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<Appendix Label="DIMACS">
  <Heading>
    DIMACS: Graph Format for Clique and Coloring Problems
  </Heading>

  <Section>
    <Heading>
      Note from the Digraphs authors
    </Heading>
    The contents of this appendix were originally available in a PostScript file
    <URL Text="on the Carnegie Mellon University
      website">http://mat.gsia.cmu.edu/COLOR/general/ccformat.ps.
    This file is still <URL Text="accessible through the Wayback
      Machine">https://web.archive.org/web/20220120085844/http://mat.gsia.cmu.edu/COLOR/general/ccformat.ps.
    We reproduce its contents here for convenience, without adjustments
      beyond minor re-formatting.
  </Section>

  <Section>
    <Heading>
      Preamble
    </Heading>

    <B>Last revision of this document: May 08, 1993.</B>

    <P/>

    <E>
      This paper outlines a suggested graph format.  If you
  have comments on this or other formats or you have information you
      think should be included, please send a note to
    </E>
    <Email>challenge@dimacs.rutgers.edu</Email>.
  </Section>

  <Section>
    <Heading>
      Introduction
    </Heading>

    One purpose of the DIMACS Challenge is to ease the effort required to
    test and compare algorithms and heuristics by providing a common
    testbed of instances and analysis tools.  To facilitate this effort, a
    standard format must be chosen for the problems addressed.  This
    document outlines a format for graphs that is suitable for those
    looking at graph coloring and finding cliques in graphs.  This format
    is a flexible format suitable for many types of graph and network
    problems.  This format was also the format chosen for the First
    Computational Challenge on network flows and matchings.

    <P/>

    This document describes three problems:  unweighted clique, weighted
    clique, and graph coloring.  A separate format is used for
    satisfiability.
  </Section>

  <Section>
    <Heading>
      File Formats for Graph Problems
    </Heading>

    This section describes a standard file format for graph inputs and outputs.
    There is no  requirement that participants follow these specifications;
    however, compatible implementations will be able to make full use of
    DIMACS support tools.  (Some tools assume that output is appended
    to input in a single file.)

    Participants are welcome to develop translation programs to convert
    instances to and from more convenient, or more compact, representations;
    the Unix <B>awk</B> facility is recommended as especially suitable
    for this task.

    All files contain ASCII characters.  Input and output files contain
    several types of <E>lines</E>, described below.  A line is terminated
    with an end-of-line character.  Fields in each line are separated by
    at least one blank space.  Each line begins with a one-character
    designator to identify the line type.

    <Subsection>
      <Heading>
        Input Files
      </Heading>

      An input file contains all the information about a graph needed to define
      either a clique problem or a coloring problem.  Some information may be
      included that is not relevant to one problem (for instance, node weights
      are not needed for coloring problem) so that information may be ignored.

      In this format, nodes are numbered from 1 up to <M>n</M>.  There are
      <M>m</M> edges in the graph.

      Files are assumed to be well-formed and internally consistent: node
      identifier values are valid, nodes are defined uniquely, exactly <M>m</M>
      edges are defined, and so forth.  A input checker will be made available
      to ensure compatibility with this standard.

      <List>

        <Mark>Comments</Mark>
        <Item>
          Comment lines give human-readable information about the file and are
          ignored by programs.  Comment lines can appear anywhere in the file.
          Each comment line begins with a lowercase character <B>c</B>.

          <Log>c This is an example of a comment line.</Log>
        </Item>

        <Mark>
          Problem line
        </Mark>
        <Item>
          There is one problem line per input file.  The problem line must
          appear before any node or arc descriptor lines.  For network
          instances, the problem line has the following format.

          <Log>p FORMAT NODES EDGES</Log>

          The lowercase character <C>p</C> signifies that this is the problem
          line.  The <C>FORMAT</C> field is for consistency with the previous
          Challenge, and should contain the word <M>edge</M>.  The <C>NODES</C>
          field contains an integer value specifying <M>n</M>, the number of
          nodes in the graph.  The <C>EDGES</C> field contains an integer value
          specifying <M>m</M>, the number of edges in the graph.
        </Item>

        <Mark>Node Descriptors</Mark>
        <Item>
          For this Challenge, a node descriptor is required only for the
          weighted clique problem.  These lines will give the weight assigned to
          a node in the clique.  There is one node descriptor line for each
          node, with the following format.  Nodes without a descriptor will take
          on a default value of 1.

          <Log>n ID VALUE</Log>

          The lowercase character <C>n</C> signifies that this is a node
          descriptor line.  The <C>ID</C> field gives a node identification
          number, an integer between 1 and <M>n</M>.  The <C>VALUE</C> gives the
          objective value for having this node in the clique.  This value is
          assumed to be integer and can be either positive or negative (or
          zero).
        </Item>

        <Mark>Edge Descriptors</Mark>
        <Item>
          There is one edge descriptor line for each edge the graph, each with
          the following format.  Each edge <M>(v,w)</M> appears exactly once in
          the input file and is not repeated as <M>(w,v)</M>.

