예제 #1
0
 /**
  * run as <CODE>
  *  java -cp &lt;classpath&gt; tests.DDETest &lt;params_file&gt; [random_seed] [maxfuncevals]
  * </CODE>. The params_file must contain lines of the following form:
  *
  * <ul>
  *   <li>class,dde.function, &lt;fullclasspathname&gt; mandatory, the java class name defining the
  *       function to be optimized, which must accept as arguments either <CODE>double[]</CODE> or
  *       <CODE>VectorIntf</CODE> objects.
  *   <li>class,dde.localoptimizer, &lt;fullclasspathname&gt; optional the java class name of an
  *       object implementing the LocalOptimizerIntf defined in the popt4jlib.GradientDescent
  *       package, to be used as further optimizer of the best solution found by the DE process.
  *   <li>dde.numdimensions, $num$ mandatory, the dimension of the domain of the function to be
  *       minimized.
  *   <li>dde.numtries, $num$ optional, the total number of "tries", default is 100.
  *   <li>dde.numthreads, $num$ optional, the number of threads to use, default is 1.
  *   <li>rndgen,$num$,$num2$ mandatory, specifies the starting random seed to use for each of the
  *       $num2$ threads to use (the value num2 must equal the number given in the line for
  *       dde.numthreads).
  *   <li>dde.popsize, $num$ optional, the total population size in each iteration, default is 10.
  *   <li>dde.w, $num$ optional, the "weight" of the DE process, a double number in [0,2], default
  *       is 1.0
  *   <li>dde.px, $num$ optional, the "crossover rate" of the DE process, a double number in [0,1],
  *       default is 0.9
  *   <li>dde.minargval, $num$ optional, a double number that is a lower bound for all variables of
  *       the optimization process, i.e. all variables must satisfy x_i &ge; $num$, default is
  *       -infinity
  *   <li>dde.maxargval, $num$ optional, a double number that is an upper bound for all variables
  *       of the optimization process, i.e. all variables must satisfy x_i &le; $num$, default is
  *       +infinity
  *   <li>dde.minargval$i$, $num$ optional, a double number that is a lower bound for the i-th
  *       variable of the optimization process, i.e. variable must satisfy x_i &ge; $num$, default
  *       is -infinity
  *   <li>dde.maxargval$i$, $num$ optional, a double number that is an upper bound for the i-th
  *       variable of the optimization process, i.e. variable must satisfy x_i &le; $num$, default
  *       is +infinity
  *   <li>dde.de/best/1/binstrategy, $val$ optional, a boolean value that if present and true,
  *       indicates that the DE/best/1/bin strategy should be used in evolving the population
  *       instead of the DE/rand/1/bin strategy, default is false
  *   <li>dde.nondeterminismok, $val$ optional, a boolean value indicating whether the method
  *       should return always the same value given the same parameters and same random seed(s).
  *       The method can be made to run much faster in a multi-core setting if this flag is set to
  *       true (at the expense of deterministic results) getting the CPU utilization to reach
  *       almost 100% as opposed to around 60% otherwise, default is false
  *   <li>dde.countfuncevals, $val$ optional, a boolean value indicating whether or not the process
  *       should be counting the number of function evaluations it performs, default is false
  *   <li>&lt;"dde.dmpaddress", String location&gt; optional, if existing, specifies the location
  *       of a distributed Msg-Passing server that implements the basic send/recv operations as
  *       specified in <CODE>parallel.distributed.DActiveMsgPassingCoordinatorLongLivedConnSrv[Clt]
  *       </CODE> default is null
  *   <li>&lt;"dde.dmpport", Integer port&gt; optional, if existing, specifies the port number of a
  *       distributed Msg-Passing server implementing the basic send/recv operations as specified
  *       in <CODE>parallel.distributed.DActiveMsgPassingCoordinatorLongLivedConnSrv[Clt]</CODE>
  *       default is null
  *   <li>&lt;"dde.dmpthisprocessid", Integer myid&gt; optional, if existing, it indicates the id
  *       of this process (this is the number to use in a recvData(myid) call on the
  *       DActiveMsgPassingCoordinatorLongLivedConnClt object) default is null
  *   <li>&lt;"dde.dmpnextprocessid", Integer id&gt; optional, if existing, it indicates the id of
  *       the process to which this process should be sending "migrants" to; (this is the number to
  *       use as the "send address" in a sendData(myid, id, data) DActiveblahblah call); default is
  *       null
  *   <li>&lt;"dde.numgensbetweenmigrations", Integer num&gt; optional, if existing, it indicates
  *       the number of generations that must pass between two successive "migrations" between DDE
  *       island-processes; default is 10
  *   <li>&lt;"dde.nummigrants",Integer num&gt; optional, if it exists, indicates how many migrants
  *       will be sent and received from each dde process; default is 10
  *   <li>&lt;"dde.reducerhost", String host&gt; optional, if existing, it indicates the address in
  *       which the reducer server resides; default is null
  *   <li>&lt;"dde.reducerport", Integer port&gt; optional, if existing, it indicates the address
  *       in which the reducer server process listens at; default is -1
  * </ul>
  *
  * <p>Notice that in case of running DDE in a distributed manner, there are two important
  * constraints:
  *
  * <ul>
  *   <li>First of all, the various processes participating in the distributed DDE process must
  *       have their dde.dmpthisprocessid and dde.dmpnextprocessid parameters set up so that the
  *       flow of migration forms an exact ring, e.g. for a 3-process DDE we have that DDE_0 sends
  *       migrants to DDE_1 which sends migrants to DDE_2 which sends migrants to DDE_0.
