4.10 Polyhedral Comparison
4.11.3 Branch-and-Cut Results
Tables 4.4–4.7 show results for all considered instance sets when solving the RDCST problem to optimality using the formulations within a branch-and-cut system. We report obtained average gaps between the best primal and dual bounds, the median runtime to reach these bounds, and the number of instances solved to optimality within the time limit. Dashes in gap and time columns represent 100% and 10 000 seconds, respectively. It may be surprising that for some instances the time to find the optimal solution is even less than the corresponding time for just solving the
averagegapin%mediantimeinseconds#optimalsolutions(outof5) SetBM1M2BPFLP1P2L1L2L3M1M2BPFLP1P2L1L2L3M1M2BPFLP1P2L1L2L3 R560.00.00.04.73.00.00.00.00.07182-12980000555245555 80.00.00.03.06.80.00.00.00.0323303655836241000555335555 100.02.50.01.46.70.00.00.00.037306377629982000545435555 120.00.00.01.32.50.00.00.00.010744632671000555445555 C560.00.00.01.51.50.00.00.00.03122318716200000555335555 81.50.00.00.95.60.00.00.00.08323536125-1000455325555 104.63.30.01.44.90.00.00.00.0--7--42211115215555 125.12.70.00.54.30.00.00.00.0--72715-72743015305555 E560.01.60.06.515.20.00.00.00.092449548--9111545005555 811.310.70.06.720.41.50.00.00.0--18--163663005004555 1014.211.50.01.916.95.60.00.00.0--50---27436005101555 1214.211.50.02.314.17.10.70.30.0--157---19327422005200445 R10100.00.00.02.64.60.00.00.00.0271013-10421000555245555 150.01.40.04.27.10.00.00.00.0826864-35322000545135555 200.01.60.01.23.00.00.00.00.02912524206620814000545445555 250.00.00.00.00.70.00.00.00.01116842081395111555545555 C10100.00.00.02.50.00.00.00.00.015894-9510000555155555 152.11.60.01.36.10.00.00.00.0255357245337-11112335315555 205.23.20.00.03.40.00.00.00.0--62816-84152110105525555 253.80.90.00.31.20.00.00.00.0-59542715752329531253422135435555 E10102.62.00.04.913.20.00.00.00.0-62537--5111235105555 1514.212.50.06.920.57.00.60.00.0--43---253010005000455 2013.512.00.03.515.68.81.02.00.0--405---81869283005000435 2512.010.80.03.013.311.96.94.70.0--1425-----317005200105 Table4.4:Branch-and-cutresultsforinstancesbyGouveiaetal.[68](B:delay-bound,gap:gapbetweenbestprimalanddualbound, M1:Miller-Tucker-Zemlinapproach,M2:M1withconnectioncuts,BP:stabilizedbranch-and-price,FL:flowapproach,P1:path-cut approach,P2:P1withliftedcuts,L1:layeredgraphapproach,L2:L1withconnectioncutsonG0,L3:L1withconnectioncutsonG0 L, bestresultsareprintedbold).
averagegapin%mediantimeinseconds#optimalsolutions(outof5)SetBM1M2BPFLP1P2L1L2L3M1M2BPFLP1P2L1L2L3M1M2BPFLP1P2L1L2L3
R1001006.06.10.07.912.01.30.00.00.0852634-268824554452345551507.17.70.04.09.77.02.50.00.032177415565391910104454444552004.24.20.01.95.50.00.00.00.047286638763211819194454455552500.00.00.00.00.00.00.00.00.0783644472515962555555555C1001008.33.60.05.912.20.90.00.00.0--8--1335652280250045551507.34.90.01.56.90.00.00.00.9--10--7147613676270152055542006.02.30.00.23.71.22.31.22.1--134334-1384567104677470154233432503.40.70.00.01.11.02.41.33.9-2428145942217-2987-235533130E10010013.013.30.07.616.31.20.00.00.0--16--88441959159300500355515015.112.80.06.417.37.47.85.42.1--40-----916200500100320014.011.40.03.912.810.411.68.810.4--245------00500000025012.99.50.01.610.510.010.37.811.7--425------005200000
R100010009.118.50.09.923.30.00.00.00.0974-12--6272306245315105555150015.118.50.08.821.00.03.03.12.4--18--13243704117958611511544320006.75.30.00.07.43.20.00.00.9241231620796611552028553632354833553455425001.01.90.00.02.10.020.01.820.0173422588911581556036099445545444C100010004.43.80.04.79.81.22.01.62.7-771417--54584786018235114443150011.26.10.02.212.60.04.03.16.8--29--52---005205120200010.37.10.02.610.34.626.528.013.2--39------00520000025005.82.30.00.61.71.364.264.564.5--33677195296---025433000E1000100012.99.90.07.617.23.36.06.08.9--26--449---015013111150015.511.60.05.817.26.410.315.116.1--50------005102000200015.713.40.02.415.612.831.249.433.4--88------005100000250013.810.00.00.512.110.6-----2188067-----005400000
Table4.5:Branch-and-cutresultsforinstancesbyGouveiaetal.[68](B:delay-bound,gap:gapbetweenbestprimalanddualbound,M1:Miller-Tucker-Zemlinapproach,M2:M1withconnectioncuts,BP:stabilizedbranch-and-price,FL:flowapproach,P1:path-cutapproach,P2:P1withliftedcuts,L1:layeredgraphapproach,L2:L1withconnectioncutsonG 0,L3:L1withconnectioncutsonG 0L,bestresultsareprintedbold).
