[Paper Review] A Compositional Approach to Network Algorithms

A. J. Kfoury|arXiv (Cornell University)|May 19, 2018

Advanced Graph Theory Research40 references3 citations

TL;DR

This paper introduces a compositional typing theory for flow networks using polyhedral analysis and convex optimization to formally characterize feasible, maximum, and minimum flows. It enables safe modular design of network algorithms by defining type systems that ensure correctness when composing subnetworks, with applications to max-flow and min-flow problems in planar networks with multiple sources and sinks.

ABSTRACT

We present elements of a typing theory for flow networks, where "types", "typings", and "type inference" are formulated in terms of familiar notions from polyhedral analysis and convex optimization. Based on this typing theory, we develop an alternative approach to the design and analysis of network algorithms, which we illustrate by applying it to the max-flow problem in multiple-source, multiple-sink, capacited directed planar graphs.

Motivation & Objective

• To develop a formal, compositional framework for designing and analyzing network algorithms that supports modularity, incremental development, and order-oblivious composition.
• To address the challenge of safely composing network components with multiple inputs and outputs while preserving flow conservation and capacity constraints.
• To extend traditional network algorithm design by introducing a typing system grounded in convex optimization and polyhedral analysis to encode feasible flow behavior.
• To provide a foundation for reasoning about network components in isolation and composing them safely, even under demonic non-determinism.
• To demonstrate the approach on the max-flow and min-flow problems in capacitated, directed, planar graphs with multiple sources and sinks.

Proposed method

• Define network typings as algebraic characterizations of feasible flows using interval constraints on input and output arcs, derived from polyhedral analysis.
• Introduce principal typings as canonical representations of all feasible flows in a network, capturing the full range of input-output behavior.
• Develop a notion of 'strong subtyping' (≪:) to ensure safe substitution of one network component for another under demonic non-determinism.
• Use type inference to automatically derive typings from network structure and capacity bounds, enabling compositional reasoning.
• Apply the typing framework to decompose complex networks into subnetworks, analyze them independently, and reassemble them with guaranteed correctness.
• Leverage convex optimization techniques to compute interval bounds on flows and validate type constraints across network boundaries.

Experimental results

Research questions

• RQ1 How can network components be formally encapsulated and composed such that safety invariants—like flow conservation and capacity bounds—are preserved across composition?
• RQ2 What formal type system can capture the full range of feasible, maximum, and minimum flows in a network with multiple sources and sinks?
• RQ3 How can type safety be ensured when composing networks with non-deterministic behavior, particularly under demonic non-determinism?
• RQ4 Can a compositional approach based on typing and type inference outperform or complement traditional monolithic max-flow algorithms in terms of modularity and correctness?
• RQ5 What are the conditions under which one network component can be safely substituted for another in a larger system, and how can this be formally verified?

Key findings

• The proposed typing theory enables formal, modular reasoning about network flows by encoding feasible input-output behaviors as interval constraints derived from polyhedral analysis.
• Strong subtyping (≪:) provides a formal criterion for safe component substitution in the presence of demonic non-determinism, ensuring that input and output behaviors are preserved.
• The framework supports the simultaneous analysis of max-flow and min-flow problems in planar networks with lower- and upper-bound capacities on arcs.
• By decomposing a network into subnetworks and analyzing them with principal typings, the method allows for incremental and order-oblivious system construction.
• The approach is applicable beyond max-flow, extending to other network algorithm problems with different optimization objectives, even when run-time complexity is not linear.
• The method ensures correctness of composition through formal type inference and constraint satisfaction, with concrete examples showing how adjustments to capacities can make subnetworks safely substitutable.