  *   <li>The constraint dde.numgensbetweenmigrations &le; dde.numtries must hold.
  * </ul>
  *
  * Otherwise, there is no way for processes to block in at least one migration cycle (and thereby
  * have the DReduceSrv know the total number of processes before the final reduce operation), and
  * therefore the distributed reduce operation afterwards is not guaranteed to work properly.
  *
  * <p>if the second optional argument is passed in, it overrides the random seed specified in the
  * params_file (specified as the 1st arg in cmd-line).
  *
  * <p>The optional third argument, if present, overrides any max. limit set on the number of
  * function evaluations allowed. After this limit, the function will always return
  * Double.MAX_VALUE instead, and won't increase the evaluation count. Obviously, for this limit to
  * be enforced, the "dde.countfuncevals" flag must be set to true in the params_file (1st arg in
  * the command line)
  *
  * @param args String[]
  */
 public static void main(String[] args) {
   try {
     long start_time = System.currentTimeMillis();
     HashMap params = utils.DataMgr.readPropsFromFile(args[0]);
     if (args.length > 1) {
       long seed = Long.parseLong(args[1]);
       RndUtil.getInstance().setSeed(seed); // updates all extra instances too!
     }
     if (args.length > 2) {
       long num = Long.parseLong(args[2]);
       params.put("maxfuncevalslimit", new Long(num));
     }
     FunctionIntf func = (FunctionIntf) params.get("dde.function");
     if (params.containsKey("dde.countfuncevals")
         && ((Boolean) params.get("dde.countfuncevals")).booleanValue() == true) {
       FunctionBase wrapper_func = new FunctionBase(func);
       params.put("dde.function", wrapper_func);
       func = wrapper_func;
     }
     DDE opter = new DDE(params);
     utils.PairObjDouble p = opter.minimize(func);
     VectorIntf arg = (VectorIntf) p.getArg();
     System.out.print("best soln found:[");
     for (int i = 0; i < arg.getNumCoords(); i++) System.out.print(arg.getCoord(i) + " ");
     System.out.println("] VAL=" + p.getDouble());
     // final local optimization
     utils.PairObjDouble p2 = null;
     LocalOptimizerIntf lasdst = (LocalOptimizerIntf) params.get("dde.localoptimizer");
     if (lasdst != null) {
       VectorIntf x0 = arg.newInstance(); // arg.newCopy();
       params.put("gradientdescent.x0", x0);
       lasdst.setParams(params);
       p2 = lasdst.minimize(func);
       VectorIntf xf = (VectorIntf) p2.getArg();
       System.out.print("Optimized (via a GradientDescent local method) best soln found:[");
       for (int i = 0; i < xf.getNumCoords(); i++) System.out.print(xf.getCoord(i) + " ");
       System.out.println("] VAL=" + p2.getDouble());
     }
     if (func instanceof FunctionBase)
       System.err.println("total function evaluations=" + ((FunctionBase) func).getEvalCount());
     long dur = System.currentTimeMillis() - start_time;
     double val =
         (p2 == null || p2.getDouble() >= Double.MAX_VALUE) ? p.getDouble() : p2.getDouble();
     System.out.println("total time (msecs): " + dur);
     if (func instanceof FunctionBase)
       System.out.println(
           "VVV,"
               + val
               + ",TTT,"
               + dur
               + ",NNN,"
               + ((FunctionBase) func).getEvalCount()
               + ",PPP,DDE,FFF,"
               + args[0]); // for parser program to extract from output
   } catch (Exception e) {
     e.printStackTrace();
     System.exit(-1);
   }
 }
예제 #2
0
  /**
   * run as <CODE>
   * java -cp &lt;classpath&gt; graph.packing.DBBGASPPacker &lt;graphfilename&gt; &lt;paramsfilename&gt; [maxnumBBnodes] [numthreads]
   * </CODE>. <br>
   * args[0]: graph file name must adhere to the format specified in the description of the method
   * <CODE>utils.DataMgr.readGraphFromFile2(String file)</CODE> <br>
   * args[1]: params file name may define parameters in lines of the following form:
   *
   * <ul>
   *   <li>acchost, $string$ optional, the internet address of the DAccumulatorSrv that will be
   *       accumulating incumbents. Default is localhost.