averagegapin%mediantimeinseconds#optimalsolutions(outof30) SetBM1M2BPFLP1P2L1L2L3M1M2BPFLP1P2L1L2L3M1M2BPFLP1P2L1L2L3 T10160.00.00.00.00.30.00.00.00.0110741000303030302930303030 308.212.70.00.025.310.70.00.00.063-2147-64311120133030717303030 5014.723.40.00.633.621.40.00.00.05469-3749--777156302916303030 10022.231.70.02.138.829.10.40.90.0--51805--76727691302414292930 T30160.00.80.02.91.60.00.00.00.04725538583000302830192730303030 3014.418.80.015.134.014.50.00.00.0--9---22283300012303030 5026.232.60.019.543.127.10.00.00.0--20---2021340030003303030 10036.537.60.020.747.336.70.40.60.0--47---1421293620030000292830 T50160.00.00.011.41.60.00.00.00.010255-501400030303012430303030 3016.619.10.019.534.06.60.00.00.0--19--59044421300019303030 5028.429.00.020.243.822.70.00.00.0--44---2320330030006303030 10036.336.90.042.147.036.10.70.80.0--202---3202827680030000272730 T70160.00.40.013.13.00.00.00.00.014578-1090100030283002130303030 3016.019.00.021.032.82.70.00.00.0--31--26244400300023303030 5026.130.00.033.942.417.30.00.00.0--79---2326500030009303030 10033.336.20.195.447.032.50.70.70.9--241---2543188730028000282729 T99160.20.50.017.04.10.00.00.00.04619515-7462000028273001530303030 3014.615.60.030.329.20.30.00.00.0--66--1733400300029303030 5023.326.90.0-39.76.30.00.00.0--150--40122244400300018303030 10031.631.80.3-42.916.50.40.30.6--526---2542507190028002272828 Table4.6:Branch-and-cutresultsforrandominstancesfromSection3.13.1(B:delay-bound,gap:gapbetweenbestprimalanddual bound,M1:Miller-Tucker-Zemlinapproach,M2:M1withconnectioncuts,BP:stabilizedbranch-and-price,FL:flowapproach,P1: path-cutapproach,P2:P1withliftedcuts,L1:layeredgraphapproach,L2:L1withconnectioncutsonG0 ,L3:L1withconnectioncuts onG0 L,bestresultsareprintedbold).
