Electrical flows have attracted a lot of attention in the theoretical computer science community in the past few years: they have been used to speed up the computation of maximum flow , develop the theory of Laplacian solvers [8, 12, 20, 11], graph sparsification [35, 36, 9], graph embeddings [14, 1, 20, 2] and even bound the integrality gap of the asymmetric traveling salesman problem ATSP . In this paper, we consider the network reconfiguration problem which seeks to minimize the energy of electrical flows over rooted trees, motivated by the operational requirements of electricity distribution networks. Distribution networks contain low-voltage power lines connecting substations to end consumers. They are typically built as mesh networks, i.e., containing cycles, but operated as radial networks, i.e., power is sent along trees rooted at a substation with leaves corresponding to end consumers. This is achieved by turning switches on or off so that there are no cycles, in other words by adding or deleting edges to configure a specific tree111The tree structure is desirable for the security of an electricity distribution network—a fault can be more easily isolated when the downstream branch from the fault is independent from the root, while it would otherwise compromise the entire network.. A key challenge is to find a configuration or a tree such that the power losses due to the electrical flow are minimized. We give a formal description of the network reconfiguration problem next.
Given an undirected graph (, ), root , resistances for each edge222Edges are referred to as lines, root is the location of the substation, and demands are often referred to as loads in the power systems community. , demands for each node supplied by the root node, thus , the network reconfiguration problem is to minimize the energy of the feasible flow such that the support of the flow is acyclic:
where and denote the sets of incoming and outgoing edges of (after fixing an arbitrary orientation on the edges), is any feasible flow satisfying the demands (1), the support of is constrained to be acyclic (and therefore, a tree rooted at ) (2), and the objective (P0) is to minimize the energy of the flow. While there may be more than one feasible flow satisfying the demands in general, an electrical flow minimizes the energy subject to meeting the demands. Moreover, given an -rooted tree, there is a unique flow on the tree that satisfies the demands: , where is the set of nodes that connect to the root through . Also, note that any -rooted tree can be augmented to be a spanning tree in the graph at no additional cost333Contract the support of the flow and add edges with 0 flow to construct a spanning tree.. Therefore, the network reconfiguration problem (P0) can then be equivalently written as444Our paper presents a simplified objective for the reconfiguration problem in power systems, which in reality needs to include both real and reactive power. Our approximation guarantees hold for this more realistic objective as well, as we discuss in Appendix D.:
where is the set of all spanning trees of . Although distribution grid practitioners and power systems researchers have used a broad range of heuristics to get reasonable solutions in practice (see e.g., , , ), little is known about provable bounds for the network reconfiguration problem. We next present an overview of the related work on this problem, and why known techniques do not easily generalize to give bounds for this setting.
Khodabakhsh et al.  revisited555This paper appeared in HICSS 2018 and won the runner-up for the best paper award. the network reconfiguration problem in 2018 and were the first to show that (P1) is NP-hard even for the uniform case where all the resistances and demands are equal to one, i.e., for all and for all . They showed that (P1) can be cast as a supermodular minimization problem subject to a matroid base constraint. Inspired by submodular maximization, a local search algorithm was also proposed in , however this did not provide a (multiplicative) approximation guaranteee.
Relaxing the tree constraint (2) results in the well-studied problem of computing electrical flows as they uniquely minimize energy [12, 20, 11]. However, existing results in spectral sparsification [35, 36] do not extend to our setting since they change the resistances on the remaining edges to compensate for edge-deletions. Many existing heuristics involve iterative edge-deletion (e.g., ) using the electrical flow values in the resultant graph, but offer no provable guarantees, to the best of our knowledge. We show that breadth-first trees, depth-first trees and even low-stretch trees  can have a , and gaps from the optimal solution for specific instances, respectively. Although we would like a “bicriteria” tree that can connect demands to the root via shortest paths that are as disjoint as possible, i.e., degree of the nodes (except the root) in the tree must be small, numerous results on bicriteria tree approximations (e.g., , , , ) seem to not extend to our setting due to the dependence of the approximation factor on the resultant flow in the trees. We give a more detailed review of related work in Appendix A.
Ours is the first paper to provide a provable approximation guarantee for the network reconfiguration problem. We delve deeper into the problem structure to construct novel lower bounds and give new ways of graph sparsification while maintaining original edge-resistances.
Summary of Contributions:
We summarize our contributions in three categories: randomized edge-deletion using flow relaxation, new lower bounds to improve the gap from the optimal solution, and a constant factor approximation for the reconfiguration problem over grid graphs with uniform demands.