   *   <li>accport, $num$ optional, the port to which the DAccumulatorSrv listens. Default is 7900.
   *   <li>cchost, $string$ optional, the internet address of the DConditionCounterSrv that will be
   *       listening for distributed condition-counter requests. Default is localhost.
   *   <li>ccport, $num$ optional, the port to which the DAccumulatorSrv listens. Default is 7899.
   *   <li>pdahost, $string$ optional, the internet address of the PDAsynchBatchTaskExecutorSrv that
   *       will be listening for distributed tasks execution requests. Default is localhost.
   *   <li>pdaport, $num$ optional, the port to which the asynch distributed executor server
   *       listens. Default is 7981.
   *   <li>tightenboundlevel, $num$ optional, the depth in the B&amp;B tree constructed at which a
   *       stronger computation of the upper bound will be started, default is 0.
   *   <li>cutnodes, $boolean$ optional, if true, then when the BBQueue of nodes is full, any new
   *       nodes created will be discarded instead of processed on the same thread. Default is
   *       false.
   *   <li>localsearch, $boolean$ optional, if true, then when an incumbent solution is found that
   *       cannot be further improved, a local search kicks in to try to improve it using (unless
   *       there is another explicit specification) the (default) N1RXP(FirstImproving) neighborhood
   *       concept that basically attempts to remove a single node from the solution and then see
   *       how many other nodes it can add to the reduced solution. This local search can become
   *       quite expensive and for this reason it is only applied to final incumbent solutions in
   *       the B &amp; B-tree construction process. Default is false.
   *   <li>class,localsearchtype, &lt;fullclassname&gt;[,optionalarguments] optional if present,
   *       will utilize in the local-search procedure the <CODE>AllChromosomeMakerClonableIntf
   *       </CODE> specified in the classname, constructed using the arguments specified (if
   *       present). For example, the line could be:
   *       <PRE>
   * class,localsearchtype,graph.packing.IntSetN2RXPGraphAllMovesMaker,1
   * </PRE>
   *       which would be of use with MWIS problems, for random graphs in the class C(n,p),
   *       producing G_{|V|,p} type random graphs. On the other hand, by default, the <CODE>
   *       IntSetN1RXPFirstImprovingGraphAllMovesMakerMT</CODE> moves maker applies both for 1- and
   *       2-packing problems local-search, which is also better suited when solving MWIS problems
   *       arising from duals of disk graphs (arising from wireless ad-hoc networks etc.) Currently
   *       works only with MWIS (k=1) type problems and is ignored for 2-packing problems.
   *   <li>usemaxsubsets, $boolean$ optional, if false, then each GASP process augmenting candidate
   *       packings will augment these sets one node at a time, leading to the possibility that many
   *       active nodes in the B&amp;B tree will represent the same packing. In such a case, a
   *       "recent-nodes" queue will be used to safe-guard against the possibility of having the
   *       same nodes created and processed within a "short" interval. Default is true.
   *   <li>sortmaxsubsets, $boolean$ optional, if true, then the max subsets generated in method
   *       <CODE>getBestNodeSets2Add()</CODE> will be sorted in descending weight order so that if
   *       children <CODE>BBNode*</CODE> objects are "cut", they will be the "least" heavy-weight.