averagegapin%mediantimeinseconds#optimalsolutions(outof18/10/5/5/5)SetBM1M2BPFLP1P2L1L2L3M1M2BPFLP1P2L1L2L3M1M2BPFLP1P2L1L2L3
B-Ran3140.00.00.00.00.00.00.00.00.00000000001818181818181818184270.00.00.00.00.00.00.00.00.0000000000181818181818181818B-Cor400.00.00.00.00.00.00.00.00.0000000000181818181818181818540.00.00.00.00.00.00.00.00.0000000000181818181818181818C-Ran3970.90.80.021.82.70.30.00.00.01212318-771411288104791010105411.92.40.021.83.92.50.00.00.0193048541510689931067710577101010C-Cor500.30.40.021.30.70.10.00.00.093823-52151128810479101010680.00.00.010.10.30.00.00.00.0724833313311821109108910101010D-Ran5540.32.00.040.20.30.00.00.00.01111076-105443333352355557550.10.10.01.90.10.00.00.00.17248145-95204424811725435245554D-Cor660.00.00.020.90.00.00.00.00.061711315723616222555355555900.00.00.00.00.00.00.00.00.011199999108273232555455555Berlin-Ran190.00.00.00.00.00.00.00.00.0000000000555555555260.00.00.00.00.00.00.00.00.0000300000555555555Berlin-Cor1650.00.00.00.00.00.00.00.00.02112109475555555552250.00.00.00.00.00.02.40.03.35814311-1155155555555254Brazil-Ran200.00.00.00.00.00.00.00.00.00001510000555555555270.00.00.00.00.00.00.00.00.0251401164000555555555Brazil-Cor39797.60.00.00.00.00.033.457.039.4-2441131---055555000542510.30.80.00.00.00.0----19167142548---035555000
Table4.7:Branch-and-cutresultsforinstancesbyLeggierietal.[111](B:averagedelay-bound,gap:gapbetweenbestprimalanddualbound,M1:Miller-Tucker-Zemlinapproach,M2:M1withconnectioncuts,BP:stabilizedbranch-and-price,FL:flowapproach,P1:path-cutapproach,P2:P1withliftedcuts,L1:layeredgraphapproach,L2:L1withconnectioncutsonG 0,L3:L1withconnectioncutsonG 0L,bestresultsareprintedbold).
LP relaxation, see Section 4.11.2. This is because of the built-in features of modern MIP solvers like CPLEX and SCIP, e.g. sophisticated presolving methods, general purpose mixed integer cuts, and efficient primal heuristics, see [120] for a general overview of acceleration techniques for MIP solvers.
The overall picture looks similar to the LP results in the previous Section 4.11.2: When considering dense graphs with large sets of achievable edge delays and high delay-bounds in Table 4.5 the branch-and-price approach significantly outperforms all other methods. However, regarding some of these instances path-cut approach P2 is close on BP’s heels, sometimes it is even slightly faster. All other formulations including layered graph models are out of question, here.
However, on instance sets with small delay-bounds, see Tables 4.4, the layered graph ap-proaches seem to be more suitable than BP. Here, the small size of corresponding layered graphs allow both extremely tight bounds and efficient computation of them. It is interesting to see that e.g. for the instances in Table 4.4 it definitely makes sense to add directed connection cut inequalities (4.50) on layered graphG0Lto tighten the LP bounds, but in Tables 4.5–4.7 variants L1 and L2 are superior to L3. Here, the additional overhead of cut separation and the repeated LP resolvings does not pay off.
The picture is not clear for the experiments in Table 4.7: BP is the only one which can solve all instances to optimality within the given time limit. However, other approaches like P2, L1-3, or even M1, have lower average computation times. But if we take the slightly worse general performance of SCIP in comparison to CPLEX into account, runtimes get similar again. In other words, no clear winner can be determined for these instances.
To conclude, the formulations based on MTZ and simple path-cut inequalities provide too weak bounds to be competitive and the multi-commodity flow formulation typically includes too many variables. The rest of the approaches – branch-and-price, layered graph approaches, and the path-cut formulation with lifted inequalities – all have their strengths and weaknesses:
BP provides robust performance throughout all tests. The formulations based on layered graphs provide the tightest bounds and are leading on sets with small delay-bounds but are unusable for large delays and bounds. Finally, the lifted path-cut approach P2 does not obtain the best LP bounds but due to the small number of variables and the fact that the model stays quite small throughout the branch-and-cut process this approach is still competitive in some cases.
4.12 Future Work
We have seen that the size of the corresponding layered graph is crucial for the performance of these approaches. Thus, we aim at extending our preprocessing methods to further reduce graph GL. It is definitely worth to spend more runtime here applying more sophisticated and complex reduction rules which may additionally consider costs to eliminate suboptimal arcs and nodes.
A different approach to deal with large graph sizes will be discussed in Chapter 6.
The simple formulation based on lifted infeasible path inequalities yields surprisingly good results. Thus, we want to find further strengthenings of these inequalities or maybe find other similar sets of valid inequalities. Currently, edge delays are not incorporated in the inequalities but considered in a more abstract way in the definition of infeasible paths. However, we also
may sum up the delays of used edges within the constraint and then ensure the satisfaction of the delay-bound. We already considered inequalities of this type but they are still in a preliminary and rather weak stage.
