Reducibility of joint relay positioning and flow optimization problem

Reducibility of joint relay positioning and flow optimization problem

\authorblockNMohit Thakur \authorblockAInstitute for Communications Engineering,
Technische Universität München,
80290, München, Germany.
mohit.thakur@tum.de
   \authorblockNNadia Fawaz \authorblockATechnicolor Research Center,
Palo Alto, CA, USA.
nadia.fawaz@technicolor.com
   \authorblockNMuriel Médard \authorblockAResearch Laboratory for Electronics,
Massachusetts Institute of Technology,
Cambridge, MA, USA.
medard@mit.edu
Abstract

This paper shows how to reduce the otherwise hard joint relay positioning and flow optimization problem into a sequence a two simpler decoupled problems. We consider a class of wireless multicast hypergraphs mainly characterized by their hyperarc rate functions, that are increasing and convex in power, and decreasing in distance between the transmit node and the farthest end node of the hyperarc. The set-up consists of a single multicast flow session involving a source, multiple destinations and a relay that can be positioned freely. The first problem formulates the relay positioning problem in a purely geometric sense, and once the optimal relay position is obtained the second problem addresses the flow optimization. Furthermore, we present simple and efficient algorithms to solve these problems.

I Introduction

We consider a version of network planning problem under a relatively simple construct of a single session consisting of a source , a destination set and an arbitrarily positionable relay , all on a -D Euclidean plane. The problem can then be stated as: What is optimal relay position that maximizes the multicast flow from to ? Similarly, we can also ask: What is the optimal relay position that minimizes the cost (in terms of total network power) for a target multicast flow ?

A fairly general class of acyclic hypergraphs are considered. The hypergraph model is characterized by the following rules of construction of the hypergraph :

  1. consists of finite set of nodes positioned on on a -D Euclidean plane and a finite set of hyperarcs .

  2. Each hyperarc in emanates from a transmit node and connects a set of receivers (or end nodes) in the system. Also, each hyperarc is associated with a rate function that is convex and increasing in transmit node power and decreasing in distance between the transmit node and the farthest node spanned by the hyperarc in the system.

  3. Each end node spanned by the hyperarc can decode the information sent over the hyperarc equally reliably, i.e. all the end nodes of an hyperarc get equal rate.

In relation to the special case of our hypergraph model, the authors addressed the first question (max-flow) in the context of Low-SNR Broadcast Relay Channel in [1].

This paper has two major contributions. Firstly, we solve the general joint relay positioning and max-flow optimization problem for our hypergraph model. Secondly, we address the min-cost flow problem and establish a relation of duality between the max-flow and min-cost problems. An efficient algorithm that solves the joint relay positioning and max-flow problem is presented, in addition to an algorithm that solves an important special case of the min-cost problem.

The relay positioning problem has been studied in various settings [2, 3, 4]. In most cases, the problem is either heuristically solved due to inherent complexity, or approximately solved using simpler methods but compromising accuracy. We reduce the non-convex joint problem into easily solvable sequence of two decoupled problems. The first problem solves for optimal relay position in a purely geometric sense with no flow optimization involved. Upon obtaining the optimal relay position, the second problem addresses the flow optimization. The decoupling of the joint problem comes as a consequence of the convexity (in power) of hyperarc rate functions.

The next section develops the wireless network model. Section III presents the key multicast flow concentration ideas for max-flow and min-cost flow that are central to the reducibility of the joint problem. In Section IV, we present the algorithms and Section V contains an example where the results of this paper are applied. Finally, we conclude in Section VI.

Ii Preliminaries and Model

Consider a wireless network hypergraph consisting of nodes placed on a -D Euclidean plane with number of hyperarcs and the only arbitrarily positionable node as the relay . The node set consists of a source node , a relay and an ordered destination set (in increasing distance from ). Their positions on the -D Euclidean plane are denoted by the set of two-tuple vector .

