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The root page 15DOTs60ia13:Tutorial could not be found in space 15.S60 SSIM: Software Tools for Operations Research.

TSPLIB performance

Run your new code (through main, with both the lazy and userCut options) on the same TSPLIB problems as before and compare the performance. My running times were

Unknown macro: {chart}

Strategy

eil51

bier127

ch130

ch150

d198

Without User Cuts

0.13

1.23

1.96

5.01

8.58

With User Cuts

0.40

3.76

12.52

83.62

103.93

We see that the running time increased on every example! What is going on here? Lets add a few timely print statements to our UserCutCallback to try and better understand how the performance is worse.

			V source = graph.getVertices().iterator().next();
			double best = Double.MAX_VALUE;
			System.out.print("looking for cut... ");
			for(V sink : graph.getVertices()){
				if(sink != source){
					Set<V> cutNodes = minCutSolver.computeMinCut(source, sink);
					Set<E> cutEdges = tspInstance.cutEdges(cutNodes);
					double cutVal = Util.calcSum(cutEdges,edgeWeights);
					best = Math.min(best, cutVal);
					if(cutVal < minCutVal){
						cutSets.add(cutEdges);
					}
				}
			}
			System.out.println("found " + cutSets.size() + " cuts, best " +	best);

Lets run the code again on d198 and take a closer look at the output.

Reading file d198... complete!
Building problem... complete!
Solving TSP...Lazy constraint(s) or lazy constraint callback is present.
    Disabling dual reductions (CPX_PARAM_REDUCE) in presolve.
    Disabling non-linear reductions (CPX_PARAM_PRELINEAR) in presolve.
Tried aggregator 1 time.
Presolve time = 0.01 sec. (9.91 ticks)
Warning: Control callbacks may disable some MIP features.
Probing time = 0.01 sec. (3.35 ticks)
MIP emphasis: balance optimality and feasibility.
MIP search method: traditional branch-and-cut.
Parallel mode: none, using 1 thread.
Root relaxation solution time = 0.01 sec. (13.55 ticks)

        Nodes                                         Cuts/
   Node  Left     Objective  IInf  Best Integer    Best Bound    ItCnt     Gap         Variable B NodeID Parent  Depth

