## 1 Introduction

Network design problems are generally about finding a minimum cost subgraph that satisfies certain “connectivity” requirements. The most well studied problem is the survivable network design problem [22, 1, 23, 25, 19], where the requirement is to have a specified number of edge-disjoint paths between every pair of vertices . Other combinatorial requirements are also well studied in the literature, including vertex connectivity [29, 15, 5, 10, 30, 7] and shortest path distances [12, 11].

Some spectral requirements are also studied, including spectral expansion [28, 2], total effective resistances [21, 35], and mixing time [4], but in general much less is known about these problems. See Section 1.1 for more discussions of previous work.

In this paper, we study a basic problem in designing networks with a spectral requirement – the effective resistance between two vertices.

###### Definition 1.1 (The - effective resistance network design problem).

The input is an undirected graph , two specified vertices , and a budget . The goal is to find a subgraph of with at most edges that minimizes , where denotes the effective resistance between and in the subgraph . See Section 2.2 for the definition of effective resistance and Section 3.1 for a mathematical formulation of the problem.

The - effective resistance is an interpolation between - shortest path distance and - edge connectivity. To see this, let be a unit - flow in and define the -energy of as , and let - be the minimum -energy of a unit - flow that the graph can support. Thomson’s principle (see Section 2.2) states that , so that a graph of small - effective resistance can support a unit - flow with small -energy. Note that the shortest path distance between and is (as the -energy of a flow is just the average path length and is minimized by a shortest - path), and so a graph with small has a short path between and . Note also that the edge-connectivity between and is equal to the reciprocal of (because if there are edge-disjoint - paths, we can set the flow value on each path to be ), and so a graph with small has many edge-disjoint - paths. As is between and , the objective function takes both the - shortest path distance and the - edge-connectivity into consideration.

A simple property suggests that -energy may be even more desirable than and as a connectivity measure. Conceptually, adding an edge to would make and more connected. For and , however, adding would not yield a better energy if does not improve the shortest path and the edge connectivity respectively. In contrast, the -energy would typically improve after adding an edge, and so -energy provides a smoother quantitative measure that better captures our intuition how well and are connected in a network.

Traditionally, the effective resistance has many useful probabilistic interpretations, such as the commute time [6], the cover time [34]

, and the probability of an edge in a random spanning tree

[27]. These interpretations suggest that the effective resistance is a useful distance function and have applications in the study of social networks. Recently, effective resistance has found surprising applications in solving problems about graph connectivity, including constructing spectral sparsifiers [40] (by using the effective resistance of an edge as the sampling probability), computing maximum flow [9], finding thin trees for ATSP [3], and generating random spanning trees [33, 39].Thomson’s principle also states that the electrical flow between and is the unique flow that minimizes the -energy. So, designing a network with small - effective resistance has natural applications in designing electrical networks [13, 21, 24]. One natural formulation is to keep at most wires in the input electrical network to minimize , so that the electrical flow between and can still be sent with small energy while we switch off many wires in the electrical network.

Based on the above reasons, we believe that the effective resistance is a nice and natural alternative connectivity measure in network design. More generally, it is an interesting direction to develop techniques to solve network design problems with spectral requirements.

### 1.1 Main Results

Unlike the classical problems of shortest path and min-cost flow (corresponding to the and versions of the problem), the - effective resistance network design problem is NP-hard.

###### Theorem 1.2.

The - effective resistance network design problem is NP-hard.

On the other hand, we analyze a natural convex programming relaxation for the problem (Section 3.1), and use it to design a constant factor approximation algorithm for the problem.

###### Theorem 1.3.

There is a convex programming based -approximation randomized algorithm for the - effective resistance network design problem.

The algorithm crucially uses a nice characterization of the optimal solutions to the convex program (Lemma 3.2) to design a randomized path-rounding procedure (Section 3.2) for Theorem 1.3.

A simple example shows that the integrality gap of the convex program is at least two. When the budget is much larger than the length of a shortest - path, we show how to achieve an approximation ratio close to two with a randomized “short” path rounding algorithm (Section 3.5).

