Approximation algorithms and an integer program for multi-level graph spanners
Reyan Ahmed, Keaton Hamm, Mohammad Javad Latifi Jebelli, Stephen Kobourov, Faryad Darabi Sahneh, Richard Spence
AApproximation algorithms and an integerprogram for multi-level graph spanners
Reyan Ahmed , Keaton Hamm , Mohammad Javad Latifi Jebelli , StephenKobourov , Faryad Darabi Sahneh , and Richard Spence University of Arizona, USA
Abstract.
Given a weighted graph G ( V, E ) and t ≥
1, a subgraph H isa t –spanner of G if the lengths of shortest paths in G are preserved in H up to a multiplicative factor of t . The subsetwise spanner problem aims topreserve distances in G for only a subset of the vertices. We generalize theminimum-cost subsetwise spanner problem to one where vertices appearon multiple levels, which we call the multi-level graph spanner (MLGS)problem, and describe two simple heuristics. Applications of this prob-lem include road/network building and multi-level graph visualization,especially where vertices may require different grades of service.We formulate a 0–1 integer linear program (ILP) of size O ( | E || V | ) forthe more general minimum pairwise spanner problem , which resolvesan open question by Sigurd and Zachariasen on whether this problemadmits a useful polynomial-size ILP. We extend this ILP formulation tothe MLGS problem, and evaluate the heuristic and ILP performance onrandom graphs of up to 100 vertices and 500 edges. Keywords:
Graph spanners · Integer programming · Multi-level graphrepresentation
Given an undirected edge-weighted graph G ( V, E ) and a real number t ≥
1, asubgraph H ( V, E (cid:48) ) is a (multiplicative) t –spanner of G if the lengths of shortestpaths in G are preserved in H up to a multiplicative factor of t ; that is, d H ( u, v ) ≤ t · d G ( u, v ) for all ( u, v ) ∈ V × V , where d G ( u, v ) is the length of the shortest pathfrom u to v in G . We refer to t as the stretch factor of H . Peleg et al. [12] show thatdetermining if there exists a t –spanner of G with m or fewer edges is NP–complete.Further, it is NP–hard to approximate the (unweighted) t –spanner problem towithin a factor of O (log | V | ), even when restricted to bipartite graphs [15].In the pairwise spanner problem [11], distances only need to be preserved fora subset P ⊆ V × V of pairs of vertices. Thus, the classical t –spanner problem isa special case of the pairwise spanner problem where P = V × V . The subsetwisespanner problem is a special case of the pairwise spanner problem where P = S × S for some S ⊂ V ; that is, distances need only be preserved between vertices in S [11]. The case t = 1 is known as the pairwise distance preserver or sourcewisedistance preserver problem, respectively [10]. The subsetwise spanner problemwhere t is arbitrarily large is known as the Steiner tree problem on graphs. a r X i v : . [ c s . D M ] A p r Ahmed et al.
Fig. 1: An interactive road map serves as a good analogy for the MLGS problem,where the top level graph G (cid:96) represents the network of major highways, andzooming in to G (cid:96) − shows a denser network of smaller roads. In many network design problems, vertices or edges come with a natural notionof priority, grade of service, or level; see Fig. 1. For example, consider the caseof rebuilding a transportation infrastructure network after a natural disaster.Following such an event, the rebuilding process may wish to prioritize connectionsbetween important buildings such as hospitals or distribution centers, makingthese higher level terminals, while ensuring that no person must travel an excessivedistance to reach their destination. Such problems have been referred to by namessuch as hierarchical network design, grade of service problems, multi-level, multi-tier, and have applications in network routing and visualization.Similar to other graph problems which generalize to multiple levels or gradesof service [8], we extend the subsetwise spanner problem to the multi-level graphspanner (MLGS) problem:
Definition 1. [Multi-level graph spanner (MLGS) problem] Given a graph G ( V, E ) with positive edge weights c : E → R + , a nested sequence of terminals, T (cid:96) ⊆ T (cid:96) − ⊆ . . . ⊆ T ⊆ V , and a real number t ≥ , compute a minimum-cost sequenceof spanners G (cid:96) ⊆ G (cid:96) − ⊆ . . . ⊆ G , where G i is a subsetwise ( T i × T i ) –spannerfor G with stretch factor t for i = 1 , . . . , (cid:96) . The cost of a solution is defined asthe sum of the edge weights on each graph G i , i.e., (cid:80) (cid:96)i =1 (cid:80) e ∈ E ( G i ) c e . We refer to T i and G i as the terminals and the graph on level i . A moregeneral version of the MLGS problem can involve different stretch factors on eachlevel or a more general definition of cost, but for now we use the same stretchfactor t for each level.An equivalent formulation of the MLGS problem which we use interchangeablyinvolves grades of service : given G = ( V, E ) with edge weights, and requiredgrades of service R : V → { , , . . . , (cid:96) } , compute a single subgraph H ⊆ G withvarying grades of service on the edges, with the property that for all u, v ∈ V , if u and v each have required a grade of service greater than or equal to i , then there pproximation algorithms and an ILP for multi-level graph spanners 3 Fig. 2:
Left:
Input graph G with edge weights, (cid:96) = 2, | T | = 4, | T | = 3, and t = 3. Required grades of service R ( v ) are shown in red. Center:
A validMLGS G ⊆ G ⊆ G is shown. Right:
The equivalent solution, where darkedges e have y e = 2 and light edges have y e = 1. The cost of this solution is2 × (4 + 2 + 5) + 1 × (1 + 2) = 25.exists a path in H from u to v using edges with a grade of service greater than orequal to i , and whose length is at most t · d G ( u, v ). Thus, T (cid:96) = { v ∈ V | R ( v ) = (cid:96) } , T (cid:96) − = { v ∈ V | R ( v ) ≥ (cid:96) − } , and so on. If y e denotes the grade of edge e (orthe number of levels e appears in), then the cost of a solution is equivalently (cid:80) e ∈ H c e y e , that is, edges with a higher grade of service incur a greater cost.This interpretation makes it clear that more important vertices (e.g., hubs) areconnected with higher quality edges; see example instance and solution in Fig. 2.If t is arbitrarily large, the MLGS problem reduces to the multi-level Steinertree (MLST) problem [1]. However it is worth noting that the problem of com-puting or approximating spanners is significantly harder than that of computingSteiner trees, and that a Steiner tree of G may be an arbitrarily poor spanner; acycle on | V | vertices with one edge removed is a possible Steiner tree of G , but isonly a ( | V | − G . The techniques used here have similarities to thoseused in the MLST problem, but more sophisticated methods are needed as well,including the use of approximate distance preservers and a new ILP formulationfor the pairwise spanner problem. Spanners and variants thereof have been studied for at least three decades, so wefocus on results relating to pairwise or subsetwise spanners. Alth¨ofer et al. [2]provide a simple greedy algorithm that constructs a multiplicative r –spannergiven a graph G and a real number r >
0. The greedy algorithm sorts edges in E by nondecreasing weight, then for each e = { u, v } ∈ E , computes the shortestpath P ( u, v ) from u to v in the current spanner, and adds the edge to the spannerif the weight of P ( u, v ) is greater than r · c e . The resulting subgraph H is a r –spanner for G . The main result of [2] is that, given a weighted graph G and t ≥
1, there is a greedy (2 t + 1)–spanner H containing at most n (cid:100) n /t (cid:101) edges,and whose weight is at most w ( M ST ( G ))(1 + n t ) where w ( M ST ( G )) denotesthe weight of a minimum spanning tree of G .Sigurd and Zachariasen [17] present an ILP formulation for the minimum-weight pairwise spanner problem (see Section 3), and show that the greedy Ahmed et al. algorithm [2] performs well on sparse graphs of up to 64 vertices. ´Alvarez-Mirandaand Sinnl [3] present a mixed ILP formulation for the tree t ∗ –spanner problem,which asks for a spanning tree of a graph G with the smallest stretch factor t ∗ .Dinitz et al. [13] provide a flow-based linear programming (LP) relaxation toapproximate the directed spanner problem. Their LP formulation is similar to thatin [17]; however, they provide an approximation algorithm which relaxes their ILP,whereas the previous formulation was used to compute spanners to optimality.Additionally, the LP formulation applies to graphs of unit edge cost; they latertake care of it in their rounding algorithm by solving a shortest path arborescenceproblem. They provide a ˜ O ( n )–approximation algorithm for the directed k –spanner problem for k ≥
1, which is the first sublinear approximation algorithmfor arbitrary edge lengths. Bhattacharyya et al. [6] provide a slightly differentformulation to approximate t –spanners as well as other variations of this problem.They provide a polynomial time O (( n log n ) − k )–approximation algorithm for thedirected k –spanner problem. Berman et al. [5] provide an alternative randomizedLP rounding schemes that lead to better approximation ratios. They improvedthe approximation ratio to O ( √ n log n ) where the approximation ratio of thealgorithm provided by Dinitz et al. [13] was O ( n ). They have also improved theapproximation ratio for the important special case of directed 3–spanners withunit edge lengths.There are several results on multi-level or grade-of-service Steiner trees,e.g., [1, 4, 8, 9, 16], while multi-level spanner problems have not been studied yet. Here, we assume an oracle subroutine that computes an optimal ( S × S )–spanner,given a graph G , subset S ⊆ V , and t . The intent is to determine if approximatingMLGS is significantly harder than the subsetwise spanner problem. We formulatesimple bottom-up and top-down approaches for the MLGS problem. The approach is as follows: compute a minimum subsetwise ( T × T )–spanner of G with stretch factor t . This immediately induces a feasible solution to the MLGSproblem, as one can simply copy each edge from the spanner to every level (or, interms of grades of service, assign grade (cid:96) to each spanner edge). We then pruneedges that are not needed on higher levels. It is easy to show that the solutionreturned has cost no worse than (cid:96) times the cost of the optimal solution. LetOPT denote the cost of the optimal MLGS G ∗ (cid:96) ⊆ G ∗ (cid:96) − ⊆ . . . ⊆ G ∗ for a graph G . Let MIN i denote the cost of a minimum subsetwise ( T i × T i )–spanner forlevel i with stretch t , and let BOT denote the cost computed by the bottom-upapproach. If no pruning is done, then BOT = (cid:96) MIN . Theorem 1.
The oracle bottom-up algorithm described above yields a solutionthat satisfies BOT ≤ (cid:96) · OPT. pproximation algorithms and an ILP for multi-level graph spanners 5
Fig. 3:
Left:
Tightness example of the top-down approach. Consider the latticegraph G with pairs of vertices of grade (cid:96) ( | T (cid:96) | = 2), (cid:96) −
1, and so on. The edgeconnecting the two vertices of grade i has weight 1, and all other edges haveweight ε , where 0 < ε (cid:28)
1. Set t = 2. The top-down solution (middle) has costTOP ≈ (cid:96) + ( (cid:96) −
1) + . . . + 1 = (cid:96) ( (cid:96) +1)2 , while the optimal solution (bottom) hascost OPT ≈ (cid:96) . Right:
Tightness example of the bottom-up approach. Consider acycle G containing two adjacent vertices of grade (cid:96) , and the remaining verticesof grade 1. The edge connecting the two vertices of grade (cid:96) is 1 + ε , whilethe remaining edges have weight 1. Setting t = | E | yields BOT = (cid:96) | E | whileOPT = (1 + ε ) (cid:96) + 1( | E | − ≈ | E | + (cid:96) . Proof.