      0     0    11793.0000    28                  11793.0000      282         
      0     0    11823.0000    18                     Cuts: 9      291         
      0     0    11837.0000    21                     Cuts: 5      295         
      0     0    11845.1667    10                     Cuts: 5      303         
      0     0    11846.0000    10                     Cuts: 5      306         
      0     0    11846.7500    14                     Cuts: 3      309         
      0     0    11848.0000     8                     Cuts: 3      311         
      0     0    11858.0000    10                     Cuts: 5      317         
      0     0    11860.8750    18                     Cuts: 5      323         
      0     0    11869.5000    12                     Cuts: 3      329         
      0     0    12442.0000    22                     Cuts: 3      331         
      0     0    12450.0000    20                     Cuts: 8      337         
      0     0    12456.0000    14                     Cuts: 5      351         
      0     0    12461.5000    14                     Cuts: 6      358         
      0     0    12462.5000    22                     Cuts: 3      367         
      0     0    12463.0000    14                 ZeroHalf: 1      370         
looking for cut... found 2 cuts, best 0.0
      0     0    12463.0000    22                     User: 2      372         
looking for cut... found 1 cuts, best 0.0
      0     0    12463.0000    33                     User: 1      374         
looking for cut... found 3 cuts, best 0.0
      0     0    12468.5000    20                     User: 3      389         
looking for cut... found 3 cuts, best 0.0
      0     0    12468.5000    20                     User: 3      392         
looking for cut... found 3 cuts, best 0.0
      0     0    12469.7500    60                     User: 3      399         
looking for cut... found 2 cuts, best 0.0
      0     0    12471.0000    40                     User: 2      401         
looking for cut... found 3 cuts, best 0.0
      0     0    12479.5000    44                     User: 3      413         
looking for cut... found 3 cuts, best 0.0
      0     0    12494.0000    49                     User: 3      428         
looking for cut... found 1 cuts, best 0.0
      0     0    12494.0000    49                     User: 1      429         
looking for cut... found 3 cuts, best 0.0
      0     0    12980.0000    34                     User: 3      449         
looking for cut... found 4 cuts, best 0.0
      0     0    12981.2500    61                     User: 4      454         
looking for cut... found 8 cuts, best 0.0
      0     0    12997.5000    14                     User: 8      471         
looking for cut... found 2 cuts, best 0.0
      0     0    13000.5000    29                     User: 2      474         
looking for cut... found 6 cuts, best 0.0
      0     0    13003.3750    57                     User: 6      479         
looking for cut... found 11 cuts, best 0.0
      0     0    13006.5000    22                    User: 11      487         
looking for cut... found 2 cuts, best 0.0
      0     0    13008.5000    73                     User: 2      495         
looking for cut... found 8 cuts, best 0.0
      0     0    13023.2500    62                     User: 8      516         
looking for cut... found 6 cuts, best 0.0
      0     0    14782.0000     8                     User: 6      543         
looking for cut... found 1 cuts, best 0.0
      0     0    14782.0000    28                     User: 1      547         
looking for cut... found 2 cuts, best 0.0
      0     0    14782.0000     8                     User: 2      551         
looking for cut... found 1 cuts, best 0.0
      0     0    14782.6667    45                     User: 1      555         
looking for cut... found 7 cuts, best 0.0
      0     0    14795.5000    68                     User: 7      572         
looking for cut... found 5 cuts, best 0.0
      0     0    14846.0000    40                     User: 5      588         
looking for cut... found 3 cuts, best 0.0
      0     0    14871.5000    42                     User: 3      596         
looking for cut... found 7 cuts, best 0.0
      0     0    14873.0000    14                     User: 7      598         
looking for cut... found 2 cuts, best 0.0
      0     0    14873.4000    52                     User: 2      602         
looking for cut... found 6 cuts, best 0.0
      0     0    14902.5000    38                     User: 6      613         
looking for cut... found 3 cuts, best 0.0
      0     0    14906.0000    16                     User: 3      619         
looking for cut... found 2 cuts, best 0.0
      0     0    14906.8000    54                     User: 2      623         