###### Theorem 1.4.

There is a -approximation algorithm for the - effective resistance network design problem, when where is the length of a shortest - path.

### 1.2 Other Results

We consider some variants of the - effective resistance network design problem, including the weighted version, the dual version, and the problem on special graphs.

There is a natural weighted generalization of the - effective resistance network design problem, where we associate a cost and resistance to each edge of the input graph.

###### Definition 1.5 (The weighted - effective resistance network design problem).

The input is an undirected graph where each edge has a non-negative cost and a non-negative resistance , two specified vertices , and a cost budget . The goal is to find a subgraph of that minimizes subject to the constraint that the total edge cost of is at most . In the following, we may refer to this problem as the weighted problem for simplicity.

In the weighted problem, the integrality gap of the convex program (Section 3.1) becomes unbounded, even when the cost on the edges are the same ( for all ). This suggests that the weighted version may be strictly harder. Indeed, we show stronger hardness result for the weighted problem assuming the small-set expansion conjecture [37, 38].

###### Theorem 1.6.

Assuming the small-set expansion conjecture, it is NP-hard to approximate the weighted - effective resistance network design problem within a factor of for any , even when for every edge .

On the other hand, when the cost on the edges are the same, the following approximation follows from the randomized path rounding algorithm in a black box manner.

###### Corollary 1.7.

There is a convex programming based -approximation randomized algorithm for the weighted - effective resistance network design problem when for every edge , where is the ratio between the maximum and minimum resistance.

As our problem is related to electrical network design, it is natural to consider the special case when the input graph is a series-parallel graph. In this setting, we can use dynamic programming to design an exact algorithm for the original problem, and a fully polynomial time approximation scheme (FPTAS) for the weighted problem.

###### Theorem 1.8.

There is an exact algorithm for the - effective resistance network design problem with running time when the input graph is a series-parallel graph.

There is a -approximation algorithm for the weighted - effective resistance network design problem when the input graph is a series-parallel graph. The running time of the algorithm is where is the ratio between the maximum and minimum resistance. By a simple preprocessing scaling step, we can assume that is bounded by a polynomial, and so the algorithm is a FPTAS for the weighted problem.

We note that the integrality gap examples in Section 3.1 are actually series-parallel graphs, and so the dynamic programming algorithms go beyond the limitation of the natural convex program. We leave it as an open problem whether the weighted problem admits a constant factor approximation algorithm (possibly by combining these techniques).

We also consider the “dual” problem where we set the effective resistance as a hard constraint, and the objective is to minimize the number of edges in the solution subgraph. We present similar results as the original problem in Section 3.6.

### 1.3 Related Work

In the survivable network design problem, we are given an undirected graph and a connectivity requirement for every pair of vertices , and the goal is to find a minimum cost subgraph such that there are at least edge-disjoint paths for all . This problem is extensively studied and captures many interesting special cases [22, 1, 23, 19]. The best approximation algorithm for this problem is due to Jain [25], who introduced the technique of iterative rounding to design a -approximation algorithm. His result has been extended in various directions, including element-connectivity [16, 8], directed graphs [18, 19], and with degree constraints [31, 14, 17, 32].

Other combinatorial connectivity requirements were also considered. A natural variation is to require internally vertex disjoint paths for every pair of vertices . This problem is much harder to approximate [29, 30], but there are good approximation algorithms for global connectivity [15, 7] and when the maximum connectivity requirement is small [5, 10]. Another natural problem is to require a path of length between every pair of vertices . This problem is also hard to approximate in general but there are better approximation algorithms when every edge has the same cost and the same length [12].

Spectral connectivity requirements were also studied, including spectral gap [20, 28] (closely related to graph expansion), total effective resistances [21], and mixing time [4]

. Some of the earlier works only proposed convex programming relaxations and heuristic algorithms. Approximation guarantees are only obtained in two recent papers for the more general experimental design problem. When every edge has the same cost, there is a

-approximation algorithm for minimizing the total effective resistance when the budget is at least [35], and there is a -approximation algorithm for maximizing the spectral gap when the budget is at least [2]. For our problem, the interesting regime is when is much smaller than , where the techniques in [2, 35] do not apply. We have developed a set of new techniques for analyzing and rounding the solutions to the convex program that will hopefully find applications for solving related problems.### 1.4 Techniques

Our main technical contribution is in designing rounding techniques for a convex programming relaxation of our problem. There is a natural convex programming relaxation, by using the conductance of the edges as variables, and writing the - effective resistance as the objective function and noting that it is convex with respect to the variables (Section 3.1).