All hyperarcs in are denoted by , where is the transmit node and is the ordered set (in increasing distance ) of end nodes of the hyperarc, and . The hyperarcs emanating from a transmitter node are constructed in order of increasing distances of the receivers from the transmitter (refer Figure 1). This construction rule captures the distance based approach and is analogous to time sharing for broadcasting. Note that, this is one technique to construct the hypergraph , our model allows arbitrary styles of hypergraph construction that follow the above three mentioned rules. Although, since time sharing is optimal for broadcasting we will stick to this technique as the main example in this paper. All the nodes in the set receive the information transmitted over the hyperarc equally reliably. Any hyperarc is associated with a rate function , where and denotes the fraction of the total transmit node power allocated for the hyperarc and the Euclidean distance between transmit node and the farthest end node , respectively.

Fig. 1: Hyperarcs are constructed in increasing order of distance from the transmitter. (a)-(c): node system. (a): Source hyperarc set - . (b): Relay hyperarc set - . (c): Hypergraph . (d)-(f): node system with such that and . (d) Source hyperarc set - . (e) Relay hyperarc set - . (f): Hypergraph .

The hyperarc rate function is increasing and convex in power and decreasing in . Furthermore, without loss of generality, we write the hyperarc rate function into two separable functions of power and distance

(1)

where is increasing and convex and is increasing. Mainly, we will be concerned with the first equation in (1). Moreover, to comply with standard physical wireless channel models we assume that

(2)

. If the functions and are not differentiable entirely in , then Inequality 2 can be rewritten with partial sub-derivatives, implying that differentiability is not imperative.

Denote the convex hull of the nodes in by . For a given relay position , let be the set of paths from to a destination and let be the set of paths from that span all the destination set , therefore . Moreover, any path in the system consists of either a single hyperarc or at most two hyperarcs as there are only two transmitters in the system. Let and denote the total given power of source and relay, respectively, and denote their ratio, where . Denote with and the flow over the path (for ) and the total flow to the destination , respectively, such that . Define to be the the multicast flow from to the destination set as the minimum among the total flows to each destination, then for a given relay position the multicast max-flow problem can be written as,

subject to: (3)
(4)

The hyperarc rate constraints and node sum-power constraints are denoted by the set in Program (A) for simplicity. Program (A) in general is non-convex, as the path flow function can be non-convex, e.g. let the path be , ( in Figure 1(f)), then .

Now we define the notion of cost for a given hyperarc rate . The cost of rate is given by the total power consumed by the hyperarc to achieve

(5)

where is the inverse function of that maps its range to its domain. Therefore, the total cost of multicast flow is simply the sum of powers of all the hypearcs in the system. Note that the function is increasing and concave, and if is convex then from Inequality (2), increasing and convex. So for a given relay position , the min-cost problem minimizing the total cost for setting up the multicast session with a target flow can be written as,

subject to: (6)
(7)

Constraint (6) makes sure that any destination receives a minimum of flow . Like in Program (A), we denote with the set the hyperarc rate and power constraints.

Finally, define the point , that will be crucial in developing algorithms in later sections, as

(8)

where, and . An easy way to understand is that if then is the circumcenter of two or more nodes in the set . Note that the program in Equation (8) is a convex program. Also, denote the optimal value of the objective function in Equation (8) as .

Hereafter, we represent with and the joint relay positioning and flow optimization problem instances that maximizes the multicast flow and minimizes the total cost for a the target flow , and with and denote the optimal relay positions, respectively.

Iii Multicast Flow Properties And Reduction

In this section we develop fundamental multicast flow properties that govern the multicast flow in the wireless network hypergraphs that we consider in this paper. First, we briefly note the main hurdles in jointly optimizing the problem. For a given problem instance different relay positions can result in different hypergraphs, which makes the use of standard graph-based flow optimization algorithms difficult. Moreover, the hyperarc rate function can be non-convex itself.

We will show that the joint problems and can be reduced to solving a sequence of two decoupled problems. The reduced problems are decoupled in the sense that the first problem is purely a geometric optimization problem and involves no flow optimization and vice versa for the second problem. At the same time, they are not entirely decoupled because the two problems need to be solved in succession and cannot be solved separately. Now we present a series of results that are fundamental to the reducibility of the joint problem.