Similarly as Gouveia et al. [70] did for the HCMST problem, we should further think about the reasons why layered graph formulations are that strong. Maybe we are able to find new sets of valid inequalities projected from the layered graph space into the space of original arc variables. These additional inequalities can then be used to further strengthen formulationPC.
Within a branch-and-bound system it is highly important for the overall performance to provide strong primal bounds, i.e. feasible solutions. Whenever an incumbent solution is found it could be improved by heuristics, e.g. by a variable neighborhood descent. The hopefully better bound could then be used as new cut-off value for subtree pruning and the corresponding tree as guiding solution for MIP solver internal heuristics.
There are already many articles presenting heuristic approaches for the RDCST problem.
However, it might be worth to adapt our methods presented in Chapter 3 for the RDCMST problem accordingly and apply them on the RDCST problem. This is not a trivial task since successful heuristics for the RDCST problem usually consider potential Steiner nodes in a spe-cial way, e.g. it is common for genetic algorithms for Steiner tree problems to use a binary vector for the optional nodes deciding which optional nodes are included in the tree and which not. Clearly, even if we know the exact set of nodes included in the solution the problem to decode one specific vector to an according optimal delay-constrained tree is stillN P-hard in our case since then we are actually faced with the RDCMST problem.
CHAPTER 5
Rooted Delay- and Delay-Variation-Constrained Steiner Tree Problem
This chapter discusses several exact mixed integer programming approaches for solving the rooted delay- and delay-variation-constrained Steiner tree (RDDVCST) problem. Section 5.1 formally defines the problem and Section 5.2 summarizes previous related work. Modifications and extensions to the reduction techniques for the RDCST problem from Section 4.3 are pro-posed in Section 5.3. In Section 5.4 we present a formulation based on multi-commodity flows for the RDDVCST problem. The transformation to a layered graph and a corresponding model is described in Section 5.5 and 5.6, respectively. The presented models are compared theoreti-cally in Section 5.7 and practitheoreti-cally in Section 5.8. Finally, Section 5.9 discusses open problems and possible future research directions. Most parts are based on the published article [165].
5.1 Problem Definition
The RDDVCST problem is a generalization of the RDCST problem discussed in Chapter 4 in which we have to satisfy an additional constraint relating the path-delays to different terminal nodes.
More formally, we are given an undirected graphG= (V, E)with node setV, a fixed root nodes∈V, setR⊆V \ {s}of terminal or required nodes, setS =V \(R∪ {s})of potential Steiner nodes, edge setE, a cost functionc:E →Z+0, a delay functiond:E → Z+, a delay boundB ∈ Z+, and a delay-variation-bound D ∈ Z+0. A feasible solution to the RDDVCST problem is a Steiner tree T = (V0, E0), s ∈ V0, R ⊂ V0 ⊆ V, E0 ⊆ E, satisfying the delay-constraints
dTv = X
e∈PT(s,v)
de≤B, ∀v∈R, (5.1)
wherePT(s, v)denotes the unique path from the specified root nodesto terminal nodev∈Rin Steiner treeT anddTv the corresponding total delay on this path. We further limit the difference between the path-delays to any two terminal nodes by the constraint
u,v∈Rmax |dTu −dTv| ≤D. (5.2)
Finally, we define the cost function
cT = X
e∈E0
ce, (5.3)
summing up the cost values of all edges in a solution T. An optimal solution T∗ to the RD-DVCST problem is a feasible solution with minimal total edge costs, i.e.cT∗ ≤cT, ∀T.
Similarly to the RDCST problem, we define a directed variant of this problem on graph G0 = (V, A)with arc setA ={(s, v) : {s, v} ∈ E} ∪ {(u, v),(v, u) : {u, v} ∈ E, u, v 6=s}
consisting of two opposite arcs for each edge in graphGexcept for edges incident to root node s, for which we include only the corresponding arc going out from s. A feasible solution to the directed variant is a Steiner arborescenceT0 = (V0, A0), s∈ V0, R ⊂V0 ⊆V, A0 ⊂A, directed out of root nodes. It can be easily seen that each feasible Steiner treeT bijectively corresponds to a feasible Steiner arborescenceT0.