We show that optimal solutions of this convex program enjoy some nice properties^{6}^{6}6We can also show that there exists an optimal solution such that the fractional edges form a forest, but this is not included in the paper as we have not used this property in the rounding algorithm.. Given an optimal fractional solution and the unit - electrical flow supported in , we derive from the KKT optimality conditions that there is a flow-conductance ratio such that for every fractional edge with and for every integral edge with .
The flow-conductance ratio is crucial in the rounding algorithm and the analysis.

The rounding techniques in the two recent papers on experimental design [2, 35]

considered each edge/vector as a unit. In

[2], a potential function as in spectral sparsification is used to guide a local search algorithm to swap two edges/vectors at a time to improve the current solution. In [35], a probability distribution on the edges/vectors is carefully designed for an independent randomized rounding. These techniques are only known to work in the case when the solutions form a spanning set so that the “contribution” of each individual edge/vector is well-defined. This is basically the reason why the results in

[2, 35] only apply when the budget is at least .Our approach is based on a randomized rounding procedure on - paths. Given , we compute the unit - electrical flow supported in , and decompose as a convex combination of - paths. The rounding algorithm has iterations (recall that is the flow-conductance ratio of the optimal solution ), where we pick a random path from the convex combination in each iteration, and return as our solution. One difference from the previous techniques in the literature is that each unit in the rounding algorithm is a - path, so in particular and are always connected in our solution. Another difference is that our problem has some extra structure, so that we can compute the electrical flow to guide our rounding procedure, where the variables are not in the convex program. These allow us to obtain a constant factor approximation algorithm for all budget (note that when there is no feasible integral solution).

In the analysis, we prove in Lemma 3.6 that the expected number of edges in is at most , and in Lemma 3.7 that the expected effective resistance is . To bound the expected effective resistance, we use Thomson’s principle and construct a unit - flow to show that . To construct the unit - flow , we keep the flow-conductance ratio and send units of flow on each sampled path (i.e. and ). The flow-conductance ratio plays a crucial role in the proofs of both lemmas. This is because the rounding algorithm is based on the flow variables , and thus the performance guarantees are in terms of , but the ratio allows us to relate them back to the variables in the convex program. Combining the two lemmas give us a constant factor bicriteria approximation algorithm for the problem. This can be turned into a true approximation algorithm by scaling down the budget to and run the bicriteria approximation algorithm with some additional claims (Section 3.4).

The improvement on the approximation ratio when budget is large comes from two observations. The first is that if is much larger than the length of the shortest - path, then the number of independent iterations in the rounding scheme is large (Lemma 3.3). The second is that we can ignore some - paths in the flow decomposition with many fractional edges without affecting the performance much. Combining these, we can apply a Chernoff-Hoeffding bound to show that the number of edges is at most with high probability. Then it is not necessary to scale down the budget by a factor of and we can prove a stronger bound that the effective resistance is at most times the optimal value.

### 1.5 Organization

In Section 2, we define the notations used in this paper and cover background knowledge on effective resistances. We present the convex programming relaxation and our two rounding procedures in Section 3, and the dynamic programming algorithm in Section 4. The NP-hardness and small set expansion hardness results are provided in Section 5.

## 2 Preliminaries

We introduce the notations and definitions for graphs and matrices in Section 2.1, and then define electrical flow and effective resistance and state some basic results in Section 2.2.

### 2.1 Graphs and Matrices

Let be an undirected graph with edge weight on each edge . The number of vertices and the number of edges are denoted by and . For a subset of edges , the total weight of edges in is . For a subset of vertices , the set of edges with one endpoint in and one endpoint in is denoted by . For a vertex , the set of edges incident on a vertex is , and the weighted degree of is . The volume of a set is defined as the sum of the weighted degrees of vertices in . The conductance of a set is defined as the ratio of the total weight on the boundary of to the total weighted degrees in . For two subsets , the set of edges with one endpoint in and one endpoint in is denoted by .