looking for cut... found 6 cuts, best 0.0
      0     0    14935.2500    44                     User: 6      643         
looking for cut... found 2 cuts, best 0.0
      0     0    15356.5000    32                     User: 2      647         
looking for cut... found 2 cuts, best 0.0
      0     0    15360.0000    52                     User: 2      660         
looking for cut... found 5 cuts, best 0.0
      0     0    15388.7500    74                     User: 5      679         
looking for cut... found 5 cuts, best 0.0
      0     0    15401.7500    70                     User: 5      682         
looking for cut... found 5 cuts, best 0.0
      0     0    15422.2500    66                     User: 5      693         
looking for cut... found 5 cuts, best 0.0
      0     0    15508.2500    64                     User: 5      703         
looking for cut... found 5 cuts, best 0.0
      0     0    15549.7500    58                     User: 5      712         
looking for cut... found 5 cuts, best 0.0
      0     0    15560.0000    20                     User: 5      720         
looking for cut... found 2 cuts, best 0.0
      0     0    15560.0000    20                     User: 2      722         
looking for cut... found 2 cuts, best 0.0
      0     0    15560.0000    20                     User: 2      723         
looking for cut... found 2 cuts, best 0.0
      0     0    15560.0000    20                     User: 2      724         
looking for cut... found 2 cuts, best 0.0
      0     0    15561.0000    42                     User: 2      732         
looking for cut... found 5 cuts, best 0.0
      0     0    15561.0000    38                     User: 5      735         
looking for cut... found 5 cuts, best 0.0
      0     0    15562.7500    67                     User: 5      738         
looking for cut... found 7 cuts, best 0.0
      0     0    15564.7500    71                     User: 7      743         
looking for cut... found 8 cuts, best 0.5
      0     0    15610.0000    47                     User: 8      761         
looking for cut... found 1 cuts, best 0.0
      0     0    15675.0000    47                     User: 1      771         
looking for cut... found 2 cuts, best 0.0
      0     0    15701.2500    61                     User: 2      797         
looking for cut... found 1 cuts, best 0.0
      0     0    15726.7500    71                     User: 1      801         
looking for cut... found 0 cuts, best 1.9999999999999998
      0     2    15726.7500    71                  15740.2500      801                                 0             0
Elapsed time = 23.90 sec. (10515.28 ticks, tree = 0.00 MB, solutions = 0)
looking for cut... found 0 cuts, best 1.9999999999999998
looking for cut... found 1 cuts, best 0.0
looking for cut... found 0 cuts, best 1.9999999999999998
      2     4    15753.0000    47                  15741.2500      816                   x16243 N      2      1      2
looking for cut... found 0 cuts, best 1.9999999999999998
looking for cut... found 1 cuts, best 1.0
looking for cut... found 0 cuts, best 1.9999999999999998
looking for cut... found 1 cuts, best 0.0
looking for cut... found 0 cuts, best 1.9999999999999998
      5     7    15762.0000    49                  15741.2500      834                   x25486 N      5      4      4
looking for cut... found 1 cuts, best 1.0
looking for cut... found 0 cuts, best 1.9999999999999998
looking for cut... found 1 cuts, best 1.5
looking for cut... found 0 cuts, best 1.9999999999999998
looking for cut... found 0 cuts, best 1.9999999999999998
looking for cut... found 0 cuts, best 1.9999999999999998
looking for cut... found 1 cuts, best 0.0
looking for cut... found 1 cuts, best 0.0
looking for cut... found 1 cuts, best 0.0
looking for cut... found 1 cuts, best 0.0
looking for cut... found 1 cuts, best 0.0
looking for cut... found 1 cuts, best 0.0
looking for cut... found 1 cuts, best 0.0
looking for cut... found 1 cuts, best 0.0
looking for cut... found 1 cuts, best 0.0
looking for cut... found 1 cuts, best 0.0
looking for cut... found 1 cuts, best 0.0
looking for cut... found 1 cuts, best 0.0
looking for cut... found 1 cuts, best 0.0
looking for cut... found 0 cuts, best 1.9999999999999998
*    10+   10                        15900.0000    15741.2500      887    1.00%
     10    12    15776.0000    23    15900.0000    15741.2500      887    1.00%           x5629 N     10      9      7
*    12    12      integral     0    15787.0000    15741.2500      891    0.29%           x6893 U     12     11      9
looking for cut... found 0 cuts, best 2.0
looking for cut... found 1 cuts, best 1.3333333333333335
looking for cut... found 0 cuts, best 1.9999999999999998