Flow Relaxation: We first relax the spanning tree constraint to reduce the problem to finding a minimum energy electrical flow; we call this the flow relaxation. In Section 2, we construct instances that have a gap between the energy of the optimal tree and the flow relaxation of the order over grid graphs and in general graphs, where and are the number of nodes and maximum degree in the graph (which can be as large as ) respectively. Further, we propose a randomized iterative edge-deletion algorithm, RIDe
(that may be of independent interest) that sparsifies a graph while preserving the original resistances of edges. The algorithm deletes edges sampled according to a probability distribution that depends on theeffective resistances of remaining edges. We show that this method can guarantee gap from the flow relaxation. This can be useful for obtaining approximations for sparse graphs.
The randomized iterative edge-deletion algorithm, RIDe, gives a approximation in expectation with respect to the cost of the flow relaxation on the given graph . In particular, where is the energy of the feasible flow on the final tree computed by the algorithm.
New Lower Bounds: We next exploit the combinatorial structure of the graph to obtain better novel lower bounds in Section 3. First, we decompose the total loss into the contributions of each vertex and cross-product terms arising from the quadratic objective. We show that if one ignores the cross-product terms, then a shortest-paths tree (with respect to resistances) is optimal and prove that this gives an approximation. Second, we show that we can further improve this approximation factor by finding a laminar family of cuts. In particular, for grid instances that have uniform edge resistances, a selection of laminar cuts gives -approximation. Informally, our second main result is:
Theorem 2 (informal).
Given an instance of the network reconfiguration problem, we construct two lower bounds using paths and cuts in the graph:
Shortest Path Trees: the shortest-path tree with respect to resistances is an -approximation. Moreover, there exist instances where this approximation is tight.
Packing of Cuts: If there exists a family of cuts such that each edge appears in at most cuts and the union of the cut-edges supports a rooted-arborescence with edges directed away from the root, then is an approximation, where is the size of the maximum cut in .
Constant-factor Approximation: In Section 4, we consider an grid with uniform resistances and a root at one of the corners of the grid. We propose a novel combinatorial algorithm, Min-Min, that gives a constant factor approximation when the demands are uniform. The key idea of the Min-Min algorithm is to divide the grid into two triangles along the main diagonal, route the lower-triangle loads via disjoint paths up to the diagonal, and start merging these paths in upper triangle part while keeping them balanced in a greedy manner. We show that the Min-Min algorithm has an asymptotic approximation ratio of 2. It also extends to the setting where demands are general but lie in a fixed interval and we analyze the corresponding approximation factor in that case.
Theorem 3 (informal).
The Min-Min algorithm is a -approximation algorithm for the network reconfiguration problem over an grid with uniform loads and resistances, with the root at a corner of the grid. In particular, as , the approximation factor goes to 2. For the case when all demands lie in an interval , the approximation factor increases by , where .
As a preview, in Section 2 we discuss our iterative deletion algorithm using electrical flows that achieves a factor gap from optimal, in Section 3 we present novel lower bounds using paths and cuts which give -approximation for the network reconfiguration problem and -approximation for a grid with uniform demands, and finally in Section 4 we present an asymptotic 2-approximation for the case of uniform demands and resistances over grids. We discuss open questions and possible approaches in Section 5. We defer to the appendix for details about the related work (Section A), preliminaries on electrical flows and technical details of RIDe (Section B), and missing proofs (Section C).
2 Electrical Flows
Relaxing the spanning tree constraint (2) in (P0), we get the flow relaxation that aims to find a feasible flow that minimizes the energy . The (unique) optimal flow is in fact the electrical flow. There has been extensive work done in the area of electric flows and their applications as subroutines in many graph algorithms. Moreover, electrical flows can be computed efficiently in near-line time [8, 12, 29, 11]. In this section, we explore the gap between the energy of the optimal spanning tree for the network reconfiguration problem and the energy of electrical flow in the original graph. We also propose a randomized iterative deletion algorithm that gives a approximation factor.
It is easy to construct instances with maximum degree666Note that the maximum degree can be as large as . such that the gap from the flow relaxation is , for example, consider a graph with two degree nodes ( is the root, and is the only node with positive demand, say 1 unit) with disjoint paths between them; see Figure 1 (left). In this case, the electrical flow sends units of flow along each of the disjoint - paths. The energy of the electric flow is then , however, the energy of the optimal spanning tree is .
Moreover, the gap of the optimal tree compared to the flow relaxation can be large even for graphs with constant degree. Consider a grid (with nodes) where the root is in the top left corner, all edges have unit resistances, the node in the bottom right corner, call it , has a demand of one, and all other nodes (excluding the root) have zero demand; see Figure 1 (middle) for an example. The potential drop between and on sending one unit of current from to is often referred to as the effective resistance between and which we denote by (see Appendix B.1.4 for more details). In this instance, it is known that (see Proposition 9.17 in ) , and using Ohm’s Law777Ohm’s Law says that the electrical flow on any edge is equal to the potential difference divided by the resistance of the edge (or equivalently multiplied by the conductance)., this provides an upper bound on the the energy of the electrical flow. Furthermore, the cost of any optimal tree888The path from r-t can be grown into a spanning tree without incurring any additional cost. is , since any - path has hop-length . Combining these two facts, we get the gap for grid instances is .