In this paper, an undirected graph with non-negative edge weights is interpreted as an electrical network, where each edge is a resistor with conductance (not to be confused with the conductance of a set as defined above), or equivalently with resistance . The adjacency matrix of the graph is defined as for all . The Laplacian matrix of the graph is defined as where is the diagonal degree matrix with for all . For each edge , let where is the vector with one in the -th entry and zero otherwise. The Laplacian matrix can also be written as

Let

be the eigenvalues of

with corresponding orthonormal eigenvectors

so that . It is well-known that the Laplacian matrix is positive semidefinite and with as the corresponding eigenvector, and if and only if is connected. The pseudo-inverse of the Laplacian matrix of a connected graph is defined aswhich maps every vector orthogonal to to a vector such that .

### 2.2 Electrical Flow and Effective Resistance

Before defining - electrical flow, we first define the standard unit - flow. For each edge , we have two variables and with , where is positive if the flow is going from to and negative otherwise. A unit - flow satisfies the following flow conservation constraints:

Given a unit - flow , we overload the notation and define its undirected flow vector with for each edge . A unit - electrical flow is a unit - flow that also satisfies the Ohm’s law: There exists a potential vector such that for all ,

The effective resistance between and is defined as

which is the potential difference between and when one unit of electrical flow is sent from to . The - effective resistance can be interpreted as the resistance of the whole graph as a big resistor when an electrical flow is sent from to .

One can write the effective resistance in terms of the Laplacian matrix. For , let , where is the unit vector with in the -th entry and in other entries. Combining the flow conservation constraint and the Ohm’s law, it can be checked that the potential vector of a unit - electrical flow is a solution to the linear system

Note that is a solution, and if is connected then any solution is given by for . Therefore, we can write

The effective resistance can also be characterized by the energy of a flow. The energy of an - flow is defined as

Thomson’s principle [26] states that the unit - electrical flow is the unique unit - flow that minimizes the energy. This can be verified by writing down the optimality condition of the minimization problem. Moreover, this energy is exactly the - effective resistance. To see this, note that the flow value on edge in the unit - electrical flow satisfies and thus

To summarize, we will use the following result from Thomson’s principle.

###### Fact 2.1 (Thomson’s principle [26]).

Let be the unit electrical - flow in . Then

A corollary of Thomson’s principle is the following intuitive result known as the Rayleigh’s monotonicity principle.

###### Fact 2.2 (Rayleigh’s monotonicity principle).

The - effective resistance cannot increase if the resistance of an edge is decreased.

We will also use the following result to write a convex programming relaxation of our problem.

###### Fact 2.3 ([21]).

The - effective resistance is a convex function with respect to the conductance of the edges.

## 3 Convex Programming Algorithm

In this section, we analyze a convex programming relaxation for our problem. We first describe the convex program and prove a characterization of the optimal solutions in Section 3.1. We then present a randomized rounding algorithm using flow decomposition in Section 3.2, and show that it is a constant factor bicriteria approximation algorithm in Section 3.3. Then, we show how to convert the bicriteria approximation algorithm into a true approximation algorithm in Section 3.4, and how to modify the algorithm slightly to achieve a better approximation guarantee when the budget is large in Section 3.5. Finally, we discuss the dual problem of minimizing the cost while satisfying the effective resistance constraint in Section 3.6.

### 3.1 Convex Programming Relaxation

The formulation is for the weighted problem, where each edge has a weight . We introduce a variable for each edge to indicate whether is chosen in our subgraph. Let

be the Laplacian matrix of the fractional solution , and be the - effective resistance of the graph with conductance on edge . The following is a natural convex programming relaxation for the problem.

(CP) | ||||||

subject to | ||||||

This is an exact formulation if for all . The objective function is convex in by Fact 2.3. The convex program can be solved in polynomial time by the ellipsoid method to inverse exponential accuracy, or by the techniques described in [2] to inverse polynomial accuracy, which are both sufficient for the rounding algorithm.