Proposition 1

The optimal relay positions and lie inside the convex hull .

Refer Appendix A for the proof. Proposition 1 tells us that only the points inside the polygon need to be considered. This brings us to the following fundamental theorem.

Theorem 1 (Flow Concentration)

Given :

  1. the maximized multicast flow concentrates over at most two paths from to the destination set .

  2. for any target flow the min-cost multicast flow concentrates over at most two paths from to .

The proof is detailed in Appendix B. Theorem 1 is central to the two questions we aim to answer and reduces the complexity of joint optimization greatly by considering only two paths instead of many. Essentially, Theorem 1 tells that for a given relay position , the multicast flow must go only over the paths that span all the destination set , i.e. set . Furthermore, among the paths in , the maximized multicast flow goes over only two paths, namely the path that has the highest min-cut among all the paths through the relay , and path , which is the biggest hyperarc from spanning all the destination set , where and . The same holds for the min-cost case for a given relay position . Consequently, it is also true for the optimal relay positions and . Hereafter, we only need to consider the flow over paths and (corresponding to the relay position in consideration).

Iii-a Max-flow Problem -

Fig. 2: The solid piecewise linear segment in examples (a) and (b) marks the set of points for different values of . Each point corresponds to for some . The piecewise linear segment breaks beyond the dashed circle as . (a): E.g. with , . Same goes for the example in (b).

Assuming that the transmitted signal propagates omnidirectionally, we can geometrically represent the hyperarcs of the path by circles and centered at and with radii and (where and ), respectively. Similarly, the path can be represented by the circle with radius . Also, denotes the union region of the two circles. Then using Theorem 1, Program (A) can be re-written as,

where, and are the powers for hyperarcs of the paths and , respectively. The radii variables and correspond to path for the relay position such that and .

Although Program (C) is reduced, it is still a non-convex optimization problem. The objective function is non-convex and different positions of the relay result in different end node sets and for the hyperarcs of path .

On the other hand, we know that the relay position is sensitive only to the flow over path . In addition, as there always exist a relay position such that the min-cut of path is higher than that of path , then this also holds true for . Therefore, optimizing the relay position to maximize the flow over path results in global optimal relay position solving the original problem . This motivates the decoupling of computation of optimal relay position from the flow maximization over the path .

Proposition 2

For a given problem instance , if , then .

Refer Appendix C for the detailed proof. At point , in general the following holds (from Equation (8)), thus making it naturally a good candidate for . Proposition 2, essentially proves that if the relay is positioned at and we get , and if maximizing the flow over the path results in no spare source power (i.e. ), then and . Furthermore, the joint problem in Program (C) can be reduced to solving in sequence the computation of the optimal relay position by solving Equation (8) and then calculating the max-flow . But this is not true when . We cover this case in the section of algorithms.

Let us now see the problem in a different way. Consider the radius and construct the hyperarc . Denote by , the set of destination nodes that lie outside the hyperarc circle . Then compute the point such that

and position the relay at (here is the point in such that the maximum among the distances to the nodes in the set from is minimized). If , then we contract the hyperarc to , else we simply re-denote it with . Finally, we can construct the hyperarc ( note that ). The set of points computed in this way for different values of are the optimal relay positions solving for some . Figure 2(a) captures this interesting insight of the relationship between the points and . Note that the set of points is a discontinuous piecewise linear segment.

Iii-B Min-cost Problem And Duality

The min-cost problem can be written as

(9)
(10)

In the non-convex Program (D), the path correspond to the relay position which is implicitly represented in the distance variables and . From Theorem 1, we know that paths and carry all the min-cost target multicast flow . In this sub-section we refer the path as the cheapest path for a unit flow among all the paths through in for given position of relay.