In general, it seems one cannot obtain better than performance using electrical flows, unless the flow relaxation is strengthened using new inequalities. Even existing work on analyzing the stretch999The stretch of a tree is a metric used to analyze how well a tree preserves distances between the endpoints of edges in the original graph. of trees that has been used to bound the energy of a tree with respect to the flow relaxation , does not give compelling approximation bounds, since there exist instances with stretch at least ; we refer the reader to Appendix A for more details. We next present a randomized iterative edge-deletion algorithm, RIDe, for sparsifying graphs while maintaining edge-resistances. This algorithm achieves an expected gap of , and may be of independent interest.
The key idea is to delete edges iteratively, while maintaining the graph connectivity, until the resultant graph is a spanning tree. This is done as follows: sample an edge at random to delete from the graph with a probability proportional to , where is the conductance of an edge. The normalization constant of this probability distribution is
, which follows from the fact that vector, is in Edmond’s spanning tree polytope (see Appendix B.2.2). Moreover, the graph Laplacian and effective resistances can be efficiently updated after every deletion. We give the complete description of the algorithm in Algorithm 1.
Intuitively, the quantity fully characterizes the graph’s ability to reconfigure the flow upon deleting edge (see Lemma 7, Appendix B.2.1). In particular, the smaller the (thus, the larger probability of deleting the edge), the better is the graph’s ability to re-route the flow of edge upon deleting the edge, without significantly increasing the energy. Moreover, this sampling procedure ensures that connectivity is maintained, since for any bridge edge , as in such a case we have .
To analyze this algorithm, we note that deleting a single edge results in a rank-one update to the graph Laplacian, and therefore, one can obtain a closed-form expression for the psuedoinverse of the Laplacian. This helps characterize the resultant energy of the flow in the new graph. We can show that in expectation RIDe gives an gap from the flow relaxation. Moreover, this matches the optimal energy for some instances (e.g., Figure 1 (right)), as we discuss in Section B.2.3 in the appendix.
The randomized iterative edge-deletion algorithm, RIDe, gives a approximation in expectation with respect to the cost of the flow relaxation on the given graph . In particular,
In light of this discussion, it is imperative to note that our approximation factor is with respect to the flow relaxation. The optimal spanning tree must also pay for edge deletions but it is unclear how we can incorporate that in the analysis of our randomized algorithm. However, this does not preclude the possibility of obtaining a better lower bound and approximation using electrical flows and this remains an open question. To strengthen the lower bound and consequently obtain better approximation factors, we proceed by exploiting the combinatorial structure in certain graphs as well as demand scenarios.
3 New Lower Bounds
As discussed in Section 2, relaxing the spanning tree constraint can lead to a weak lower bound, mainly because we allow the demand of a single node to be delivered via multiple paths from root to node . In this section we ask: “In what ways can we strengthen the flow relaxation by exploiting the tree constraints?” We first answer this question by accounting for the minimum loss each node creates to get connected to the root, in the absence of all other nodes, and show how to use this lower bound to achieve an -approximation algorithm. Next, we consider cuts in a graph to lower bound the energy of an optimal tree by considering the demand it separates. We show that by constructing a laminar family of cuts, one can derive a new lower bound by accounting for the loss of a spanning tree which is balanced over all these cuts. We then use this lower bound to get a -approximation for grid graphs.
The major source of hardness in (P1
) is the quadratic loss function, which introduces a cross-term for any two nodes that share an edge on their path to the root. If the loss was a linear function of the flow, and in the absence of cross-terms, the problem would decompose intodisjoint problems, which can be solved via shortest path trees. Note that, however, we can relate the quadratic loss to the linear case as follows:
where the first inequality is by the non-negativity assumption, and the second one is due to the Cauchy-Schwarz inequality. Now looking at as the new demand of each node , our quadratic loss lies between the two linear objectives, for which it is easy to show that the shortest-path tree rooted at node solves the corresponding optimization problem, namely101010See the problem SymT in  for example.: . This immediately implies the following result.
The shortest-path tree (with respect to resistances) rooted at is an -approximation solution for problem (P1). Moreover, there exist graph instances for which the cost of a shortest-path tree (or BFS tree) is at least times the cost of an optimal tree.
The approximation factor follows immediately from the above discussion. We refer the reader to Appendix C.2 for the lower bound instances.
We now consider special graphs with a particular set of cuts that yield improved approximation factors. For the rest of this section we assume that all the edges of the graph have the same resistances, and without loss of generality we set for all .