   *       Default is false.
   *   <li>maxitersinGBNS2A, $num$ optional, if present and also the "usemaxsubsets" key is true,
   *       then the number represents the max number of iterations the <CODE>getBestNodeSets2Add()
   *       </CODE> method of the <CODE>DBBNode1</CODE> class will be allowed to go through. Default
   *       is 100000 (specified in <CODE>DBBTree</CODE> class.)
   *   <li>useGWMIN2criterion, $boolean$ optional, if true, then when computing the best nodes to
   *       consider as a partial solution is being expanded, the "GWMIN2-heuristic" criterion
   *       (described in Sakai et. al. 2003: "A note on greedy algorithms for the MWIS problem",
   *       Discr. Appl. Math., 126:313-322) will be used for nodes selection in 1-packing problems.
   *       Default is false.
   *   <li>maxnodechildren, $num$ optional, specify an upper bound on the number of children any
   *       node is allowed to create. Default is Integer.MAX_VALUE.
   *   <li>class,dbbnodecomparator, &lt;fullclassname&gt; optional, the full class name of a class
   *       implementing the <CODE>graph.packing.DBBNodeComparatorIntf</CODE> that is used to define
   *       the order in which B&amp;B nodes in the tree are picked for processing. Default is <CODE>
   *       graph.packing.DefDBBNodeComparator</CODE>.
   *   <li>ff, $num$ optional, specify the "fudge factor" used in determining what constitutes the
   *       list of "best nodes" in the 1-packing problem (a.k.a. the MWIS problem) where it makes
   *       much more sense to have a "fudge factor" by which to multiply the best cost in order to
   *       determine if a node is "close enough" to the best cost to be included in the
   *       best-candidate-nodes list. Default value is <CODE>DBBNode1._ff</CODE> (currently set to
   *       0.85). The smaller this value, the longer it will take for the search to complete, with
   *       potentially better solutions found.
   *   <li>minknownbound, $num$ optional, a known bound to the problem at hand, which will be used
   *       to fathom B&amp;B nodes having smaller bounds than this number. Currently only applies to
   *       1-packing problems. Default is -infinity.
   *   <li>expandlocalsearchfactor, $num$ optional, if present, then when a solution is found within
   *       the specified factor of the best known solution, a local search kicks in. Default is 1.0
   *       (only when a best solution is found does local search kicks in). Currently only applies
   *       to 1-packing problems.
   * </ul>
   *
   * <br>
   * args[2]: [optional] override max num nodes in params_file to create in B&amp;B procedure
   *
   * <p>This implementation writes the solution in a text file called "sol.out" in the current
   * directory, whose lines contain one number each, the id of each "active" node in the solution
   * (id in the set {1,...graph_num_nodes}).
   *
   * @param args String[]
   */
  public static void main(String[] args) {
    try {
      if (args.length < 2) {
        System.err.println(
            "usage: java -cp <classpath> graph.packing.DBBGASPPacker <graphfilename> <paramsfilename> [maxnumBBnodes]");
        System.exit(-1);
      }
      long start = System.currentTimeMillis();
      Graph g = DataMgr.readGraphFromFile2(args[0]);
      // print out total value weight of the nodes
      {
        double totw = 0.0;
        for (int i = 0; i < g.getNumNodes(); i++) {
          Double w = g.getNode(i).getWeightValue("value");
          totw += w == null ? 1.0 : w.doubleValue();
        }
        System.err.println("Graph total nodes' weight=" + totw);
      }
      HashMap params = DataMgr.readPropsFromFile(args[1]);
      int maxnodes = -1;
      if (args.length > 2) maxnodes = Integer.parseInt(args[2]); // override max num nodes

      Graph[] graphs = null;
      if (g.getNumComponents() > 1) {
        // graphs = g.getGraphComponents();
        System.err.println(
            "Distributed BBGASPPacker does not currently support breaking graphs into disconnected components...");
        System.exit(-1);
      } else { // optimize when there is only one component in the graph
        graphs = new Graph[1];
        graphs[0] = g;
      }