*    15    11      integral     0    15781.0000    15741.2500      895    0.25%           x6545 U     15     14     11
looking for cut... found 0 cuts, best 1.9999999999999998
looking for cut... found 1 cuts, best 1.0
looking for cut... found 0 cuts, best 1.9999999999999998
looking for cut... found 0 cuts, best 1.9999999999999998
looking for cut... found 0 cuts, best 2.0
looking for cut... found 1 cuts, best 0.0
looking for cut... found 0 cuts, best 1.9999999999999998
looking for cut... found 0 cuts, best 1.9999999999999998
looking for cut... found 1 cuts, best 1.0
looking for cut... found 0 cuts, best 1.9999999999999998
looking for cut... found 1 cuts, best 1.0
looking for cut... found 1 cuts, best 1.3333333333333335
looking for cut... found 1 cuts, best 0.0
looking for cut... found 0 cuts, best 1.9999999999999998
looking for cut... found 0 cuts, best 1.9999999999999998
looking for cut... found 1 cuts, best 1.0
looking for cut... found 0 cuts, best 1.9999999999999998
looking for cut... found 0 cuts, best 1.9999999999999998
looking for cut... found 4 cuts, best 1.5
looking for cut... found 1 cuts, best 1.0
looking for cut... found 0 cuts, best 1.9999999999999998
     29    16    15777.0000    47    15781.0000    15755.7500      971    0.16%           x4489 N     29     27     10
looking for cut... found 0 cuts, best 1.9999999999999998
looking for cut... found 0 cuts, best 1.9999999999999998
looking for cut... found 0 cuts, best 1.9999999999999998
looking for cut... found 1 cuts, best 0.0
looking for cut... found 0 cuts, best 1.9999999999999998
looking for cut... found 0 cuts, best 1.9999999999999998
looking for cut... found 0 cuts, best 1.9999999999999998
looking for cut... found 1 cuts, best 0.0
looking for cut... found 0 cuts, best 1.9999999999999998
looking for cut... found 0 cuts, best 1.9999999999999998
looking for cut... found 0 cuts, best 1.9999999999999998
looking for cut... found 1 cuts, best 0.0
looking for cut... found 0 cuts, best 1.9999999999999998
looking for cut... found 0 cuts, best 1.9999999999999998
looking for cut... found 0 cuts, best 1.9999999999999998
looking for cut... found 0 cuts, best 1.9999999999999998
looking for cut... found 0 cuts, best 1.9999999999999998
looking for cut... found 0 cuts, best 1.9999999999999998
looking for cut... found 0 cuts, best 1.9999999999999998
looking for cut... found 0 cuts, best 1.9999999999999998
looking for cut... found 0 cuts, best 2.0
     57    21    15776.5000     8    15781.0000    15763.0000     1076    0.11%           x2871 D     57     56      9
looking for cut... found 0 cuts, best 2.0
*    59    20      integral     0    15780.0000    15763.0000     1078    0.11%           x2847 D     59     58     11
looking for cut... found 1 cuts, best 0.0
looking for cut... found 1 cuts, best 0.0
looking for cut... found 0 cuts, best 1.9999999999999998
looking for cut... found 0 cuts, best 1.9999999999999998
looking for cut... found 0 cuts, best 1.9999999999999998
looking for cut... found 0 cuts, best 1.9999999999999998
looking for cut... found 0 cuts, best 2.0
looking for cut... found 0 cuts, best 2.0
looking for cut... found 0 cuts, best 2.0
looking for cut... found 0 cuts, best 2.0
looking for cut... found 0 cuts, best 1.9999999999999998
looking for cut... found 0 cuts, best 2.0
looking for cut... found 0 cuts, best 2.0
looking for cut... found 0 cuts, best 2.0
looking for cut... found 1 cuts, best 1.0
looking for cut... found 0 cuts, best 1.9999999999999998
looking for cut... found 2 cuts, best 0.6666666666666667
looking for cut... found 1 cuts, best 1.0
looking for cut... found 0 cuts, best 1.9999999999999998
looking for cut... found 1 cuts, best 1.3333333333333333
looking for cut... found 0 cuts, best 1.9999999999999998
looking for cut... found 0 cuts, best 1.9999999999999998
looking for cut... found 0 cuts, best 2.0
looking for cut... found 0 cuts, best 2.0
looking for cut... found 0 cuts, best 2.0
     91    18        cutoff          15780.0000    15769.5000     1232    0.07%           x2871 U     91     65      9
looking for cut... found 0 cuts, best 1.9999999999999998
looking for cut... found 0 cuts, best 2.0
looking for cut... found 0 cuts, best 2.0
looking for cut... found 0 cuts, best 1.9999999999999998
looking for cut... found 0 cuts, best 2.0
looking for cut... found 1 cuts, best 1.0
looking for cut... found 0 cuts, best 2.0
looking for cut... found 0 cuts, best 1.9999999999999998
looking for cut... found 0 cuts, best 2.0
looking for cut... found 0 cuts, best 2.0
looking for cut... found 0 cuts, best 2.0
looking for cut... found 0 cuts, best 2.0
looking for cut... found 0 cuts, best 1.9999999999999998
looking for cut... found 0 cuts, best 1.9999999999999998
looking for cut... found 0 cuts, best 2.0
looking for cut... found 0 cuts, best 1.9999999999999998