Consider a graph with root , and . Assume that has a family of cuts (with ) such that:
each edge appears in at most cuts, i.e., for all ,
the union of the cuts supports a spanning arborescence rooted at such that (directed) edge only if for all such that .
Then, the approximation factor of the arborescence is at most .
The key idea is to bound the cost of the tree and restricted to each cut in the family. Let be one of the cuts with , and let be the size of the cut. Assume that the total demand outside the cut is . Then the tree has a cost at most over this cut, which happens if the entire demand is routed via a single edge of this cut. On the other hand, has to pay at least , which happens when the load is equally distributed among all edges of the cut. This justifies the ratio of between the costs over this particular cut , and would simply extend to the entire objective if the cuts were disjoint. However, we may double count what the optimal tree is paying since the cuts are not disjoint, but we know that each edge will be counted at most times. Analyzing the directionality of edges used in the arborescence is more involved, and we defer to Appendix C.3 for the complete proof.
We use Theorem 6 for an grid (where we let the number of nodes be ), and consider diagonal cuts as shown in Figure 2. Note that these cuts partition the edges of the grid, and their size is less than . Any spanning tree that has edges only going to the right or the bottom, satisfies the second requirement of Theorem 6, and thus has cost at most times the optimal spanning tree.
There exists an -approximation algorithm for minimizing the loss on an grid with nodes, when all the edges have the same resistance and the root is located at the corner of the grid.
Constructing the above described set of cuts for general graphs remains an open question; planar graphs would be a natural candidate. By the planar separator theorem , we know that any -node planar graph has a vertex separator of size that splits the graph into two (almost) equal parts. However, it is not clear how to find the desired family of cuts by using the planar separator oracle. This is due to the second requirement of the cuts in Theorem 6 which fixes a natural direction on any edge once it appears in a cut. This direction should be respected in future cuts that include this edge; however, the separator oracle is oblivious to edge directions.
4 A 2-Approximation for Uniform Grid
We now propose a constant-factor approximation algorithm for an uniform grid () with the root at a corner of the grid (see Figure 2), in which all demands are equal ( for all ) and all resistances are equal ( for all ). The key idea is to help us understand better the structure of optimal solutions through combinatorial techniques. For the sake of brevity, we will present our novel Min-Min algorithm for a square grid, while the results hold for rectangular grids as well (and more general demands).
We let be the size of the grid, while the total number of nodes is . We consider the diagonal cuts as shown in Figure 2 and we name them , for the upper triangle, and for the lower triangle. Note that the diagonal cuts cover all the edges and each edge appears in only one cut. Therefore we can divide the cost of any spanning tree (either optimal or approximate tree) into the costs from each cut. Let and denote the cost of edges that cross cut in the optimal and approximate solution, respectively. Similarly we define and for the lower triangle. Finally, let and . Then we have . Similarly, we define and .
4.1 Lower bounds
Let and be the number of nodes below cuts and , respectively. Then:
Upper triangle cuts
By a quick look at the structure of the grid, we can observe that there are edges that cross the cut . These edges are connected to nodes on the root side of the cut, and nodes on the other side. Call these nodes on the root side and below the cut. We call an edge of the tree outgoing, if is the parent of in the tree. We claim that the tree can have at most outgoing edges over this cut, although it can have all the edges. This is because if we have more than outgoing edges, then by the pigeonhole principle, a node will have two parents and this creates a loop in the tree.
In addition, all the nodes below the cut are connected to the root through (at least) one of these outgoing edges over this cut. This is true because we can traverse the path from the root to that node, and at some point we must cross the cut through an outgoing edge. It is possible to have multiple outgoing edges on that path if we cross the same cut multiple times, but all we need is that each node below the cut is counted as a successor for at least one of the outgoing edges. So the aggregate number of successors for the outgoing edges of cut is at least , while there are at most such edges. Recall that when the summation of a number of variables is fixed, their sum of squares is minimized when all of them are equal. Hence, in the most balanced way any tree (including the optimal tree) has to pay the following cost over this cut:
By summing the above lower bound over different cuts, we can also get the following lower bound on the energy of any spanning tree over the entire upper triangle. The proof can be found in Appendix C.4.
The cost of the optimal tree over the upper triangle part of the grid is lower bounded by:
Lower triangle cuts
The argument here is exactly same as in the upper triangle except that there are nodes on the root side and nodes on the other side. Therefore, in the most balanced case when all the outgoing edges carry the same flow, we have outgoing edges with flows , paying the total cost of:
4.2 Min-Min algorithm
The Min-Min algorithm builds a spanning tree which contains disjoint paths with different lengths over the lower triangle; see the blue paths in Figure 4 (right). Then in each cut of the upper triangle, exactly one pair of subtrees merge together. As the name suggests, we merge the two subtrees with the minimum number of successors in each step, and call it the merging step. However, this requires those two subtrees to be next to each other.