The RDDVCST problem is N P-hard because the RDCST problem, whereD = B, is an N P-hard special case, see Chapter 4. Furthermore, Rouskas and Baldine [160] showed that even the problem of finding a feasible solution without considering the costs isN P-hard.
A lower bound to the optimal cost value clearly is provided by relaxing the delay- and/or the delay-variation-constraints, e.g. resulting in a minimal-cost Steiner treeTlwithout considering any constraints on delays. If such a treeTlis feasible for the RDDVCST problem, i.e. satisfies the delay- and delay-variation-constraints, thenTlalso is an optimal solution for it. However, finding an optimal Steiner treeTl is stillN P-hard. In contrast to the RDCMST and RDCST problem, we are not able to construct a trivial feasible solution here, due to theN P-hardness result mentioned above.
Instead of usingc{u,v}andd{u,v}to denote cost and delay values assigned to edge{u, v} ∈ E, we use the better readable notationcuvandduv, respectively. The same holds for arcs(u, v)∈ Ain directed graphG0. Variabledv, v ∈V, refers to the node delay with respect to one specific treeT. In case of multiple solutions the considered tree is explicitly included in the notation, i.e.
dTv, v ∈V.
5.2 Related Work
Rouskas and Baldine [159, 160] introduce a variant of the RDDVCST problem called delay-and delay-variation-bounded multicast tree (DVBMT) problem. In it the aim is to just find a feasible tree satisfying both the delay- and delay-variation-constraints without considering edge costs at all. To solve the DVBMT problem the authors present a construction heuristic with rela-tively high runtime complexity again adapting the concept by Takahashi and Matsuyama [177], cf. Section 4.2: They start with a feasible path to one terminal node and iteratively connect the rest of the terminals in feasible ways as long as possible by computing k-shortest-delay-paths. Haberman and Rouskas [77] tackle the RDDVCST problem for the first time and present
(1,2)
Figure 5.1: Example graphs withB = 4, D = 0, and squared nodes representing terminal nodes: (a) Removing edge{1,2} according to reduction rule (3.4) results in an infeasible in-stance. (b) Similarly, removing edge{1,2}according to Theorem 3.3.1 also leads to infeasibil-ity.
a heuristic similar to the one in [160] but additionally considering edge costs. Lee et al. [109]
provide another construction heuristic: first, the shortest-delay-paths to all terminals are com-bined to form a tree naturally satisfying the delay-constraint. Second, tree costs are reduced possibly violating delay- and delay-variation-constraints. Not feasibly connected terminals are then removed and re-added to the tree by low-delay paths. Low et al. [121] present a two phase construction approach: in the first phase a tree is obtained by only considering the costs and the delay-constraint. If the delay-variation-constraint is violated in this solution the second phase searches for alternative paths in a distributed way. Sheu et al. [171] improve the worst-case time complexity of the heuristic in [160] for the DVBMT problem still obtaining high quality solu-tions in the sense that the delay-variation is quite low. A further construction heuristic avoiding drawbacks of previously proposed approaches can be found in [10]. Zhang et al. [108] propose a simulated annealing approach for the RDDVCST problem using a path-based solution encoding scheme and a path-exchange neighborhood only allowing feasible moves. Several of the men-tioned articles also discuss the complexity of the solution modification if nodes join or leave the multicast group.
To the best of our knowledge only one MIP formulation exists so far for another problem variant in which the delay-variationDis minimized while satisfying the delay-constraints with-out considering edge costs: Sheu et al. [172] present an MCF formulation, which we revise and adapt to the RDDVCST problem in Section 5.4.
5.3 Preprocessing
Clearly, we can apply all procedures to eliminate infeasible edges described in Section 3.3.1 and 4.3 for the RDCMST and RDCST problem, respectively. However, unfortunately it is not feasible here to use methods removing suboptimal edges as defined in Section 3.3.2. In some cases we may have to choose more expensive paths with higher delay to a terminal node to satisfy the delay-variation-constraint. In other words, smaller delay combined with smaller cost does not imply a better connection anymore. Figure 5.1 shows two of these situations where preprocessing based on root edges (3.4) and alternative triangles subject to Theorem 3.3.1 is
invalid, respectively. A similar counterexample can easily be found for the reduction procedure based on alternative paths.