Cover cuts applied:  3
Zero-half cuts applied:  11
User cuts applied:  362

Root node processing (before b&c):
  Real time             =   23.88 sec. (10502.64 ticks)
Sequential b&c:
  Real time             =   49.95 sec. (2187.41 ticks)
                          ------------
Total (root+branch&cut) =   73.83 sec. (12690.05 ticks)
 complete!
opt is: 15780.0

Running the code with and without user cuts, judging from the print statements (which are not entirely reliable), it seems that the bottle neck is actually solving the \( |V|-1 \) max flow min cut problems. While we know there are better ways to compute a global minimum cut from our previous discussion, instead we will explore some heuristics to vastly improve performance without writing a lot of code.

Look at the first column of CPLEX's output, which gives the number of nodes generated thus far. When this column is zero, that means we are still working on the root node. Here are a few observations about our cuts:

  • More often then not, when there is a violated cut, the most violated cut has weight zero.
  • This is particularly true towards the end of the optimization when we are doing branch and bound (perhaps the extreme action of forcing a variable to take the value one or zero causes the solution to make loops).
  • The value of the root LP is significantly larger (which is better) with the cutset constraints, as seen below
  • The total number of nodes created using the cutset constraints is much smaller, as seen below.

These final two bullet points indicate that we do stand to gain a lot from the cutset formulation if we can get the running time of our separation routine under control. We now give a few heuristics to improve performance.

Unknown macro: {chart}

Problem Name

eil51

bier127

ch130

ch150

d198

Without User Cuts

99.6

99.1

99.7

98.1

78.9

With User Cuts

99.7

99.5

99.7

99.4

99.7
Unknown macro: {chart}

Problem Name

eil51

bier127

ch130

ch150

d198

Without User Cuts

16

88

142

1765

1536

With User Cuts

2

19

44

429

138

Check Connected Components First

The first heuristic is as follows: on the subgraph using the edges with only nonnegative weight (which we have already formed) check for connectivity. If you have more than one connected component, then any cut separating connected components will have weight zero and hence be minimal. As we can check for connected components in \( O(|V|+|E|) \) time, we will save a lot of time when there are multiple connected components. Viewing the log file above, we this is frequently the case (as conversely, whenever the best cut is zero, there must have been multiple connected components). Notice that the class MinCutSolver (which is holding the subgraph using only edges with positive weight) already has a method built in:

Method Name

Return Type

Arguments

Description

checkConnectedComponents

Set<Set<V>>

 

Returns the nodes of the subgraph using only edges with positive weight, partitioned into connected components.

To integrate this with the UserCutCallback, simply add the code

			MinCutSolver<V,E> minCutSolver = new MinCutSolver<V,E>(graph,edgeWeights);
			Set<Set<E>> cutSets = new HashSet<Set<E>>();
			//add here
			Set<Set<V>> connectedComponents = minCutSolver.checkConnectedComponents();
			if(connectedComponents.size() > 1){				
				for(Set<V> connectedComponent: connectedComponents){
					cutSets.add(tspInstance.cutEdges(connectedComponent));				
				}
				System.out.println("trivial cuts exist, found " + cutSets.size());
			}
			else{
				//old
				V source = graph.getVertices().iterator().next();
				//...
				System.out.println("found " + cutSets.size() + " cuts, best " +	best);
				//add extra brace
			}			
			for(Set<E> cutEdges: cutSets){
				this.add(cplex.ge(Util.integerSum(cplex, edgeVariables, cutEdges), 2));
			}

Observe the large improvement in running time below.

Unknown macro: {chart}

Problem Name

eil51

bier127

ch130

ch150

d198

Without User Cuts

0.13

1.23

1.96

5.01

8.58

With User Cuts

0.40

3.76

12.52

83.62

103.93

Connected Components First

0.38

2.42

5.10

30.39

45.74

However, we are still not faster than our solver without user cuts, and we still can't practically solve d493 (although we get a lower bound that is 99.6% of the optimum in about 60 seconds). Our separation routine is still too slow, and is notably ineffective towards the end of the optimization. We give a quick fix in the next section.

Use Only Connected Components After Root Node

Recall our observation from the log file that as we progressed in the branch and bound tree, it became rare that we encountered violated cutset constraints where the graph was not disconnected into multiple peices (i.e. constraints violated by 2.0). A coarse but effect solution for this problem is to simply stop solving max flow problems after a certain point in the optimization, and rely on the heuristic from the previous section (as a user cut) in combination with the lazy constraints. A convenient (and reasonable) place to stop attempting these max flow problems is when we solve the root node and start branching. That we chose the root node to stop looking for expensive cuts was not a coincidence: we see in was not a coincidence, our earlier computations on the quality of the cutset LP relaxation suggest that once we have solved the root node, our lower bound will be within 1% of optimal, leaving little room for further improvement by more cutting planes. Using the method getNnodes64(), we can modify MinCutCallback with a single line of code as follows:

			//...
			if(connectedComponents.size() > 1){				
				for(Set<V> connectedComponent: connectedComponents){
					cutSets.add(tspInstance.cutEdges(connectedComponent));				
				}
				System.out.println("trivial cuts exist, found " + cutSets.size());
			}
			else if(this.getNnodes64() == 0){// <--- the only line that changed!				
				V source = graph.getVertices().iterator().next();
			//...

We see in the chart below that this new variation beats out all the other user cut heuristics thus far and is competitive with just using lazy constraints on small instances.