We know MIN ≤ OPT, since the lowest-level graph G ∗ is a ( T × T )–spanner whose cost is at least MIN . Further, we have BOT = (cid:96) MIN if nopruning is done. Then MIN ≤ OPT ≤ BOT = (cid:96) · MIN , so BOT ≤ (cid:96) · OPT. (cid:3)
The ratio of (cid:96) is asymptotically tight; an example can be constructed byletting G be a cycle containing t vertices and all edges of cost 1. Let two adjacentvertices in G appear in T (cid:96) , while all vertices appear in T , as shown in Figure 3.As t → ∞ , the ratio BOTOPT approaches (cid:96) . Note that in this example, no edges canbe pruned without violating the t –spanner requirement.We give a simple heuristic that attempts to “prune” unneeded edges withoutviolating the t –spanner requirement. Note that any pruning strategy may notprune any edges, as a worst case example (Figure 3) cannot be pruned. Let G be the ( T × T )–spanner computed by the bottom-up approach. To computea ( T × T )–spanner G using the edges from G , we can compute a distancepreserver of G over terminals in T . One simple strategy is to use shortest pathsas explained below.Even more efficient pruning is possible through the distant preserver literature[7, 10]. A well-known result of distant preservers is due to the following theorem: Theorem 2 ( [10]).
Given G = ( V, E ) with | V | = n , and P ⊂ (cid:0) V (cid:1) , there existsa subgraph G (cid:48) with O ( n + √ n | P | ) edges such that for all ( u, v ) ∈ P we have d G (cid:48) ( u, v ) = d G ( u, v ) . The above theorem hints at a sparse construction of G simply by letting P = T × T . Given G , let G i be a distance preserver of G i − over the terminals Ahmed et al. T i , for all i = 2 , . . . , (cid:96) . An example is to let G be the union of all shortest paths(in G ) over vertices v, w ∈ G . The result is clearly a feasible solution to theMLGS problem, as the shortest paths are preserved exactly from G , so each G i is a ( T i × T i )–spanner of G with stretch factor t . A simple top-down heuristic that computes a solution is as follows: let G (cid:96) be theminimum-cost ( T (cid:96) × T (cid:96) )–spanner over terminals T (cid:96) with stretch factor t , and costMIN (cid:96) . Then compute a minimum cost ( T (cid:96) − × T (cid:96) − )–spanner over T (cid:96) − , and let G (cid:96) − be the union of this spanner and G (cid:96) . Continue this process, where G i is theunion of the minimum cost ( T i × T i )–spanner and G i +1 . Clearly, this produces afeasible solution to the MLGS problem.The solution returned by this approach, with cost denoted by TOP, is notworse than (cid:96) +12 times the optimal. Define MIN i and OPT as before. Define OPT i to be the cost of edges on level i but not level i + 1 in the optimal MLGS solution,so that OPT = (cid:96) OPT (cid:96) + ( (cid:96) − (cid:96) − + . . . + OPT . Define TOP i analogously. Theorem 3.
The oracle top-down algorithm described above yields an approxi-mation that satisfies the following:(i) TOP (cid:96) ≤ OPT (cid:96) , (ii) TOP i ≤ OPT i + OPT i +1 + . . . + OPT (cid:96) , i = 1 , . . . , (cid:96) − , (iii) TOP ≤ (cid:96) +12 OPT . Proof.
Inequality (i) is true by definition, as we compute an optimal ( T (cid:96) × T (cid:96) )–spanner whose cost is TOP (cid:96) , while OPT (cid:96) is the cost of some ( T (cid:96) × T (cid:96) )-spanner.For (ii), note that TOP i ≤ MIN i , with equality when the minimum-cost ( T i × T i )–spanner and G i +1 are disjoint. The spanner of cost OPT i + OPT i +1 + . . . + OPT (cid:96) is a feasible ( T i × T i )–spanner, so MIN i ≤ OPT i + . . . + OPT (cid:96) , which shows (ii).To show (iii), note that (i) and (ii) implyTOP = (cid:96) TOP (cid:96) + ( (cid:96) − (cid:96) − + . . . + TOP ≤ (cid:96) OPT (cid:96) + ( (cid:96) − (cid:96) − + OPT (cid:96) ) + . . . + (OPT + OPT + . . . + OPT (cid:96) )= (cid:96) ( (cid:96) + 1)2 OPT (cid:96) + ( (cid:96) − (cid:96) (cid:96) − + . . . + 1 ·
22 OPT ≤ (cid:96) + 12 OPT , as by definition OPT = (cid:96) OPT (cid:96) + ( (cid:96) − (cid:96) − + . . . + OPT . (cid:3) The ratio (cid:96) +12 is tight as illustrated in Figure 3, left. pproximation algorithms and an ILP for multi-level graph spanners 7
Again, assume we have access to an oracle that computes a minimum weight( S × S )–spanner of an input graph G with given stretch factor t . A simple combinedmethod, similar to [1], is to run the top-down and bottom-up approaches for theMLGS problem, and take the solution with minimum cost. This has a slightlybetter approximation ratio than either of the two approaches. Theorem 4.
The solution whose cost is min(
TOP , BOT ) is not worse than (cid:96) + 23 times the cost OPT of the optimal MLGS. The proof is given in Appendix A.
So far, we have assumed that we have access to an optimal subsetwise spannergiven by an oracle. Here we propose a heuristic algorithm to compute subsetwisespanner. The key idea is to apply the greedy spanner to an auxiliary completegraph with terminals as its vertices and the shortest distance between terminalsas edge weights. Then, we apply the distance preserver discussed in Theorem 2to construct a subsetwise spanner.
Theorem 5.