Now, we claim that . This is true because given the hyperarc of path with optimal radius , the second hyperarc must be centered at the point that minimizes the maximum among the distances to all the destination nodes not spanned by the hyperarc from itself, as this minimizes the cost over the hyperarc . Therefore, (like ) always lie on on the curve . This observation motivates an interesting fundamental relationship between and .

Theorem 2 (Max-flow/Min-cost Duality)

For ,

(11)

where and .

Theorem 2 establishes the underlying duality relation between the max-flow problem and the min-cost problem and says that the point (or ) lies on the segment (, respectively) of the curve . Implying that the optimal relay position solving the problem is also the optimal relay position solving the problem for some . The proof of Theorem 2 is presented in Appendix D.

However, the max-flow is not always reducible to a sequence of decoupled problems. This is mainly due to the fact that the path can be cheaper than path for a unit flow corresponding to the optimal position , i.e.

This information is not easy to get a priori. In contrast, we can safely assume that

(12)

as almost all wireless network models that comply with our model result in the hyperarc cost function being the increasing convex function of distance that satisfy Inequality (12). If Inequality (12) holds, then similar to the Max-flow problem the joint optimal relay positioning and min-cost flow optimization problem in Program (D) can be reduced to a sequence of decoupled problems of computing the optimal relay position and then optimizing the hyperarc powers to achieve the min-cost flow in the network using the similar arguments as in previous subsection. For a special of the min-cost problem , we present the Min-cost Algorithm that sequentially solves and outputs the optimal relay position and powers to achieve the target flow in Section IV-B.

Iv Algorithms

In this section we present the general max-flow and the special case min-cost algorithms that solve the sequence of decoupled problems.

Iv-a Max-flow Algorithm

 

0:  Problem instance .  
1:  Compute , if , output , and quit, else go to .
2:  Construct the set (ordered in increasing distance from ) and compute . If , declare and and quit, else go to Step .
3:  Compute the points and , and maximized multicast flow and , respectively. Declare before quitting,
 
3:   and .

 

Fig. 3: Max-flow Algorithm.

The Max-flow Algorithm in Figure 3, is a simple and non-iterative step algorithm that outputs the optimal relay position and the maximized multicast flow. The first step is essentially Proposition 2, in case it is not satisfied the second step filters the redundant nodes that are too close to the source and can be ignored. If the conditions of first or second step are not met, then the third step divides the computation of into two regions of and computes the optimal relay position and for these two regions and outputs the better one. The proof of optimality is provided in Appendix E.

Iv-B Min-cost Algorithm

In this subsection, we assume that the Inequality (12) is satisfied and the target flow goes over the path (corresponding to the optimal relay position ) only. Min-cost Algorithm in Figure 4, unlike the Max-flow algorithm, is an iterative algorithm. In the first step the geometric feasibility region is constructed and in the second step this region is divided into at most sub-regions. The optimal relay position is computed for all the sub-regions and the one minimizing the cost among them is declared global optimal. Computing the optimal relay position for the sub-regions is a simple geometric convex program that can be solved efficiently and the number of such iterations is upper bounded by . The proof of optimality is presented in Appendix F.

 

0:  Problem instance and .  
1:  Compute , and build the set . If , then recompute , calculate and to go to Step .
2:  Build the set (ordered in increasing distance from ), compute the points
and calculate the cost of unit flow over the path corresponding to the relay position , . Declare
where , and and quit.  
2:  , and .

 

Fig. 4: Min-Cost Algorithm.

V Example: Low-SNR Achievable Network Model

In this section we present an example from the interference delimited network model that was originally presented in [1].

V-a Low-SNR Broadcast and MAC Channel Model

Consider the AWGN Low-SNR (wideband) Broadcast Channel with a single source and multiple destinations (arranged in the order of increasing distance from ). From [5] and [6], we know that the superposition coding is equivalent to time sharing, which is optimal. Implying that the broadcast communication from a single source to multiple receivers can be decomposed into communication over hyperarcs sharing the common source power. Therefore, we get the set of hyperarcs .

