2. 2.

    \(\widetilde{O}(.)\)-notation hides poly-log terms.

  3. 3.

    The notions of max-min fairness and lexicographical ordering of vectors are defined in Sect. 2.1.

  4. 4.

    Note that we treat Eq. (2) as a linear constraint, since the considered problems focus on maximizing \(\lambda _{i, t}\)’s (under the max-min fairness criterion), and (2) can be replaced by \(b_{i, t+1}\le B\) and \(b_{i, t+1}\le b_{i, t}+e_{i,t}- (c_{\text {rt}}f^{\Sigma }_{i, t} + c_{\text {st}}\lambda _{i, t})\) while leading to the same solution.

  5. 5.

    Notice that this is consistent with the definition of a descendant in a routing tree.

  6. 6.

    P is determined by linear equalities and inequalities, which implies that it is convex.


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The authors are grateful to Mihalis Yannakakis and David Johnson for useful discussions.

Author information

Correspondence to Jelena Marašević.

Additional information

A partial and preliminary version of this paper appeared in Proc. ACM MobiHoc’14. This research was supported in part by NSF Grants CCF-1349602, CCF-1421161, CCF-09-64497, and CNS-10-54856 and the People Programme (Marie Curie Actions) of the European Union’s Seventh Framework Programme (FP7/2007-2013) under REA Grant Agreement No. [PIIF-GA-2013-629740].11.

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Marašević, J., Stein, C. & Zussman, G. Max-min Fair Rate Allocation and Routing in Energy Harvesting Networks: Algorithmic Analysis. Algorithmica 78, 521–557 (2017). https://doi.org/10.1007/s00453-016-0171-6

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  • Energy harvesting
  • Energy adaptive networking
  • Network flows
  • Sensor networks
  • Routing
  • Fairness