# Hochbaum’s Pseudoflow (HPF) Algorithm for (Linear) Fully Parametric Minimum Cut

This package provides an fully parametric implementation of pseudoflow for minimum cut on directed graphs. In the parametric minimum cut problem, the capacity of source-adjacent arcs is monotone non-decreasing in the parameter lambda whereas the capacity of sink-adjacent arcs is monotone non-increasing in lambda. This solver requires that the capacities of source and sink adjacent arcs are linear in lambda: capacity = constant + multiplier * lambda.

This fully parametric solver finds the optimal minimum cut for all lambda values in a given range. The solution for all lambda values is represented with O(n) intervals for the parameter lambda. In each interval, the optimal minimum cut remains the same.

A simple parametric minimum cut solver that provides the optimal minimum cut for a given list of arc capacities is available here, and a non-parametric maximum flow version of pseudoflow is available here.

The package provides interfaces for Python, C, and Matlab.

This implementation uses a variant of the fully parametric HPF algorithm as described in:

DS Hochbaum (2008), The Pseudoflow algorithm: A new algorithm for the maximum flow problem. Operations Research, 58(4):992-1009.

This implementation does not use free runs nor does it use warm starts with informatiom from previous runs (see pg.15). This implementation should therefore not be used for comparison with the fully parametric HPF algorithm.

## Source Code

The latest version of this software is available on GitHub. The source code is also available in ZIP format here.

## Instructions for Python

Install the package with pip:

    pip install pseudoflow

#### Example

import networkx as nx
import pseudoflow

G = nx.DiGraph()

source = 0
sink = 2
lambda_range = [0., 2.]

breakpoints, cuts, info = pseudoflow.hpf(
G,  # Networkx directed graph.
source,  # Node id of the source node.
sink,  # Node id of the sink node.
const_cap="const",  # Edge attribute with the constant capacity.
mult_cap="mult",  # Edge attribute with the lambda multiplier.
lambdaRange=lambda_range,  # (lower, upper) bounds for the lambda parameter.
roundNegativeCapacity=False  # True if negative arc capacities should be rounded to zero.
)

# breakpoints: list of upper bounds for the lambda intervals.
# cuts: A dictionary with for each node a list indicating whether
#       the node is in the source set of the minimum cut.
print(breakpoints)  # Output: [1., 2.]
print(cuts)  # Output: {0: [1, 1], 1: [0, 1], 2: [0, 0]}

## Instructions for C

Navigate to directory src/pseudoflow/c, and compile the hpf executable with make.

To execute the solver, use:

hpf input-file.txt output-file.txt

The input file should contain the graph structure and is assumed to have the following format:

    c <comment lines>
p <# nodes> <# arcs> <lower bound> <upper bound> <round if negative>
n <source node> s
n <sink node> t
a <from-node> <to-node> <constant capacity> <lambda multiplier>

where the a line is repeated for each arc. The file should satisfy the following conditions:

• Nodes are labeled 0 .. <# nodes> - 1.
• <lambda multiplier> is non-negative if <from-node> == <source node> and <to-node> != <sink-node>.
• <lambda multiplier> is non-positive if <from-node> != <source node> and <to-node> == <sink-node>.
• <lambda multiplier> is zero if <from-node> != <source node> and <to-node> != <sink-node>.
• <round if negative> takes value 1 if the any negative capacity arc should be rounded to 0, and value 0 otherwise.

The solver will generate the following output file:

t <time (in sec) read data> <time (in sec) initialize> <time (in sec) solve>
s <# arc scans> <# mergers> <# pushes> <# relabels > <# gap >
p <number of lambda intervals = k>
l <lambda upperbound interval 1> ... <lambda upperbound interval k>
n <node-id> <sourceset indicator interval 1 > .. <indicator interval k>

The n line appears for each node. <sourceset indicator interval 1 > indicates whether the node is in the source set of the minimum cut for the first lambda interval.

See src/pseudoflow/c/example for an example.

## Instructions for Matlab

Copy the content of src/pseudoflow/matlab to your current directory.

From within Matlab, compile the mex extension with:

    mex hpfMatlab.c

The solver is accessible via the hpf function with the following signature:

    [cuts, lambdas, stats, times]  = hpf(arcmatrix, num_nodes, source, sink lambda_range, rounding);

#### Inputs:

• arcmatrix: Each row of the matrix has the following structure: [from_node, to_node, constant capacity, lambda multiplier]
• num_nodes: Number of nodes in the graph
• source_node: The numeric label of the source node
• sink_node: The numeric label of the sink node
• lambda_range: [lower bound, upper bound] for the lambda parameter.
• rounding: Set to 1 if negative arc capacities should be rounded to zero, and 0 otherwise.

#### Outputs:

• cuts: n x k matrix where A(i,j) is 1 if node i is in the source set for lambda interval j, and 0 otherwise.
• lambdas: 1 x k matrix where L(j) is the upper bound of the lambda interval j.