ColPack

ColPack is a package comprising of implementations of algorithms for the specialized vertex coloring problems discussed in the previous section as well as algorithms for a variety of related supporting tasks in derivative computation.

Coloring capabilities

the table below gives a quick summary of all the coloring problems (on general and bipartite graphs) supported by ColPack.

General Graph Coloring	Bipartite Graph one-sided coloring	Bipartite Graph Bicoloring
Distance 1 coloring	Partial distance-2 coloring	Star bicoloring
Distance 2 coloring	Partial distance-2 coloring
Star coloring
Acyclic coloring
Restricted star coloring
Triangular coloring

All of the coloring problems listed in the above table are NP-hard. Their corresponding algorithms in ColPack are greedy heuristics in the sense that the algorithms progressively extend a partial coloring by processing one vertex at a time, in some order, in each step assigning a vertex the smallest allowable color. Listed beneath each coloring problem in the table is the complexity of the corresponding algorithm in ColPack. In the cases where ColPack has multiple algorithms for a problem (these are designated by the superscript †), the complexity expression corresponds to that of the fastest algorithm. In the complexity expressions,

the complexity of the corresponding algorithm can be found here ColPack's project

Ordering techniques

The order in which vertices are processed in a greedy coloring algorithm determines the number of colors used by the algorithm. ColPack has implementations of various effective ordering techniques for each of the supported coloring problems. These are summarized in the table below.

General Graph Coloring	Bipartite Graph one-sided coloring	Bipartite Graph Bicoloring
Natural	Column Natural	Natural
Largest First	Column Largest First	Largest First
Smallest Last	Column Smallest Last	Smallest Last
Incidence Degree	Column Incidence Degree	Incidence Degree
Dynamic Largest First	Row Natural	Dynamic Largest First
Distance-2 Largest First	Row Largest First	Selective Largest First
Distance-2 Smallest Last	Row Smallest Last	Selective Smallest Last
Distance-2 Incidence Degree	Row Incidence Degree	Selective Incidence Degree
Distance-2 Dynamic Largest First

Recovery routines

Besides coloring and ordering capabilities, ColPack also has routines for recovering the numerical values of the entries of a derivative matrix from a compressed representation. In particular the following reconstruction routines are currently available:

Recovery routines for direct (via star coloring ) and substitution-based (via acyclic coloring) Hessian computation
Recovery routines for unidirectional, direct Jacobian computation (via column-wise or row-wise distance-2 coloring)
Recovery routines for bidirectional, direct Jacobian computation via star bicoloring

Graph construction routines

Finally, as a supporting functionality, ColPack has routines for constructing bipartite graphs (for Jacobians) and adjacency graphs (for Hessians) from files specifying matrix sparsity structures in various formats, including Matrix Market, Harwell-Boeing and MeTis.

ColPack : organization

ColPack is written in an object-oriented fashion in C++ heavily using the Standard Template Library (STL). It is designed to be simple, modular, extensible and efficient. Figure 1 below gives an overview of the structure of the major classes of ColPack.

Ubuntu Build Instructions

In the ColPack directory run the following:

autoreconf -vif
./configure --prefix=/path/to/install/
make -j 4   #Where "4" is the number of cores on your machine
make install

USAGE

$./ColPack <GraphName> [order_option] [coloring_option]
$./ColPack --graph <GraphName> [--order <order_option>] [--color <coloring_option>]
$./ColPack -g <GraphName> [-o <order_option>] [-c <coloring_option>]

-g or --graph can be changed to -f --file in case needed.

HELP

$./ColPack -h
$./ColPack --help

OPTIONs

order: NATURAL, LARGEST_FIRST, DYNAMIC_LARGEST_FIRST, SMALLEST_LAST, INCIDENCE_DEGREE, RANDOM
color: DISTANCE_ONE, ACYCLIC, ACYCLIC_FOR_INDIRECT_RECOVERY, STAR, RESTRICTED_STAR, DISTANCE_TWO

EXAMPLE

$./ColPack Graphs/bcsstk01.mtx
$./ColPack Graphs/bcsstk01.mtx LARGEST_FIRST DISTANCE_ONE
$./ColPack --graph Graphs/bcsstk01.mtx --order SMALLEST_LAST -d DISTANCE_TWO
$./ColPack -f Graphs/bcsstk01.mtx -o RANDOM -distance DISTANCE_ONE

EXAMPLE OUTPUT

ReadMatrixMarketAdjacencyGraph
Found file Graphs/bcsstk01.mtx
Graph of Market Market type: [matrix coordinate real symmetric]
		Graph structure and VALUES will be read

#DISTANCE_ONE Result: 
6  : (NATURAL)
6  : (LARGEST_FIRST)
6  : (DYNAMIC_LARGEST_FIRST)
6  : (SMALLEST_LAST)
6  : (INCIDENCE_DEGREE)
6  : (RANDOM)

#ACYCLIC Result: 
8  : (NATURAL)
8  : (LARGEST_FIRST)
8  : (DYNAMIC_LARGEST_FIRST)
8  : (SMALLEST_LAST)
8  : (INCIDENCE_DEGREE)
8  : (RANDOM)

#ACYCLIC_FOR_INDIRECT_RECOVERY Result: 
8  : (NATURAL)
8  : (LARGEST_FIRST)
8  : (DYNAMIC_LARGEST_FIRST)
8  : (SMALLEST_LAST)
8  : (INCIDENCE_DEGREE)
8  : (RANDOM)

#STAR Result: 
12  : (NATURAL)
12  : (LARGEST_FIRST)
12  : (DYNAMIC_LARGEST_FIRST)
12  : (SMALLEST_LAST)
12  : (INCIDENCE_DEGREE)
12  : (RANDOM)

#RESTRICTED_STAR Result: 
15  : (NATURAL)
15  : (LARGEST_FIRST)
15  : (DYNAMIC_LARGEST_FIRST)
15  : (SMALLEST_LAST)
15  : (INCIDENCE_DEGREE)
15  : (RANDOM)

#DISTANCE_TWO Result: 
15  : (NATURAL)
15  : (LARGEST_FIRST)
15  : (DYNAMIC_LARGEST_FIRST)
15  : (SMALLEST_LAST)
15  : (INCIDENCE_DEGREE)
15  : (RANDOM)

ouyangtianzuo / ColPack