Given graph G ( V, E ) , stretch factor t ≥ , and subset T ⊂ V , thereexists a ( T × T ) –spanner for G with stretch factor t and O ( n + √ n | T | t +1 ) edges.Proof. The spanner may be constructed as follows:1. Construct the terminal complete graph ¯ G whose vertices are ¯ V := T , suchthat the weight of each edge { u, v } in ¯ G is the length of the shortest pathconnecting them in G , i.e., w ( u, v ) = d G ( u, v ).2. Construct a greedy t − spanner ¯ H ( ¯ V , ¯ E (cid:48) ) of ¯ G . According to [2], this graphhas | T | t +1 edges. Let P = ¯ E (cid:48) .3. Apply Theorem 2 to obtain a subgraph H of G such that for all ( u, v ) ∈ P wehave d H ( u, v ) = d G ( u, v ). Therefore, for arbitrary u, v ∈ T we get d H ( u, v ) ≤ t d G ( u, v ).4. Finally, let shortest-path( u, v ) be the collection of edges in the shortest pathfrom u to v in H , and E = (cid:91) ( u,v ) ∈ P { e ∈ E | e ∈ shortest-path( u, v ) . } According to Theorem 2, the number of edges in the constructed spanner H ( V, E ) is O ( n + √ n | P | ) = O (cid:16) n + √ n | T | t +1 (cid:17) . (cid:3) Ahmed et al.
Hence, we may replace the oracle in the top-down and bottom-up approaches(Sections 2.1-2.2) with the above heuristic; we call the resulting algorithmsheuristic top-down and heuristic bottom-up. We analyze the performance of allalgorithms on several types of graphs.Incorporating the heuristic subsetwise spanner in our top-down and bottomup heuristics has two implications. First, the size of the final MLGS is dom-inated by the size of the spanner at the bottom level, i.e., O ( n + √ n | P | ) = O (cid:16) n + √ n | T | t +1 (cid:17) . Second, since the greedy spanner algorithm used in theabove subsetwise spanner can produce spanners that are O ( n ) more costly thanthe optimal solution, the same applies to the subsetwise spanner. Our experimen-tal results, however, indicate that the heuristic approaches are very close to theoptimal solutions obtained via our ILP. We describe the original ILP formulation for the pairwise spanner problem [17].Let K = { ( u i , v i ) } ⊂ V × V be the set of vertex pairs; recall that the t –spannerproblem is a special case where K = V × V . Here we will use unordered pairs ofdistinct vertices, so in the t –spanner problem we have | K | = (cid:0) | V | (cid:1) instead of | V | .This ILP formulation uses paths as decision variables. Given ( u, v ) ∈ K , denoteby P uv the set of paths from u to v of cost no more than t · d G ( u, v ), and denoteby P the union of all such paths, i.e., P = (cid:83) ( u,v ) ∈ K P uv . Given a path p ∈ P and edge e ∈ E , let δ ep = 1 if e is on path p , and 0 otherwise. Let x e = 1 if e isan edge in the pairwise spanner H , and 0 otherwise. Given p ∈ P , let y p = 1 ifpath p is in the spanner, and zero otherwise. An ILP formulation for the pairwisespanner problem is given below.Minimize (cid:88) e ∈ E c e x e subject to (1) (cid:88) p ∈ P uv y p δ ep ≤ x e ∀ e ∈ E ; ∀ ( u, v ) ∈ K (2) (cid:88) p ∈ P uv y p ≥ ∀ ( u, v ) ∈ K (3) x e ∈ { , } ∀ e ∈ E (4) y p ∈ { , } ∀ p ∈ P (5)Constraint (3) ensures that for each pair ( u, v ) ∈ K , at least one t –spannerpath is selected, and constraint (2) enforces that on the selected u - v path, everyedge along the path appears in the spanner. The main drawback of this ILP isthat the number of path variables is exponential in the size of the graph. Theauthors use delayed column generation by starting with a subset P (cid:48) ⊂ P of paths,with the starting condition that for each ( u, v ) ∈ K , at least one t –spanner path pproximation algorithms and an ILP for multi-level graph spanners 9 in P uv is in P (cid:48) . The authors leave as an open question whether this problemadmits a useful polynomial-size ILP.We introduce a 0-1 ILP formulation for the pairwise t –spanner problem basedon multicommodity flow, which uses O ( | E || K | ) variables and constraints, where | K | = O ( | V | ). Define t , c e , d G ( u, v ), K , and x e as before. Note that d G ( u, v )can be computed in advance, using any all-pairs shortest path (APSP) method.Direct the graph by replacing each edge e = { u, v } with two edges ( u, v ) and( v, u ) of weight c e . Let E (cid:48) be the set of all directed edges, i.e., | E (cid:48) | = 2 | E | . Given( i, j ) ∈ E (cid:48) , and an unordered pair of vertices ( u, v ) ∈ K , let x uv ( i,j ) = 1 if edge( i, j ) is included in the selected u - v path in the spanner H , and 0 otherwise. Thisdefinition of path variables is similar to that by ´Alvarez-Miranda and Sinnl [3]for the tree t ∗ –spanner problem. We select a total order of all vertices so that thepath constraints (8)–(9) are well-defined. This induces 2 | E || K | binary variables,or 2 | E | (cid:0) | V | (cid:1) = 2 | E || V | ( | V | −
1) variables in the standard t –spanner problem. Notethat if u and v are connected by multiple paths in H of length ≤ t · d G ( u, v ), weneed only set x uv ( i,j ) = 1 for edges along some path. Given v ∈ V , let In ( v ) and Out ( v ) denote the set of incoming and outgoing edges for v in E (cid:48) . In (7)–(11)we assume u < v in the total order, so spanner paths are from u to v . An ILPformulation for the pairwise spanner problem is as follows.Minimize (cid:88) e ∈ E c e x e subject to (6) (cid:88) ( i,j ) ∈ E (cid:48) x uv ( i,j ) c e ≤ t · d G ( u, v ) ∀ ( u, v ) ∈ K ; e = { i, j } (7) (cid:88) ( i,j ) ∈ Out ( i ) x uv ( i,j ) − (cid:88) ( j,i ) ∈ In ( i ) x uv ( j,i ) = i = u − i = v ∀ ( u, v ) ∈ K ; ∀ i ∈ V (8) (cid:88) ( i,j ) ∈ Out ( i ) x uv ( i,j ) ≤ ∀ ( u, v ) ∈ K ; ∀ i ∈ V (9) x uv ( i,j ) + x uv ( j,i ) ≤ x e ∀ ( u, v ) ∈ K ; ∀ e = { i, j } ∈ E (10) x e , x uv ( i,j ) ∈ { , } (11)Constraint (7) requires that for all ( u, v ) ∈ K , the sum of the weights of theselected edges corresponding to the pair ( u, v ) is not more than