          <Log>e W  V</Log>

          The lowercase character <C>e</C> signifies that this is an edge
          descriptor line.  For an edge <M>(w,v)</M> the fields <C>W</C>  and
          <C>V</C> specify its endpoints.
        </Item>

        <Mark>Optional Descriptors</Mark>
        <Item>
          In addition to the required information, there can be additional
          pieces of information about a graph.  This will typically define the
          parameters used to generate the graph or otherwise define
          generator-specific information.  The following list may be added to as
          interesting problem generators are decided on:

          <List>
            <Mark>Geometric Descriptors</Mark>
            <Item>
              One common method to generate or display graphs is to have the
              nodes be embedded in some space and to have the edges be included
              according to some function of the distance between nodes according
              to some metric.  The node information can be defined by a
              dimension descriptor and a vertex embedding descriptor.

              <Log>d DIM METRIC</Log>

              is the dimension descriptor.  <C>DIM</C> is an integer giving the
              number of dimensions of the space, while <C>METRIC</C> is a string
              representing the metric for the space.  <C>METRIC</C> is a string
              that can take a number of forms.  <B>Lp</B> (i.e. <B>L1</B>,
              <B>L2</B>, <B>L122</B>, and so on) denotes the <M>\ell_p</M> norm
              where the distance between two nodes embedded at
              <M>(x_1,x_2,\ldots,x_d)</M> and <M>(y_1,y_2,\ldots y_d)</M> is
              <M>\left(\sum_{i=1}^d |x_i-y_i|^p \right)^{1/p}</M>.  The string
              <B>LINF</B> is used to denote the <M>\ell_\infty</M> norm.  L2S
              denotes the squared euclidean norm (which can be less susceptible
              to computer differences in round-off and accuracy issues).

              <Log>v  X1  X2  X3  ... XD</Log>

              The lowercase character <C>v</C> signifies that this is a vertex
              embedding descriptor line.  The fields <C>X1, X2, ..., XD</C> give
              the <C>d</C> coordinate values for the vertex.  Note that these
              lines must appear after the <C>d</C> descriptor.
            </Item>

            <Mark>Parameter Descriptors</Mark>
            <Item>
              The parameter descriptors are used to give other information about
              how the graph was generated.  The lines are generator-specific,
              and as such it is not expected that most codes will use most (or
              any) of them.  They are included only to aid those codes
              specifically designed to attack specially structured problems.
              The general form of the parameter descriptor is:

              <Log>x PARAM VALUE</Log>

              The lowercase character <C>x</C> signifies that this is a
              parameter descriptor line.  The <C>PARAM</C> field is a string
              that gives the name of the parameter, while the <C>VALUE</C> field
              is a numeric value that gives the corresponding value.  The
              following <C>PARAM</C> values have been defined:

              <Table Align="|l|l|">
                <HorLine/>
                <Row>
                  <Item><C>PARAM</C></Item>
                  <Item>Description (Geometric Graphs)</Item>
                </Row>
                <HorLine/>
                <HorLine/>
                <Row>
                  <Item><C>MINLENGTH</C></Item>
                  <Item>
                    Edge included only if length greater than or equal to
                    <C>VALUE</C>
                  </Item>
                </Row>
                <HorLine/>
                <Row>
                  <Item><C>MAXLENGTH</C></Item>
                  <Item>
                    Edge included only if length less than or equal to
                    <C>VALUE</C>
                  </Item>
                </Row>
                <HorLine/>
              </Table>

              Note that this information is in addition to the required edge
              descriptors.
            </Item>
          </List>
        </Item>
      </List>
    </Subsection>

    <Subsection>
      <Heading>
        Output Files
      </Heading>

      Every algorithm or heuristic should create an output file.  This output
      file should consist of one or more of the following lines, depending on
      the type of algorithm and problem being solved.

      <List>
        <Mark>Solution Line</Mark>
        <Item>
          Format:
          <Log>s TYPE SOLUTION</Log>
          The lowercase character <C>s</C> signifies that this is a solution
          line. The <C>TYPE</C> field denotes the type of solution contained in
          the file.  This should be one of the following strings: <M>col</M>
          denotes a graph coloring, <M>clq</M> denotes a maximum weighted
          clique, and <M>cqu</M> denotes a maximum unweighted clique (one that
          has ignored the <C>n</C> descriptor lines).

          The <C>SOLUTION</C> field contains an integer corresponding to the
          solution value.  This is the clique size for unweighted clique, clique
          value for weighted clique, or number of colors used for graph
          coloring.
        </Item>

        <Mark>Bound Line</Mark>
        <Item>
          Format:
          <Log>b BOUND</Log>

          The lowercase character <C>b</C> signifies that this is a bound on
          the the solution.  The <C>BOUND</C> field contains an integer value
          that gives a bound on the solution value.  This bound is an upper
          bound on the maximum clique value for cliques and weighted clique and
          a lower bound on the number of colors needed for coloring the graph.
        </Item>

        <Mark>Clique Line</Mark>
        <Item>
          Format:
          <Log>v V</Log>

          The lowercase character <C>v</C> signifies that this is a clique
          vertex line.  The <C>V</C> field gives the node number for the node in
          the clique.  There will be one clique line for each node in the
          clique.
        </Item>

        <Mark>Label Line</Mark>
        <Item>
          Format:
          <Log>l V N</Log>

          The lowercase character <C>l</C> signifies that this is a label line,
          generally used for graph coloring.  The <C>V</C> field gives the node
          number for the node in the clique while the <C>N</C> field gives the
          corresponding label.  There will be one label line for each node in
          the graph.
        </Item>

      </List>
    </Subsection>
  </Section>
</Appendix>

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