      // now run the B&B algorithm for each sub-graph
      PrintWriter pw = new PrintWriter(new FileWriter("sol.out"));
      for (int j = 0; j < graphs.length; j++) {
        Graph gj = graphs[j];
        System.err.println(
            "Solving for subgraph "
                + (j + 1)
                + " w/ sz="
                + gj.getNumNodes()
                + " (/"
                + graphs.length
                + ")");
        if (gj.getNumNodes() == 3 && gj.getNumArcs() == 2) {
          _totActiveNodes += 2;
          ++_totLeafNodes;
          // figure out which node is the connecting one
          int best_node_id = -1;
          for (int m = 0; m < 3; m++) {
            Node nm = gj.getNode(m);
            if (nm.getNbors().size() == 1) {
              Double nmwD = nm.getWeightValue("value");
              double w = nmwD == null ? 1.0 : nmwD.doubleValue();
              _totActiveNodeWeights += w;
              Integer mI = (Integer) gj.getNodeLabel(m);
              best_node_id = mI == null ? m : mI.intValue(); // null mI -> g connected
              pw.println((best_node_id + 1));
            }
          }
          continue;
        } else if (gj.getNumNodes() <= 2) {
          ++_totActiveNodes;
          ++_totLeafNodes;
          // figure out max. node-weight
          int best_node_id = -1;
          double maxw = Double.NEGATIVE_INFINITY;
          for (int m = 0; m < gj.getNumNodes(); m++) {
            Double nmwD = gj.getNode(m).getWeightValue("value");
            double nmw = nmwD == null ? 1.0 : nmwD.doubleValue();
            if (nmw > maxw) {
              maxw = nmw;
              Integer mI = (Integer) gj.getNodeLabel(m);
              best_node_id = mI == null ? m : mI.intValue(); // null mI -> g connected
            }
          }
          _totActiveNodeWeights += maxw;
          pw.println((best_node_id + 1));
          continue;
        }
        // double bound = Double.MAX_VALUE;
        double bound = 0;
        String pdahost = "localhost";
        if (params.containsKey("pdahost")) pdahost = (String) params.get("pdahost");
        int pdaport = 7981;
        if (params.containsKey("pdaport")) pdaport = ((Integer) params.get("pdaport")).intValue();
        String cchost = "localhost";
        if (params.containsKey("cchost")) cchost = (String) params.get("cchost");
        int ccport = 7899;
        if (params.containsKey("ccport")) ccport = ((Integer) params.get("ccport")).intValue();
        String acchost = "localhost";
        if (params.containsKey("acchost")) acchost = (String) params.get("acchost");
        int accport = 7900;
        if (params.containsKey("accport")) accport = ((Integer) params.get("accport")).intValue();
        // DBBTree.init(g, bound, pdahost, pdaport, cchost, ccport, acchost, accport);
        // DBBTree t = DBBTree.getInstance();
        Boolean localSearchB = (Boolean) params.get("localsearch");
        // if (localSearchB != null) t.setLocalSearch(localSearchB.booleanValue());
        boolean localsearch = false;
        if (localSearchB != null) localsearch = localSearchB.booleanValue();
        AllChromosomeMakerClonableIntf maker =
            (AllChromosomeMakerClonableIntf) params.get("localsearchtype");
        // if (maker!=null) t.setLocalSearchType(maker);
        Double ffD = (Double) params.get("ff");
        /*
        if (ffD!=null) {
        	DBBNode1.setFF(ffD.doubleValue());
        	DBBNode1.disallowFFChanges();
        }
        */
        double ff = 0.85;
        if (ffD != null) ff = ffD.doubleValue();
        Integer tlvlI = (Integer) params.get("tightenboundlevel");
        /*
        if (tlvlI != null && tlvlI.intValue() >= 1) t.setTightenUpperBoundLvl(
        	tlvlI.intValue());
            */
        int tlvl = Integer.MAX_VALUE;
        if (tlvlI != null && tlvlI.intValue() >= 1) tlvl = tlvlI.intValue();
        Boolean usemaxsubsetsB = (Boolean) params.get("usemaxsubsets");
        boolean usemaxsubsets = true;
        if (usemaxsubsetsB != null)
          // t.setUseMaxSubsets(usemaxsubsetsB.booleanValue());
          usemaxsubsets = usemaxsubsetsB.booleanValue();
        int kmax = Integer.MAX_VALUE;
        Integer kmaxI = (Integer) params.get("maxitersinGBNS2A");