t · d G ( u, v ).Constraints (8)–(9) require that the selected edges corresponding to ( u, v ) ∈ K form a simple path beginning at u and ending at v . Constraint (10) enforcesthat, if edge ( i, j ) or ( j, i ) is selected on some u - v path, then its correspondingundirected edge e is selected in the spanner; further, ( i, j ) and ( j, i ) cannot bothbe selected for some pair ( u, v ). Finally, (11) enforces that all variables are binary.The number of variables is | E | + 2 | E || K | and the number of constraints is O ( | E || K | ), where | K | = O ( | V | ). Note that the variables x uv ( i,j ) can be relaxed tobe continuous in [0 , Recall that the MLGS problem generalizes the subsetwise spanner problem, whichis a special case of the pairwise spanner problem for K = S × S . Again, we useunordered pairs, i.e., | K | = (cid:0) | S | (cid:1) .We generalize the ILP formulation in (6)–(11) to the MLGS problem asfollows. Recall that we can encode the levels in terms of required grades of service R : V → { , , . . . , (cid:96) } . Instead of 0–1 indicators x e , let y e denote the grade ofedge e in the multi-level spanner; that is, y e = i if e appears on level i but notlevel i + 1, and y e = 0 if e is absent. The only difference is that for the MLGSproblem, we assign grades of service to all u - v paths by assigning grades to edgesalong each u - v path. That is, for all u, v ∈ T with u < v , the selected path from u to v has grade min( R ( u ) , R ( v )), which we denote by m uv . Note that we onlyneed to require the existence of a path for terminals u, v ∈ T , where u < v . AnILP formulation for the MLGS problem is as follows.Minimize (cid:88) e ∈ E c e y e subject to (12) (cid:88) ( i,j ) ∈ E (cid:48) x uv ( i,j ) c e ≤ t · d G ( u, v ) ∀ u, v ∈ T ; e = { i, j } (13) (cid:88) ( i,j ) ∈ Out ( i ) x uv ( i,j ) − (cid:88) ( j,i ) ∈ In ( i ) x uv ( j,i ) = i = u − i = v ∀ u, v ∈ T ; ∀ i ∈ V (14) (cid:88) ( i,j ) ∈ Out ( i ) x uv ( i,j ) ≤ ∀ u, v ∈ T ; ∀ i ∈ V (15) y e ≥ m uv x uv ( i,j ) ∀ u, v ∈ T ; ∀ e = { i, j } (16) y e ≥ m uv x uv ( j,i ) ∀ u, v ∈ T ; ∀ e = { i, j } (17) x uv ( i,j ) ∈ { , } (18)Constraints (16)–(17) enforce that for each pair u, v ∈ V such that u < v ,the edges along the selected u - v path (not necessarily every u - v path) have agrade of service greater than or equal to the minimum grade of service neededto connect u and v , that is, m uv . If multiple pairs ( u , v ), ( u , v ), . . . , ( u k , v k )use the same edge e = { i, j } (possibly in opposite directions), then the grade ofedge e should be y e = max( m u v , m u v , . . . , m u k v k ). It is implied by (16)–(17)that 0 ≤ y e ≤ (cid:96) in an optimal solution. Theorem 6.
An optimal solution to the ILP given in (6) – (11) yields an optimalpairwise spanner of G over a set K ⊂ V × V . Theorem 7.
An optimal solution to the ILP given in (12) – (18) yields an optimalsolution to the MLGS problem. We give the proofs in Appendices B and C. pproximation algorithms and an ILP for multi-level graph spanners 11
We can reduce the size of the ILP using the following shortest path tests, whichworks well in practice and also applies to the MLGS problem. Note that we areconcerned with the total cost of a solution, not the number of edges.If d G ( i, j ) < c ( i, j ), for some edge { i, j } ∈ E , then we can remove { i, j } fromthe graph, as no min-weight spanner of G uses edge { i, j } . If H ∗ is a min-costpairwise spanner that uses edge { i, j } , then we can replace { i, j } with a shorter i - j path p ij without violating the t –spanner requirement. In particular, if some u - v path uses both edge { i, j } as well as some edge(s) along p ij , then this pathcan be rerouted to use only edges in p ij with smaller cost.We reduce the number of variables needed in the single-level ILP formulation((6)–(11)) with the following test: given u, v ∈ K with u < v and some directededge ( i, j ) ∈ E (cid:48) , if d G ( u, i ) + c ( i, j ) + d G ( j, v ) > t · d G ( u, v ), then ( i, j ) cannotpossibly be included in the selected u - v path, so set x uv ( i,j ) = 0. If ( i, j ) or ( j, i )cannot be selected on any u - v path, we can safely remove { i, j } from E .Conversely, given some directed edge ( i, j ) ∈ E (cid:48) , let G (cid:48) be the directed graphobtained by removing ( i, j ) from E (cid:48) (so that G (cid:48) has 2 | E | − u, v ∈ K with u < v , if d G (cid:48) ( u, v ) > t · d G ( u, v ), then edge ( i, j ) must be in any u - v spanner path, so set x uv ( i,j ) = 1. For its corresponding undirected edge e , x e = 1. We use the Erd˝os–R´enyi [14] and Watts–Strogatz [18] models to generate randomgraphs. Given a number of vertices, n , and probability p , the model ER ( n, p )assigns an edge to any given pair of vertices with probability p . An instance of ER ( n, p ) with p = (1 + ε ) ln nn is connected with high probability for ε > n ∈ { , , , , } , and ε = 1.In the Watts-Strogatz model, WS ( n, K, β ), initially we create a ring lattice ofconstant degree K , and then rewire each edge with probability 0 ≤ β ≤ n ∈ { , , , , } , K = 6, and β = 0 . | V | , number of levels (cid:96) , and stretch factor t . As thereis randomness involved, we generated 3 instances for every choice of parameters(e.g., ER, | V | = 80, (cid:96) = 3, t = 2).We generated MLGS instances with 1, 2, or 3 levels ( (cid:96) ∈ { , , } ), whereterminals are selected on each level by randomly sampling (cid:98)| V | · ( (cid:96) − i + 1) / ( (cid:96) +1) (cid:99) vertices on level i so that the size of the terminal sets decreases linearly.As the terminal sets are nested, T i can be selected by sampling from T i − (orfrom V if i = 1). We used four different stretch factors in our experiments, t ∈ { . , . , , } . Edge weights are randomly selected from { , , , . . . , } .2 Ahmed et al.