Therefore, we need to order our disjoint paths in the lower triangle in a way that allows merging minimum load subtrees in the upper triangle; we call this the uncrossing step. In the following lemma, we show that the number of successors of edges on the main diagonal can be any permutation of the numbers .
Lemma 2 (Disjoint paths).
We can obtain a shortest path decomposition of the lower triangle of the grid for any ordering of numbers , specifying the number of successors of edges on the left diagonal of the grid.
We give a recursive construction which also proves the existence of such paths. Let be a permutation of . Put these numbers on the main diagonal. Except which is already satisfied, connect the rest of the nodes to the nodes of the next diagonal, which has nodes, in the same order. Now we have to construct a permutation of on this new diagonal, because the previous numbers should be decreased by one. We can repeat the process. An example is performed in Figure 3. ∎
Note that Lemma 2 ensures the adjacency of minimum subtrees in all upper triangle cuts. To get the right permutation, we can start from any permutation (say ) and do the Min-Min merging as shown in Figure 4 (left). Then we can do the uncrossing from top to bottom as shown in Figure 4 (middle) and this gives the desired permutation on the main diagonal. Finally, we can construct the lower part of the spanning tree corresponding to that permutation using Lemma 2, and construct the upper part of the spanning tree by merging minimum size subtrees in each layer. An example of the algorithm is shown in Figure 4 (right). The formal description of the algorithm for general rectangular grids is also tabulated under Algorithm 2, in which for the case of a rectangular grid, we use the middle part of the grid to connect the lower triangle to the upper triangle via parallel disjoint paths.
4.3 Approximation factor for the Min-Min algorithm
We first show that the cost of Min-Min algorithm is at most of the optimal tree, over the lower triangle.
The Min-Min algorithm costs at most of the optimal over the cuts in the lower triangle. In other words,
where refers to the output of Min-Min algorithm. Moreover, this implies the same approximation factor for the entire lower triangle energy, i.e., .
To analyze the Min-Min algorithm over the upper triangle, we first obtain the following relation between the cost of the algorithm over different cuts.
For , we have
Let be the number of successors for the edges of cut , in non-decreasing order (). We know that . Since we merge in the higher level, the edges of cut will have successors. Therefore the cost of cut is
Since is the smallest number, it is upper-bounded by the average, i.e. . Similarly, is the smallest among the rest, therefore . So the algorithm satisfies:
By recursively applying this upper bound, we get (5). ∎
The output of the Min-Min algorithm satisfies: .
Once we have separate approximation guarantees for and , the worse approximation factor determines the overall result.
For a rectangular grid with loads satisfying for all nodes , the Min-Min algorithm for the network reconfiguration problem with uniform resistances gives an approximation factor of , where . In particular, if the loads are uniform and as , the Min-Min algorithm gives a -approximation.
5 Conclusions and Open Questions
In this paper we studied the network reconfiguration problem from power systems (for electricity distribution networks) through the lens of approximation algorithms. We provided approximation algorithms for different scenarios with restrictions on graph structure, line resistances, and node demands. A large number of open questions still remain, including the extension of the -approximation (or even constant-factor approximation) to planar graphs, analysis for the iterative deletion of the min-flow edge (introduced by Shirmohammadi and Hong ), and the hardness of the problem for grids or planar graphs.
The authors would like to thank the participants of the Real-Time Decision Making Reunion Workshop, Mixed Integer Programming Workshop, and the IEEE Power & Energy Systems General Meeting for valuable feedback. In particular, we would like to thank David Williamson for pointing us to low-stretch spanning trees, and Alexandra Kolla for pointing us to spectrally-thin trees. A part of this work was done while the authors were visiting the Simons Institute, UC Berkeley. We gratefully acknowledge the financial support from NSF grants CCF-1733832, CCF-1331863, and CCF-1350823.
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Appendix A Detailed Overview of Existing Techniques
Related work in power systems:
The problem of reconfiguring the electric distribution network to minimize line losses was first introduced by Civanlar et al.  and Baran and Wu  where they introduced and implemented an algorithm called “Branch Exchange”, which tries to locally improve the objective by swapping two edges of the graph. Unlike branch exchange that maintains a feasible spanning tree during its execution, there are other algorithms that start with the entire graph and delete edges one by one until a feasible solution is obtained . We discussed this approach further in Section 2. Subsequently, many other heuristic algorithms were proposed for the reconfiguration problem, including but not limited to genetic algorithms 25], artificial neural networks , etc. The missing part in all these heuristics is a rigorous theoretical performance guarantee that shows why/when these algorithms perform well. To that end, Khodabakhsh et al.  recently showed that the reconfiguration problem is equivalent to a supermodular minimization problem under a matroid constraint.