#### 3.1.1 Integrality Gap Examples

We show some limitations of the convex program for general and . The following figure shows a simple example where the integrality gap is unbounded if the cost could be arbitrary.

In this graph, the top path has length where each edge has cost . The bottom path has two edges with cost . The resistance of each edge is , and the budget is . The integrality gap of this example is . To see this, the integral solution can only afford the top path, and the effective resistance is . However, the fractional solution can set for each of the two bottom edges, and the effective resistance of this fractional solution is .

The following figure shows another simple example where the integrality gap is unbounded if the edge costs are the same but the resistances could be arbitrary.

In this example, the top path has length with each edge of resistance . The bottom path has only one edge with resistance . All edges have cost and the budget . The integral solution can only afford the bottom path, with effective resistance . The fractional solution can set for each edge in the top path, with effective resistance . When , the integrality gap could be arbitrarily large.

Even in the unit-cost unit-resistance case, the integrality gap is unbounded if is smaller than the - shortest path distance. Henceforth, in view of these observations we assume the following in the rest of this section.

###### Assumption 3.1.

We assume that for every edge , which is the setting of the - effective resistance network design problem, and the budget is at least the shortest path distance between and in the input graph.

The integrality gap of the convex program is still at least two with Assumption 3.1. For a simple example, consider a graph with two vertex-disjoint - paths, each of length , and the budget is . Then the optimal integral value is while the optimal fractional value is close to , and so the integrality gap gets arbitrarily close to two.

We will show that the integrality gap of the convex program is at most with these assumptions. Note that just to connect and , then must be at least the - shortest path distance. It is interesting that this small additional assumption could reduce the integrality gap from unbounded to a constant.

#### 3.1.2 Characterization of Optimal Solutions

In the case for all edges , we will prove that the electrical flow supported in the optimal solution to (CP) satisfies a crucial property about the flow-conductance ratio .

###### Lemma 3.2 (Characterization of Optimal Solution).

Let be the input graph with for all edges . Let be an optimal solution to the convex program (CP). Let be the set of fractional edges with , and be the set of integral edges with . Let be the undirected flow vector of the unit - electrical flow supported in . There exists such that

###### Proof.

By removing edges with , we can assume for every . By removing isolated vertices, we can further assume that the nonzero edges form a connected graph. So, we can write , where has rank and the null space of is . Since , we have and , which implies that where is the all-ones matrix. Using the fact that (see e.g. [36]), we derive

where we used the assumption that for all . With this, we write down the KKT conditions for the convex program. Let be the dual variable for the budget constraint , and and be the dual variables for the upper bound and the nonnegative constraint respectively. The KKT conditions states if is an optimal solution to (CP), then there exist and such that

(Primal feasibility) | |||||

(Dual feasibility) | |||||

(Complementary slackness) | |||||

(Lagrangian optimality) |

where we used the assumption that for all . For an integral edge with , we have by the complementary slackness condition. Since , it follows from the Lagrangian optimality condition that . For a fractional edge with , we have by the complementary slackness condition, and therefore by the Lagrangian optimality condition. We can assume that . Otherwise, implies that the flow on all fractional edges are zero, and so we can delete them from the graph without affecting the - effective resistance, and we have an integral solution.

Let be a potential vector of the electrical flow supported in . For an edge ,

where the first equality is by Ohm’s law and the assumption that for all , and the second equality uses that as explained in Section 2.2. The lemma then follows from the above paragraph and writing as . ∎

The flow-conductance ratio will be crucial in the rounding algorithm and its analysis. The following lemma shows an upper bound on using the budget and the shortest path distance between and .

###### Lemma 3.3.

Under the conditions in Assumption 3.1, it holds that .