We can reduce the size of the ILP using the following shortest path tests, whichworks well in practice and also applies to the MLGS problem. Note that we areconcerned with the total cost of a solution, not the number of edges.If d G ( i, j ) < c ( i, j ), for some edge { i, j } ∈ E , then we can remove { i, j } fromthe graph, as no min-weight spanner of G uses edge { i, j } . If H ∗ is a min-costpairwise spanner that uses edge { i, j } , then we can replace { i, j } with a shorter i - j path p ij without violating the t –spanner requirement. In particular, if some u - v path uses both edge { i, j } as well as some edge(s) along p ij , then this pathcan be rerouted to use only edges in p ij with smaller cost.We reduce the number of variables needed in the single-level ILP formulation((6)–(11)) with the following test: given u, v ∈ K with u < v and some directededge ( i, j ) ∈ E (cid:48) , if d G ( u, i ) + c ( i, j ) + d G ( j, v ) > t · d G ( u, v ), then ( i, j ) cannotpossibly be included in the selected u - v path, so set x uv ( i,j ) = 0. If ( i, j ) or ( j, i )cannot be selected on any u - v path, we can safely remove { i, j } from E .Conversely, given some directed edge ( i, j ) ∈ E (cid:48) , let G (cid:48) be the directed graphobtained by removing ( i, j ) from E (cid:48) (so that G (cid:48) has 2 | E | − u, v ∈ K with u < v , if d G (cid:48) ( u, v ) > t · d G ( u, v ), then edge ( i, j ) must be in any u - v spanner path, so set x uv ( i,j ) = 1. For its corresponding undirected edge e , x e = 1. We use the Erd˝os–R´enyi [14] and Watts–Strogatz [18] models to generate randomgraphs. Given a number of vertices, n , and probability p , the model ER ( n, p )assigns an edge to any given pair of vertices with probability p . An instance of ER ( n, p ) with p = (1 + ε ) ln nn is connected with high probability for ε > n ∈ { , , , , } , and ε = 1.In the Watts-Strogatz model, WS ( n, K, β ), initially we create a ring lattice ofconstant degree K , and then rewire each edge with probability 0 ≤ β ≤ n ∈ { , , , , } , K = 6, and β = 0 . | V | , number of levels (cid:96) , and stretch factor t . As thereis randomness involved, we generated 3 instances for every choice of parameters(e.g., ER, | V | = 80, (cid:96) = 3, t = 2).We generated MLGS instances with 1, 2, or 3 levels ( (cid:96) ∈ { , , } ), whereterminals are selected on each level by randomly sampling (cid:98)| V | · ( (cid:96) − i + 1) / ( (cid:96) +1) (cid:99) vertices on level i so that the size of the terminal sets decreases linearly.As the terminal sets are nested, T i can be selected by sampling from T i − (orfrom V if i = 1). We used four different stretch factors in our experiments, t ∈ { . , . , , } . Edge weights are randomly selected from { , , , . . . , } .2 Ahmed et al. Algorithms and outputs
We implemented the bottom-up (BU) and top-down(TD) approaches from Section 2 in Python 3.5, as well as the combined approachthat selects the better of the two (Section 2.3). To evaluate the approximationalgorithms and the heuristics, we implemented the ILPs described in Section 3using CPLEX 12.6.2. We used the same high-performance computer for allexperiments (Lenovo NeXtScale nx360 M5 system with 400 nodes).For each instance of the MLGS problem, we compute the costs of the MLGSreturned using the bottom-up (BU), the top-down (TD), and the combined(min(BU, TD)) approaches, as well as the minimum cost MLGS using the ILP inSection 3.1. The three heuristics involve a (single-level) subroutine; we used boththe heuristic described in Section 2.4, as well as the flow formulation describedin Section 3 which computes subsetwise spanners to optimality. We compare thealgorithms with and without the oracle to assess whether computing (single-level)spanners to optimality significantly improves the overall quality of the solution.We show the performance ratio for each heuristic in the y -axis (defined asthe heuristic cost divided by OPT), and how the ratio depends on the inputparameters (number of vertices | V | , number of levels (cid:96) , and stretch factors t ). Finally, we discuss the running time of the ILP. All box plots show theminimum, interquartile range and maximum, aggregated over all instances usingthe parameter being compared. We first discuss the results for Erd˝os–R´enyi graphs. Figures 4–7 show the results ofthe oracle top-down, bottom-up, and combined approaches. We show the impactof different parameters (number of vertices | V | , number of levels (cid:96) , and stretchfactors t ) using line plots for three heuristics separately in Figures 4-6. Figure 7shows the performance of the three heuristics together in box plots. In Figure 4we can see that the bottom-up heuristic performs slightly worse for increasing | V | , while the top-down heuristic performs slightly better. In Figure 5 we seethat the heuristics perform worse when (cid:96) increases, consistent with the ratiosdiscussed in Section 2. In Figure 6 we show the performance of the heuristicswith respect to the stretch factor t . In general, the performance becomes worseas t increases.The most time consuming part of the experiment is the execution time of theILP for solving MLGS instances