Existing techniques in combinatorial optimization:
One may ask the question if a simple spanning tree like breadth-first or depth-first search tree will be a good solution to the reconfiguration problem. It is shown in  that if the edges are identical, the optimal tree will include all the edges incident to the root. So the BFS tree might be a better candidate. However, as we showed in Section 3, a BFS tree can have a loss of times the optimal loss. Depth-first search can be worse; for example, a cycle with spokes and root in the center has a gap of . (the optimal tree is a star with linear cost, while the DFS tree will take the cycle with cubic cost.)
Our problem looks like a bicriteria tree approximation; on the one hand, we want to connect demands to the root via shortest paths, on the other hand, we want the paths to be disjoint, i.e., degree of the nodes (except the root) in the tree must be small. Similar problems have been studied in the Computer Science literature: Könemann and Ravi  consider finding a minimum cost spanning tree subject to maximum degree at most . The variation where each node has its own specified degree bound is also studied by Fekete et al.  and Singh and Lau . Khuller et al.  also define Light Approximate Shortest-path Trees (LAST), in which a tree is -LAST if the distances to the root are increased by at most a factor of (compared to the original graph), while the cost of the tree is at most times the minimum spanning tree. However, the main difficulty in using these approximations for the reconfiguration problem is accounting for the resultant flow in the spanning trees. The cost on the edges of the tree are then no longer linear (as in the above mentioned results) or even quadratic.
Electrical energy is minimized when considering edge-disjoint paths to connect nodes . However, the complete disjointness is rarely achievable in our problem (unless the graph is a star with root in the center), and hence we want to limit the number of flows that merge together. This is more related to the Edge-Disjoint Path with Congestion (EDPwC) problem, studied by Andrews et al. , which is as follows: Given an undirected graph with nodes, a set of terminal pairs and an integer , the objective is to route as many terminal pairs as possible, subject to the constraint that at most demands can be routed through any edge in the graph. They show hardness of approximation for EDPwC problem. The main differences with our problem are that in EDPwC there is a hard constraint on the flow routed through any edge, while in our problem there is a quadratic cost associated with that flow, as well as the additional spanning tree constraint in the reconfiguration problem.
A natural question is if there are some known graph families where one can exploit existing structures to find an approximation. We consider planar graphs, also motivated by the application since many distribution networks are designed that way. As we saw in Section 3, one way to obtain useful lower bounds for the objective function is via generating a packing of cuts that are small in size, i.e., do not have many edges. A celebrated result in the theory of planar graphs is the existence of a small set of vertices, called the vertex-separator, that can disconnect the graph into components of almost equal size . This can even be applied in a recursive manner, as shown by Frederickson , to divide the graph into regions with no more than vertices each, and boundary vertices in total. However, it is unclear how to bound the cost of these cuts or regions in the reconfiguration problem, unless we have more information about the direction of the resultant flow on boundary edges.
Existing techniques in electrical flows:
Electrical flows have been an active area of research in the past two decades, due to their computational efficiency and numerous applications to graph theory problems. In particular, it was shown that one can compute an electrical flow in a graph in near-linear time [12, 20, 11]. Moreover, various novel graph algorithms involve computing an electrical flow as a subroutine. For example, Madry  and Christiano et al.  use electrical flows to obtain the fastest algorithms for the Max-flow problem so far.
If we relax the spanning tree constraint, our problem reduces down to the standard problem of computing an electrical flow in a graph that satisfies the demands. Even though the support of an electrical flow in its full generality does not form a spanning tree, still, it maybe be beneficial to use the minimum energy electrical flow in the graph as a starting point. Shirmohammadi and Hong  follow this approach and propose an iterative algorithm for the reconfiguration problem, where in each iteration they compute the electrical flow and delete the edge with smallest flow such that the graph remains connected. They demonstrated experimentally that their iterative algorithm performs well in practice but they provide no theoretical guarantees. In Section 2 and Appendix B.2, we proposed a similar iterative edge-deletion algorithm and prove its approximation bound in Theorem 4.
At the heart of these edge deletions is the question of whether we can delete edges from the graph without increasing the energy cost too much. One approach that can be used to address this question is spectral sparsification , which aims to reduce the number of edges in the graph while maintaining approximations of the Laplacian quadratic form. In their classic result, Spielman and Srivastava  show that one can construct such a sparsifier with edges. Chu et al. slightly improve upon the results of Spielman and Srivastava, bringing the number of edges down to for some specific instances. Using the fact that electric flows are fully characterized by the Laplacian quadratic form, one may conclude that by using such a sparsifier we can reduce the number of edges in the graph without significantly increasing the energy cost. However, this is not true, because to obtain such sparsifier they compensate the deletion of edges by changing the weights (i.e. resistances) on the edges. Thus, since we assume resistances are fixed, to the best of our knowledge, the existing spectral sparsification approach does not extend to our problem. This motivates the need of a novel approach to handle edge deletions without increasing the energy too much.