###### Proof.

Let be an optimal solution to (CP), and be the unit - electrical flow supported in . As , a shortest path is a feasible solution to (CP), and thus . On the other hand, by Thomson’s principle and Lemma 3.2,

where the last equality holds since we can assume for the optimal solution without loss of generality by Rayleigh’s principle (or otherwise we have an integral optimal solution). The lemma follows by combining the upper bound and the lower bound. ∎

### 3.2 Randomized Path-Rounding Algorithm

Our rounding algorithm uses the unit electrical flow supported in the optimal solution to construct an integral solution. The algorithm will first decompose the flow as a convex combination of flow paths, and then randomly choose the flow paths and return the union of the chosen flow paths as our solution.

The following lemma about flow decomposition is by the standard argument to remove one (fractional) flow path at a time, which holds for any unit directed acyclic - flow.

###### Lemma 3.4 (Flow Decomposition).

Given a unit - electrical flow , there is a polynomial time algorithm to find a set of - paths with such that the undirected flow vector can be written as a convex combination of the characteristic vectors of the paths in , i.e.

where is the characteristic vector of the path with one on each edge and zero otherwise.

With the flow decomposition, we are ready to present the rounding algorithm.

The following lemma shows that the rounding algorithm will always return a non-empty subgraph.

###### Lemma 3.5.

###### Proof.

Since we assumed that the budget is at least the length of a shortest - path, it follows from Lemma 3.3 that . This implies that

∎

### 3.3 Bicriteria Approximation

The analysis of the approximation guarantee goes as follows. First, we show that the expected number of edge in the returned subgraph is at most the budget . Then, we prove that the expected effective resistance of the returned subgraph is at most two times that of the optimal fractional solution. Both of these steps use the flow-conductance ratio crucially. These combine to show that the randomized path rounding algorithm is a constant factor bicriteria approximation algorithm.

Let be an optimal solution to (CP). Let and be the set of fractional edges and integral edges in . We assume that each edge will be included in the subgraph returned by the rounding algorithm. We focus on bounding the number of edges in that will be included in .

###### Lemma 3.6 (Expected Budget).

Let be an optimal solution to (CP) when for all edges . Let be an indicator variable of whether is included in the returned subgraph by the rounding algorithm, Then,

###### Proof.

Note that an edge is contained in with probability . By the union bound, an edge is included in the returned subgraph by the rounding algorithm with probability

where the last equality holds by the property of the flow decomposition of the electrical flow in Lemma 3.4.

The key step is to show that . To prove this, we construct a unit - flow and show that , and hence by Thomson’s principle . To construct the flow , the idea is to follow the ratio in the fractional solution and send units of flow on each path selected.

###### Lemma 3.7 (Expected Effective Resistance).

###### Proof.

Consider the undirected flow vector defined by sending units of flow on each path

chosen by the rounding algorithm, i.e. the random variable

with for each edge . We would like to upper bound the expected energy in order to upper bound .Each is a random - path sampled from the flow decomposition of the undirected flow vector of the unit - electrical flow supported in , and is its characteristic vector with expected value

Since each edge in is of conductance one, the expected energy of in is

As each path is sampled independently, for ,

For ,

where the last equality follows from the property of the flow decomposition in Lemma 3.4. Combining these two terms, it follows that

Thomson’s principle states that the is upper bounded by the energy of any one unit - flow. Note that is an - flow of units, and by Lemma 3.5. Scaling to a one unit - flow by dividing the flow on each edge by gives an upper bound on

where the second inequality follows from Lemma 3.2 that for every edge and also for every edge , and the last equality is from Thomson’s principle that . Finally, notice that as . ∎

Combining Lemma 3.6 and Lemma 3.7, it follows from a simple application of Markov’s inequality that there is an outcome of the randomized path-rounding algorithm which uses at most edges with - effective resistance at most . In the following, we apply Markov’s inequality more carefully to show that the success probability is at least . In the next subsection, we will argue that can be assumed to be and so the path-rounding algorithm is a randomized polynomial time algorithm.

###### Theorem 3.8 (Bicriteria Approximation).

###### Proof.

First, we bound the probability that the subgraph has more than edges. Let be an indicator variable of whether the edge is included in the returned subgraph . Recall that and denote the set of fractional edges and integral edges in respectively. We assume pessimistically that all edges in will be included in the subgraph returned by the rounding algorithm. Then, by Markov’s inequality and Lemma 3.6,

Comments

There are no comments yet.