optimally. The running time of the heuristicsis significantly smaller compared to that of the ILP. Hence, we first show therunning times of the exact solution of the MLGS instances in Figure 8. We showthe running time with respect to the number of vertices | V | , number of levels (cid:96) ,and stretch factors t . For all parameters, the running time tends to increase asthe size of the parameter increases. In particular, the running time with stretchfactor 4 (Fig. 8, right) was much worse, as there are many more t -spanner pathsto consider, and the size reduction techniques in Section 3.2 are less effectiveat reducing instance size. We show the running times of for computing oraclebottom-up, top-down and combined solutions in Figure 9. pproximation algorithms and an ILP for multi-level graph spanners 13(a) Bottom up (b) Top down (c) min(BU, TD) Fig. 4: Performance with oracle on Erd˝os–R´enyi graphs w.r.t. | V | . Ratio is definedas the cost of the returned MLGS divided by OPT. (a) Bottom up (b) Top down (c) min(BU, TD) Fig. 5: Performance with oracle on Erd˝os–R´enyi graphs w.r.t. the number of levels (a) Bottom up (b) Top down (c) min(BU, TD)
Fig. 6: Performance with oracle on Erd˝os–R´enyi graphs w.r.t. stretch factorFig. 7: Performance with oracle on Erd˝os–R´enyi graphs w.r.t. the number ofvertices, the number of levels, and the stretch factors
Fig. 8: Experimental running times for computing exact solutions on Erd˝os–R´enyigraphs w.r.t. the number of vertices, the number of levels, and the stretch factorsFig. 9: Experimental running times for computing oracle bottom-up, top-downand combined solutions on Erd˝os–R´enyi graphs w.r.t. the number of vertices, thenumber of levels, and the stretch factorsThe ILP is too computationally expensive for larger input sizes and thisis where the heurstic can be particularly useful. We now consider a similarexperiment using the heuristic to compute subsetwise spanners, as described inSection 2.4. We show the impact of different parameters (number of vertices | V | ,number of levels (cid:96) , and stretch factors t ) using scatter plots for three heuristicsseparately in Figures 10–12. Figure 13 shows the performance of the threeheuristics together in box plots. We can see that the heuristics perform very wellin practice. Notably when the heuristic is used in place of the ILP (Fig 24), therunning times decrease for larger stretch factors. (a) Bottom up (b) Top down (c) min(BU, TD) Fig. 10: Performance without oracle on Erd˝os–R´enyi graphs w.r.t. | V | We also analyzed graphs generated from the Watts–Strogatz model and theresults are shown in Appendix D. pproximation algorithms and an ILP for multi-level graph spanners 15(a) Bottom up (b) Top down (c) min(BU, TD)
Fig. 11: Performance without oracle on Erd˝os–R´enyi graphs w.r.t. the number oflevels (a) Bottom up (b) Top down (c) min(BU, TD)
Fig. 12: Performance without oracle on Erd˝os–R´enyi graphs w.r.t. the stretchfactorsFig. 13: Performance without oracle on Erd˝os–R´enyi graphs w.r.t. the number ofvertices, the number of levels, and the stretch factorsFig. 14: Experimental running times for computing heuristic bottom-up, top-downand combined solutions on Erd˝os–R´enyi graphs w.r.t. the number of vertices, thenumber of levels, and the stretch factors
Our final experiments test the heuristic performance on a set of largergraphs. We generated the graphs using the Erd˝os–R´enyi model, with | V | ∈{ , , , } . We evaluated more levels ( (cid:96) ∈ { , , , , } ) with stretchfactors t ∈ { . , . , , } . We show the performance of heuristic bottom-up andtop-down in Appendix E. Here, the ratio is determined by dividing the BUor TD cost by min( BU, T D ) (as computing the optimal MLGS would be tootime-consuming). The results indicate that while running times increase withlarger input graphs, the number of levels and the stretch factors seem to havelittle impact on performance.
We introduced a generalization of the subsetwise spanner problem to multiplelevels or grades of service. Our proposed ILP formulation requires only a polyno-mial size of variables and constraints, which is an improvement over the previousformulation given by Sigurd and Zachariasen [17]. We also proposed improvedformulations which work well for small values of the stretch factor t . It would beworthwhile to consider whether even better ILP formulations can be found forcomputing graph spanners and their multi-level variants. We showed that boththe approximation algorithms and the heuristics work well in practice on severaldifferent types of graphs, with different number of levels and different stretchfactors.We only considered a stretch factor t that is the same for all levels in themulti-level spanner, as well as a fairly specific definition of cost. It would beinteresting to investigate more general multi-level or grade-of-service spannerproblems, including ones with varying stretch factors (e.g., in which more impor-tant terminals require a smaller or larger stretch factors), different definitionsof cost, and spanners with other requirements, such as bounded diameters ordegrees. References
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Proof.