Similarly, in the Low-SNR (wideband) regime, interference becomes negligible with respect to noise, and all sources can achieve their point-to-point capacities analogous to Frequency Division Multiple Access (FDMA). In general, the MAC Channel consisting from sources transmitting to a common destination can be interpreted as point-to-point arcs each having point-to-point capacities. Thus, we get . Each hyperarc is associated with the rate function

(13)

where is the path loss exponent.

V-B Low-SNR Achievable Hypergraph Model

By concatenating the Low-SNR Broadcast Channel and MAC Channel models we obtain an Achievable Hypergraph Broadcast Model. For example the Broadcast Relay Channel consisting of a single source, destinations and a relay. Although, the time sharing and FDMA are capacity achieving optimal schemes in the respective models, the Achievable Hypergraph Model is not necessarily capacity achieving. In contrast and more importantly for practical use, this model is easy to scale to larger and more complex networks.

The above Low-SNR Achievable Hypergraph Model also incorporates fading [1]. The rate function in Equation (13) is linear in transmitter power and convex in hyperarc distance, hence the results from this paper can be directly applied.

Vi Conclusion

We present simple and efficient geometry based algorithms for solving joint relay positioning and flow (max-flow/min-cost) optimization problems for a fairly general class of hypergraphs. Any application that satisfies the hypergraph construction rules and can be modeled under the classical multicommodity framework can use the results presented here.

As a part of future work it would be of interest to extend the work presented here to the general multicommodity setting where multiple sessions use a common relay.

References

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Appendix A Proof of Proposition 1

Proof:

Let the set of nodes be placed on the -D Euclidean plane and denote their convex hull polygon. Let us assume that the relay node is placed outside the polygon , i.e. and be the nearest point to in the polygon . Let the line segment joining and be denoted as .

The rate over all the hyperarcs that either emanate from or is the farthest end node of the hyperarc, is relay position dependent. As the hyperarc rate is a decreasing function of distance, moving the relay closer to on the segment decreases the distance between and every point in the polygon and thus to every node in the system. This implies that for a given power allocation for the relay position dependent hyperarcs the rate can be increased as the relay gets closer to the point . Consequently, we can conclude that the optimal relay position maximizing the multicast flow for the session will lie in the convex hull polygon .

Similarly, all the relay position dependent hyperarcs will need lesser power to carry a given flow of value as the relay moves closer on the line segment to the the point . Implying, that for any target flow the optimal relay position will lie in . This concludes the proof. \qed

Appendix B Proof of Theorem 1

Before we formally prove Theorem 1, we need to establish some basic tools from convex analysis.

Let be an increasing and convex function that maps a non-negative real input to a non-negative real output. Denote with the sub-derivative of at the point and let denote the complete set of sub-derivatives at point . If the set is a singleton set, then , which is simply the derivative of the at ; else there exist a finite interval between the left and right limits of at . In addition, let us also assume that . Then the following proposition is true.

Proposition 3

If is any increasing convex function such that then

(14)
Proof:

As is increasing over the real line, for we have . Also, as is convex .

Let the slopes of line joining the points and , and be given by,

(15)

where are points on real line. From the Generalized Mean Value Theorem we know that there always exist a point and between and , such that and , respectively. This, along with the fact that is increasing and convex implies,

In general, given points with as the slope of line joining the points we get,

(16)

Consider now the four points , , and , where . From Inequality 16 we get,

(17)

Inequality 17 implies,

(18)

If , then . Using this fact it is straightforward to show that this also holds for , for any .

Without loss of generality, assume that , repeating the previous step of Inequalities 17-18 we get,

where . Similarly, repeating these times we have

(19)

if . This proves the proposition. \qed

Fig. 5: (a): Increasing convex function (possibly non-differentiable, e.g. at ). (b): Orange, blue and red lines joining the points , and , respectively.

Now let and be increasing convex functions satisfying Proposition 3 and define and as the linear compositions of the function and for , where , , and . Then consider the following program,

subject to:

Denote the set as the set of optimizers of Program (T1A). In addition, assume that

(20)

Let us denote a set of points