Unknown macro: {chart}

Problem Name

eil51

bier127

ch130

ch150

d198

Without User Cuts

0.13

1.23

1.96

5.01

8.58

With User Cuts

0.40

3.76

12.52

83.62

103.93

Connected Components First

0.38

2.42

5.10

30.39

45.74

Stop After Root

0.20

1.70

2.40

4.77

10.3

However, the real gain is on the larger, more challenging instances, which we can now actually solve. As seen in the log file, it took a little over 2 minutes to be provably within 1% of optimal, and we were able to solve the problem completely in about 12 minutes, while it took about 2 hours with only lazy constraints. Again the LP was 99.6% of optimal, so we can't make a lot of improvement with better cuts (i.e. there is more room for improvement on the primal problem of generating good integer solutions). However, we did ultimately visit over 26,000 nodes. The problem d657 is still well out of reach; after 4571 seconds (over an hour), the MIP relative gap was still 0.33%.

Unknown macro: {chart}

Problem Name

d493

d657

Stop After Root

744.36

 

Next, we try two more tricks to improve performance, but they are essentially second order corrections as compared to Christofides and Two-Opt.

(Optional) Use a Back Off Function After the Root Node

We now introduce the abstract class BackOffFunction. The idea of a back off function is as follows: suppose you repeatedly have the opportunity to take some action, but you do not want to take the action at every opportunity. Further, suppose that at as time passes, you want to take the action with a lower frequency. Suppose that \( w_k \) is the waiting time between the \( k-1 \) and \( k \) time that the action was taken. The various subclasses of BackOffFunction provided will deterministically set \( w_k \) as

Class

Constructor Arguments

 \( w_k \)

ConstantWaiting

int \( c \)

\( c \)

LinearWaiting

 

 \( k \)

QuadraticWaiting

 

 \( k^2 \)

ExponentialWaiting

 

 \( 2^k \)

Of course many other functions, perhaps more complicated, are possible. Note that ConstantWaiting technically is not backing off... In any event, back off functions are robust way to contain a potentially troublesome heuristic, as it limits how much the heuristic can hurt you, since in the worst case (when the problem takes a long time to solve), you will stop using the heuristic. The class BackOffFunction (and its subclasses) are all used through the single method

Method Name

Return Type

Arguments

Description

makeAttempt

boolean

 

Call this function at each possible time to take the action. Will return true iff you should take the action this time period.

The idea for using the back off function is as follows: we will still always solve the max flow problems at the root node, but after the root node, when the connectivity test does not produce a new cut, we will only periodically attempt to solve the max flow problems according to the back off function. I found that Quadratic Waiting seemed about right, but feel free try anything! To make the change you only need to adjust a few lines of code. First add the back off function as a field then set in the constructor (as you please):

		private double minCutVal;
		
		public MinCutCallback(){
			minCutVal = 1.99;
			this.backOff = new BackOffFunction.QuadraticWaiting();
		}

Then, in the same line of code we adjusted in the previous section, write:

			//...
			if(connectedComponents.size() > 1){				
				for(Set<V> connectedComponent: connectedComponents){
					cutSets.add(tspInstance.cutEdges(connectedComponent));				
				}
				System.out.println("trivial cuts exist, found " + cutSets.size());
			}
			else if(this.getNnodes64() == 0 || this.backOff.makeAttempt()){// <--- the only line that changed!				
				V source = graph.getVertices().iterator().next();
			//...
Unknown macro: {chart}

Problem Name

eil51

bier127

ch130

ch150

d198

Without User Cuts

0.11

0.77

1.98

5.09

17.57

With User Cuts

5

6

7

8

20

Connected Components First

5

6

7

8

20

Stop After Root

5

6

7

8

20

Backoff after root

5

6

7

8

20

If all else fails, your code should look like this:
_Toggle TspIpSolver_

(Optional) Check Random s-t Cuts

In this final section, we try to fully exploit the following idea: since user cuts are not required to find a violated inequality, it could be more efficient quickly find one with high probability and fail otherwise.

Unknown macro: {chart}

Problem Name

eil51

bier127

ch130

ch150

d198

Without User Cuts

0.11

0.77

1.98

5.09

17.57

With User Cuts

5

6

7

8

20

Connected Components First

5

6

7

8

20

Stop After Root

5

6

7

8

20

Backoff after root

5

6

7

8

20

Randomized s-t

5

6

7

8

20
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