We use the simple algebraic fact that min { x, y } ≤ αx + (1 − α ) y for all x, y ∈ R and α ∈ [0 , ≤ OPT +OPT + . . . + OPT (cid:96) , as the RHS equals the cost of G ∗ , which is some subsetwise( T × T )-spanner. Combining, we havemin(TOP , BOT) ≤ α (cid:96) (cid:88) i =1 i ( i + 1)2 OPT i + (1 − α ) (cid:96) (cid:96) (cid:88) i =1 OPT i = (cid:96) (cid:88) i =1 (cid:20)(cid:18) i ( i + 1)2 − (cid:96) (cid:19) α + (cid:96) (cid:21) ρ OPT i Since we are comparing min { TOP , BOT } to r · OPT for some approximationratio r >
1, we can compare coefficients and find the smallest r ≥ (cid:18) (cid:96) ( (cid:96) + 1)2 − (cid:96) (cid:19) α + (cid:96)ρ ≤ (cid:96)r (cid:18) ( (cid:96) − (cid:96) − (cid:96) (cid:19) α + (cid:96)ρ ≤ ( (cid:96) − r ... (cid:18) · − (cid:96) (cid:19) α + (cid:96)ρ ≤ r has a solution α ∈ [0 , (cid:96)/ (cid:96) +2 (cid:96) ≤ (cid:96)r , or r ≥ (cid:96) +23 . Also, it can be shown algebraicallythat ( r, α ) = ( (cid:96) +23 , ) simultaneously satisfies the above inequalities. This impliesthat min { TOP , BOT } ≤ (cid:96) +23 ρ · OPT. (cid:3)
B Proof of Theorem 6
Proof.
Let H ∗ denote an optimal pairwise spanner of G with stretch factor t ,and let OPT denote the cost of H ∗ . Let OPT ILP denote the minimum cost ofthe objective in the ILP (6). First, given a minimum cost t –spanner H ∗ ( V, E ∗ ),a solution to the ILP can be constructed as follows: for each edge e ∈ E ∗ , set x e = 1. Then for each unordered pair ( u, v ) ∈ K with u < v , compute a shortestpath p uv from u to v in H ∗ , and set x uv ( i,j ) = 1 for each edge along this path, and x uv ( i,j ) = 0 if ( i, j ) is not on p uv .As each shortest path p uv necessarily has cost ≤ t · d G ( u, v ), constraint(7) is satisfied. Constraints (8)–(9) are satisfied as p uv is a simple u - v path.Constraint (10) also holds, as p uv should not traverse the same edge twice inopposite directions. In particular, every edge in H ∗ appears on some shortest pproximation algorithms and an ILP for multi-level graph spanners 19 path; otherwise, removing such an edge yields a pairwise spanner of lower cost.Hence OPT ILP ≤ OPT.Conversely, an optimal solution to the ILP induces a feasible t –spanner H .Consider an unordered pair ( u, v ) ∈ K with u < v , and the set of decisionvariables satisfying x uv ( i,j ) = 1. By (8) and (9), these chosen edges form a simplepath from u to v . The sum of the weights of these edges is at most t · d G ( u, v )by (7). Then by constraint (10), the chosen edges corresponding to ( u, v ) appearin the spanner, which is induced by the set of edges e with x e = 1. HenceOPT ≤ OPT
ILP .Combining the above observations, we see that OPT = OPT
ILP . (cid:3) C Proof of Theorem 7
Proof.
Given an optimal solution to the ILP with cost OPT
ILP , construct anMLGS by letting G i = ( V, E i ) where E i = { e ∈ E | y e ≥ i } . This clearly gives anested sequence of subgraphs. Let u and v be terminals in T i (not necessarilyof required grade R ( · ) = i ), with u < v , and consider the set of all variables ofthe form x uv ( i,j ) equal to 1. By (13)–(15), these selected edges form a path from u to v of length at most t · d G ( u, v ), while constraints (16)–(17) imply that theseselected edges have grade at least m uv ≥ i , so the selected path is contained in E i . Hence G i is a subsetwise ( T i × T i )–spanner for G with stretch factor t , andthe optimal ILP solution gives a feasible MLGS.Given an optimal MLGS with cost OPT, we can construct a feasible ILPsolution with the same cost in a way similar to the proof of Theorem 6. For each u, v ∈ T with u < v , set m uv = min( R ( u ) , R ( v )). Compute a shortest path in G m uv from u to v , and set x uv ( i,j ) = 1 for all edges along this path. Then for each e ∈ E , consider all pairs ( u , v ) , . . . , ( u k , v k ) that use either ( i, j ) or ( j, i ), andset y e = max( m u v , m u v , . . . , m u k v k ). In particular, y e is not larger than thegrade of e in the MLGS, otherwise this would imply e is on some u - v path atgrade greater than its grade of service in the actual solution. (cid:3) D Experimental results on graphs generated usingWatts-Strogatz
The results for graphs generated from the Watts–Strogatz model are shown inFigures 15–23, which are organized in the same way as for Erd˝os–R´enyi.
E Experimental results on large graphs usingErd˝os-R´enyi
Figure 24 shows a rough measure of performance for the bottom-up and top-downheuristics on large graphs using the Erd˝os-R´enyi model, where the ratio is definedas the BU or TD cost divided by min(BU, TD). Figure 25 shows the aggregatedrunning times per instance, which significantly worsen as | V | is large. Fig. 15: Performance with oracle on Watts–Strogatz graphs w.r.t. the number ofvertices (a) Bottom up (b) Top down (c) min(BU, TD)
Fig. 16: Performance with oracle on Watts–Strogatz graphs w.r.t. the number oflevels (a) Bottom up (b) Top down (c) min(BU, TD)
Fig. 17: Performance with oracle on Watts–Strogatz graphs w.r.t. the stretchfactorsFig. 18: Performance with oracle on Watts–Strogatz graphs w.r.t. the number ofvertices, the number of levels, and the stretch factors pproximation algorithms and an ILP for multi-level graph spanners 21
Fig. 19: Experimental running times for computing exact solutions on Watts–Strogatz graphs w.r.t. the number of vertices, the number of levels, and thestretch factors (a) Bottom up (b) Top down (c) min(BU, TD)
Fig. 20: Performance without oracle on Watts–Strogatz graphs w.r.t. the numberof vertices (a) Bottom up (b) Top down (c) min(BU, TD)
Fig. 21: Performance without oracle on Watts–Strogatz graphs w.r.t. the numberof levels (a) Bottom up (b) Top down (c) min(BU, TD)
Fig. 22: Performance without oracle on Watts–Strogatz graphs w.r.t. the stretchfactors2 Ahmed et al.