However, we are still able to utilize the delay-variation-bound to further reduce graphGby removing all edges connecting two terminal nodes withde > Dbecause they clearly cannot appear in any feasible solution. Additionally, in graphG0we can safely remove all arcs(u, v)∈ Agoing out of a terminal nodeu∈Rwithduv> D.
5.4 Multi-Commodity Flow Formulation
The following formulation extends the multi-commodity flow formulation for the RDCST prob-lem in Section 4.6 by additionally considering the delay-variation-constraint. We use binary decision variablesxuv, ∀(u, v) ∈ A. Furthermore, real-valued flow variables fuvw, ∀(u, v) ∈ A, ∀w∈R, denote the flow on arc(u, v)from rootsto terminalw. The minimal path-delay is described by variableδmin. ModelMCF is defined as follows:
min X Classical flow constraints (5.5) describe the flow of one commodity for each terminalw ∈ R originating in roots, possibly passing any nodes in V \ {s, w}, and ending in target node w, respectively. Constraints (5.6) add up the delays on the path to a terminal and define lower and upper delay-bounds over all required nodes respecting the delay-variation D. Since variable δminis restricted to[1, B−D], inequalities (4.23) are included in (5.6) and thus delay-boundB is satisfied implicitly. Finally, linking constraints (5.8) connect flow and arc variables.
Providing edge costs are strictly positive, objective (5.4) together with constraints (5.5), (5.8) and (5.9) describe optimal Steiner trees in directed graphs, cf. [124]. However, by adding con-straints (5.6) and (5.7) detached cycles consisting of Steiner nodes may occur in an optimal solution to modelMCF, see Fig. 5.2b: arcs(0,1)and(1,2)connect both terminal nodes to the root within the given delay-boundB = 4but result in a delay-variation ofD = 3. Instead of using optimal arcs(0,1)and(0,2), see Fig. 5.2a, it is cheaper and feasible in modelMCF to add a circular flow for terminal 1 on the detached cycle(3,4,5), sof011 =f341 =f451 =f531 = 1 andf012 =f122 = 1. Due to constraints (5.6) the “path-delay” to node 1 is now increased to 4 and thusD = 0. To prevent infeasible solutions we guarantee root connectivity for all used Steiner nodes. Therefore, we add sets of flow variables and constraints for each potential Steiner node.
(1,1) (1,1)
Figure 5.2: (a) Example graphGwith edge labels(ce, de). Squared nodes denote terminal nodes and blue edges show the optimal solution forB = 4, D = 0, withcT = 7. (b) The optimal solution to modelMCF has costscT = 5but is infeasible for the RDDVCST problem.
But only if there is an incoming arc to a Steiner node the corresponding flow is activated. This finally feasible modelMCF0 extendsMCF by:
X Flow constraints (5.10) for optional nodes are similar to the counterparts (5.5) for terminal nodes but extended by indegree terms to optionally enable or disable the corresponding flows.
5.5 Transformation to Layered Graph
The transformation of graphG0 = (V, A)to a layered graphGL= (VL, AL)works exactly in the same way as for the RDCST problem, see Section 4.8 for further details. Again, we want to find an arborescenceTL= (VLT, ATL)inGLwithVLT ⊆VL, ATL ⊆AL, rooted ins∈VLT, including exactly one nodevl∈VLT for each terminal nodev ∈Rand at most one nodeul∈VLT for each potential Steiner nodeu ∈ S, having minimal costscTL =P
(uk,vl)∈ATLcuv. Additionally, we have to satisfy the transformed delay-variation-constraint
max
uk,vl∈VLT, u,v∈R
|k−l| ≤D. (5.12)
Due to its possibly huge size preprocessing inGL is even more important than inG0. Let deg−(uk)anddeg+(uk)denote the indegree and outdegree of nodeuk, respectively. The fol-lowing reduction steps are repeated as long asGLis modified by one of them:
1. A nodevl∈VL, v∈R, is removed if∃u∈R\ {v}withuk∈/ VL, ∀k∈ {l−D, l+D}, sincevlcannot be in any feasible solution.
2 3
Figure 5.3: (a) Example graph with edge labels(ce, de). Squared nodes denote terminal nodes.
(b) Corresponding layered digraph forB = 4andD = 1 (arc costs are omitted). (c) Prepro-cessed graphGL with optimal solution denoted by blue arcs. (d) Optimal treeT∗ inG with cT∗ = 17.