Uniform Spanning Trees:
To deal with the iterative edge deletions, we consider sampling from distributions over spanning trees. Random spanning trees are one of the most well-studied probabilistic combinatorial structures in graphs. Recent work has specifically considered product distributions over spanning trees where the probability of each tree is proportional to the product of its edge weights. (This is motivated by the desirable properties and numerous applications of such product distributions.) For example, Asadpour et al.  break the barrier of the ATSP problem by rounding a point in the relative interior of the spanning tree polytope by sampling from a maximum entropy distribution over spanning trees; this maximum entropy distribution turns out to be a product distribution. Moreover, a beautiful property of product distributions over spanning trees is the fact that the marginal probability of an edge being in a random spanning tree is exactly equal to the product of the edge weight and the effective resistance of the edge (see for example ). This is a fact that we exploit in our RIDe algorithm.
Finally, another relevant approach in the electric flows literature, entails low stretch trees. Given a weighted graph , a low-stretch spanning tree is a spanning tree with the additional property that it approximates distances between the endpoints of any edge in . In particular,111111We follow the definition of Elkin et al. , but this definition slightly differs in the denominator from others given in the literature. Abraham and Neiman  and Abraham et al.  define the stretch as , where is the shortest-path metric on with respect to the edge weights. Note that these definitions are equivalent if the edge weights are uniform. the stretch of an edge is the ratio of the (unique) shortest path distance between and in to (the weight of edge in ). Furthermore, the total stretch of is defined as the sum of the stretch of all edges in . Kelner et al.  show that for any tree, the gap between the energy of the flow in that tree and the flow in the original graph, is at most the total stretch of that tree. Naturally, one may wonder if there exists a low value for the (total) stretch such that all graphs have a spanning tree with that stretch. The answer to that question is unfortunately no. Abraham and Neiman  show that one can construct a spanning tree for any connected graph with total stretch at most in near-linear time (Theorem 2.11 in ); this bound is tight up to an factor because Alon et. al  show that the total stretch is for certain graph instances. Thus, this implies that the energy cost of is at most times that of the original graph. We improve upon this approximation result using our RIDe algorithm.
Appendix B Electrical Flows and Iterative Edge Deletion
b.1.1 The Graph Laplacian
Let be a connected and undirected graph with , . Each edge is also associated with a resistance . The inverse of the resistance is called conductance, defined by . We orient each edge in arbitrarily, where given a vertex , the edges incident to a vertex and oriented away from belong to the set and those oriented towards belong to . Given such an orientation, we can define the vertex-edge incidence matrix of as follows
Let be an diagonal resistance matrix where . We define the weighted Laplacian , where . Since is a positive definite and symmetric matrix, we could write , which immediately implies that is positive-semi definite, since for all .
b.1.2 The Laplacian Pseudoinverse
Let be the all-ones vector. For any matrix , denote the span of the columns of by . It is well known that if is connected, the only vector in the nullspace of the Laplacian is the all-ones vector . This means if is connected, the matrix is invertible in the subspace perpendicular to . In what follows, this inversion will be done by the so-called Moore-Penrose pseudoinverse denoted by .
Since is symmetric and positive semi-definite, we can write in terms of its eigen-decomposition
are the eigenvalues ofsorted in increasing order and
are the corresponding singular orthonormal vectors. Now, the pseudoinverse could be conveniently characterized using
Observe that and is thus a projection matrix that projects onto . In other words, for any vector such that , . This is a fact that will be used in some of the proofs later on.
b.1.3 Pseudoinverses Following Rank-one Updates
Let be a symmetric square matrix. A rank-one update to the matrix is of the form for some vectors . The following lemma provides us with a closed-form expression for the pseudoinverse of under certain conditions.
Theorem 8 (Theorem A.70 in ).
Let be an be symmetric matrix. Let be -dimensional column vectors. Assume . Then
Note that (6) is of the same form as the well-known Sherman-Morrison formula (see for example ). We recall that the Sherman-Morrison formula is used to update the inverse of a non-singular matrix following a rank-one update as long as the update does not cause to be singular (which happens if and only if ).121212The Sherman-Morrison formula applies to the case when is invertible and hence does not require .
b.1.4 An Introduction to Electrical Flows
We give a brief introduction to electrical flows from an optimization perspective; for a more detailed treatment we refer the reader to Williamson . Given a graph , a root and demands for all , we begin by assigning a demand to the root, and we collect these demands into a demand vector . For a given feasible flow satisfying the demands, we define its electrical energy to be
where is the resistance matrix. Of interest, is to find a flow in the flow polytope (defined by the undirected graph and demand vector ) that minimizes the electrical energy. This problem could be formulated as follows
Recall that is a positive definite matrix and hence the objective function is strongly-convex. Since (E) is composed of a strongly objective function over affine constraints: the optimal solution is unique and the Karush–Kuhn–Tucker (KKT) conditions are sufficient and necessary. Therefore, it is easy to show that the KKT conditions applied to the Lagrangian function of (E)
yield the following optimal solution:
Note that to obtain using the KKT conditions, we relied on that fact that by definition , i.e. , and hence the validity of using the Moore-Penrose pseudoinverse of the Laplacian to obtain .