        if (kmaxI != null && kmaxI.intValue() > 0)
          // t.setMaxAllowedItersInGBNS2A(kmaxI.intValue());
          kmax = kmaxI.intValue();
        Boolean sortmaxsubsetsB = (Boolean) params.get("sortmaxsubsets");
        boolean sortmaxsubsets = false;
        if (sortmaxsubsetsB != null)
          // t.setSortBestCandsInGBNS2A(sortmaxsubsetsB.booleanValue());
          sortmaxsubsets = sortmaxsubsetsB.booleanValue();
        Double avgpercextranodes2addD = (Double) params.get("avgpercextranodes2add");
        double apen2a = 0.0;
        if (avgpercextranodes2addD != null)
          // t.setAvgPercExtraNodes2Add(avgpercextranodes2addD.doubleValue());
          apen2a = avgpercextranodes2addD.doubleValue();
        Boolean useGWMIN24BN2AB = (Boolean) params.get("useGWMIN2criterion");
        boolean ugwm2 = false;
        if (useGWMIN24BN2AB != null)
          // t.setUseGWMIN24BestNodes2Add(useGWMIN24BN2AB.booleanValue());
          ugwm2 = useGWMIN24BN2AB.booleanValue();
        Double expandlocalsearchfactorD = (Double) params.get("expandlocalsearchfactor");
        double elsf = 1.0;
        if (expandlocalsearchfactorD != null)
          // t.setLocalSearchExpandFactor(expandlocalsearchfactorD.doubleValue());
          elsf = expandlocalsearchfactorD.doubleValue();
        double mkb = Double.NEGATIVE_INFINITY;
        Double minknownboundD = (Double) params.get("minknownbound");
        if (minknownboundD != null) // t.setMinKnownBound(minknownboundD.doubleValue());
        mkb = minknownboundD.doubleValue();
        int maxchildren = Integer.MAX_VALUE;
        Integer maxchildrenI = (Integer) params.get("maxnodechildren");
        if (maxchildrenI != null && maxchildrenI.intValue() > 0)
          // t.setMaxChildrenNodesAllowed(maxchildrenI.intValue());
          maxchildren = maxchildrenI.intValue();
        DBBNodeComparatorIntf bbcomp = (DBBNodeComparatorIntf) params.get("dbbnodecomparator");
        // if (bbcomp!=null) t.setDBBNodeComparator(bbcomp);
        DBBTree.init(
            g,
            bound,
            pdahost,
            pdaport,
            cchost,
            ccport,
            acchost,
            accport,
            localsearch,
            maker,
            ff,
            tlvl,
            usemaxsubsets,
            kmax,
            sortmaxsubsets,
            apen2a,
            ugwm2,
            elsf,
            mkb,
            maxchildren,
            bbcomp);
        DBBTree t = DBBTree.getInstance();
        t.run();
        int orsoln[] = t.getSolution();
        int tan = 0;
        double tanw = 0.0;
        for (int i = 0; i < orsoln.length; i++) {
          if (orsoln[i] == 1) {
            Integer miI = (Integer) gj.getNodeLabel(i);
            int mi = (miI == null) ? i : miI.intValue(); // null miI -> g connected
            // System.out.print( mi + " ");
            pw.println((mi + 1));
            ++tan;
            Double twD = gj.getNode(i).getWeightValue("value");
            tanw += (twD == null ? 1.0 : twD.doubleValue());
          }
        }
        // tanw == t.getBound()
        System.err.println("Total BB-nodes=" + t.getCounter());
        System.err.println("Total leaf BB-nodes=" + t.getTotLeafNodes());
        _totActiveNodes += tan;
        _totActiveNodeWeights += tanw;
        _totLeafNodes += t.getTotLeafNodes();
        System.err.println(
            "Total active nodes so far: "
                + _totActiveNodes
                + " active node weights="
                + _totActiveNodeWeights
                + " total overall trees leaf BB nodes="
                + _totLeafNodes);
        System.err.println(
            "Total #DLS searched performed: "
                + t.getNumDLSPerformed()
                + " Total time spent on DLS: "
                + t.getTimeSpentOnDLS());
      }
      pw.flush();
      pw.close();
      long time = System.currentTimeMillis() - start;
      System.out.println("Best Soln = " + _totActiveNodeWeights);
      System.out.println("\nWall-clock Time (msecs): " + time);
      System.out.println("Done.");
      System.exit(0);
    } catch (Exception e) {
      e.printStackTrace();
      System.exit(-1);
    }
  }