2. To partly prevent cycles of length two in G0 an arc (uk, vl) ∈ AL can be removed if deg−(uk) = 1∧(vm, uk)∈ALorv∈S∧deg+(vl) = 1∧(vl, um)∈AL.
3. If nodevl ∈ VL\ {s}has no incoming arcs it cannot be reached fromsand therefore is removed.
4. If nodevl ∈ VL\ {s}, v ∈ S, has no outgoing arcs it is removed since a Steiner node cannot be a leaf in an optimal solution.
A naive implementation of the first rule would examine for each layered nodevl∈VL, v ∈R, all other nodesuk ∈VL, u∈R\ {s}, k∈ {l−D, l+D}, running inO(|VL| · |R| ·D)time.
We significantly improved this performance by using a sweep algorithm basically dragging a
“window” with width2D+ 1through the layers ofGL. In each window position we update the set of active terminal nodes within that interval. If at any time the number of active nodes falls below|R|then all nodesvl∈VL, v∈R, on layerlin the middle of the current window can be removed. This improved variant runs inO(|VL|)time.
It was interesting to see in preliminary tests that the complete reduction procedure is some-times able to eliminate the whole layered graph if the delay-variation-constraint is set too tight to allow a feasible solution. On the same instances a branch-and-cut approach based on model MCF0 had to solve thousands of branch-and-bound nodes to prove infeasibility. Thus, the lay-ered graph transformation and reduction can be seen as a heuristic method for detecting infeasi-bility in an early stage of the solving process.
See Fig. 5.3 for an example of layered graph transformation, preprocessing, and solution cor-respondence. Note that example graphGis the same as in Fig. 4.2 for the RDCST problem but the cost of an optimal solution increases from 9 to 17 by imposing a delay-variation-constraint withD= 1.
5.6 Layered Graph Formulation
We extend the model presented in Section 4.9 for the RDCST problem without delay-variation-constraint. Again, we use binary variables xuv, ∀(u, v) ∈ A, to model original arcs in G0. Additionally, continuous variables yvl, ∀vl ∈ VL\ {s}, andxluv, ∀(ul, vk) ∈ AL, represent nodes and arcs in layered graphGL, respectively. ModelLAY is then defined as follows:
min X
(u,v)∈A
cuvxuv (5.13)
s.t. X
vl∈VL
yvl = 1 ∀v∈R (5.14)
X
vl∈VL
yvl ≤1 ∀v∈S (5.15)
X
(uk,vl)∈AL
xkuv=yvl ∀vl∈VL\ {s} (5.16)
X
(uk,vl)∈AL,u6=w
xkuv≥xlvw ∀(vl, wj)∈AgL (5.17) x0sv=xsv ∀(s, v)∈A (5.18) X
(uk,vl)∈AL
xkuv=xuv ∀(u, v)∈A, u6=s (5.19)
δmin ≤
B
X
l=1
l·yvl ≤δmin+D ∀v∈R (5.20)
δmin∈[1, B−D] (5.21)
xkuv≥0 ∀(uk, vl)∈AL (5.22)
yvl ≥0 ∀vl∈VL\ {s} (5.23) xuv∈ {0,1} ∀(u, v)∈A (5.24) Inequalities (5.14) and (5.15) state that from the set of layered graph nodes corresponding to one particular original node exactly one has to be chosen for required nodes and at most one for potential Steiner nodes, respectively. Indegree constraints (5.16) inGL restrict the number of incoming arcs to a layered graph nodevl in dependency ofyvl to at most one. SinceGL is acyclic inequalities (5.17) are enough to ensure connectivity. Equalities (5.18) and (5.19) link layered graph arcs to original arcs. Delay-variation-boundDis guaranteed by (5.20) and (5.21).
LAYLPdenotes the LP relaxation ofLAY.
In principle, variablesxuv andyvl are redundant since they can be substituted by Boolean layered graph arc variablesxluvusing equalities (5.16), (5.18) and (5.19). However, modelLAY is better readable by including them and branching onxuvand Booleanyvl variables turned out to be more efficient in practice than branching on variablesxluv. In fact, branching on original arcs usually is more balanced since settingxluv= 1for one particular layered graph arc in general is a stronger constraint on the set of feasible solutions than settingxuv = 1.
5.6.1 Valid Inequalities
The following sets of valid inequalities are not necessary for the feasibility of modelLAY but
The following sets of valid inequalities are not necessary for the feasibility of modelLAY but