Let be a vector of potentials defined by . Then, using (9) we have
which coincides with Ohm’s voltage law. Moreover,
Again since , we have . Using the above expression with we obtain
For any pair of vertices let be a vector with a in the coordinate corresponding to , a 1 in the coordinate corresponding to , and all other coordinates equal to 0. Also for any edge , let be the vector that has a 1 in the coordinate corresponding to and all other coordinates equal to 0. The effective resistance between a pair of vertices is given by
In other words, it is the potential difference between and when we send one unit of electrical flow from to . We define the effective resistance matrix . Observe that , and hence the diagonal of constitutes the vector of effective resistances for all edges in the graph.
For any edge , it is well known that , where equality holds if and only if is the only path between and (see for example Theorem D in ). Intuitively, if , then when sending one unit of electrical flow between the endpoints of , all that flow goes through . This means that the only path between and is the edge , since otherwise we could reroute some of the flow through another path and decrease the effective resistance, which would contradict the optimality of the electrical flow. It is also known that the effective resistance is monotonically non-increasing when adding new edges to the graph. In particular, upon adding a new edge to the graph, strictly decreases if adding the new edge to the graph creates a new path that did not exist before (which happens, for example, when the newly added edge is parallel to ). This implies that the more paths between and , the smaller . Thus, we can see that is indicative of how well and are connected in the graph.
b.2 A Randomized Iterative Deletion Algorithm
b.2.1 Iterative Deletions
We propose a randomized iterative deletion algorithm, RIDe, which randomly deletes edges one by one until it arrives at a spanning tree. A similar deterministic variant of this algorithm was first proposed by Shirmohammadi and Hong . In particular, they demonstrated that their iterative heuristic performs well in practice but they do not offer theoretical guarantees. We prove that our iterative algorithm has an approximation factor in expectation with respect to cost of the flow relaxation.
Certainly, the main component in designing and analyzing the algorithm is to determine how the electric flows change after deleting an edge from the graph. As previously shown, electric flows are fully determined by the Laplacian and its pseudoinverse. Thus, to determine how electric flows change upon edge deletions, we first obtain a closed form expression for how the Laplacian pseudoinverse changes from one iteration to the next.
Let be the weighted Laplacian blue (weighted by conductances) of a connected graph and be the Moore-Penrose pseudoinverse of . Let be the graph obtained by deleting an edge that does not disconnect . Then the Moore-Penrose pseudoinverse of the weighted Laplacian of is:
Let be the edge chosen for deletion. First, observe that deleting an edge from the graph results in a rank-one update
Recall is a square symmetric matrix and we can write
Let , , and . By construction is a column of . Therefore and are in and hence are also in . We further claim that . To see this, suppose . This implies or . In other words, when we send one unit of electrical flow between the endpoints of , all the flow goes through edge . Recall, that this implies that is a bridge, since otherwise we could reroute some of the flow through another path and decrease the effective resistance. Thus, deleting from the graph disconnects , which contradicts the assumption that was connected. Therefore the conditions of Theorem 8 are satisfied and applying (6) with , and as defined above gives the result:
We can now use Lemma 6 to obtain a closed form expression for how the energy increases from one iteration to the next. Suppose that we have performed iterations and deleted edges from the original graph to get a modified connected graph with edges. Let , and respectively be the incidence, Laplacian and effective resistance matrices of . Also, Let and respectively be the electrical flow and effective resistance for edge in iteration . Now in the iteration we wish to delete another edge from to obtain a connected graph . Of interest is to determine how much changes compared to .
Suppose that is connected. Also, suppose that is obtained by deleting an edge from such that is also connected. Then could be recursively obtained from using
Since by assumption and are connected, using Lemma 6 we have
Therefore, the new potentials are given by
We would like to remark that the expression above holds for any edge we delete from the graph as long as does not disconnect the graph as previously mentioned. At this point, there are two issues that need to be addressed: Which edge should we delete and how can we use recursive formulas to find an approximation bound for the algorithm. The problem with recursive formulas in their current form is that it is hard to find a good upper bound for the term, without assuming further information about the problem (like the structure of the graph in question). This is because the effective resistance can in some cases have a weak upper bound that worsens the analysi