Dual power assignment optimization and fault tolerance in WSNs

J Comb Optim DOI 10.1007/s10878-013-9637-5

Dual power assignment optimization and fault tolerance in WSNs Nhat X. Lam · Trac N. Nguyen · Min Kyung An · Dung T. Huynh

© Springer Science+Business Media New York 2013

Abstract Because of limited battery equipped on each sensor, power consumption is one of the crucial issues in wireless sensor networks (WSNs). It therefore has been the focus of many researchers. An important problem concerning power consumption is to minimize the number of maximum-power nodes while maintaining a desired network topology. As fault tolerance is vitally important in practice, it is desirable that the constructed network topology is k-edge-connected or k-connected. In this paper, we study the dual power assignment problem for k-edge connectivity (k E D P) and biconnectivity in WSNs. While other studies consider only the special case k = 2, our goal is to address the general problem. In addition to showing the APX-completeness of biconnectivity problem in the metric model, we also prove the NP-completeness of the k E D P problem in the geometric case and provide a 2-approximation algorithm using linear programming techniques. To the best of our knowledge, this approximation ratio is currently the best one. We also introduce a heuristic whose performance is better compared with an approximation algorithm in Wang et al. (J Comb Optim 19:174–183, 2010). Keywords NP-completeness · Dual power · Connectivity · Approximation · Heuristic · Wireless sensor network

This paper is a revised and expanded version of Lam et al. (2011). It contains the APX-completeness proof for the bi-connectivity problem in the metric model. N. X. Lam · T. N. Nguyen · M. K. An · D. T. Huynh (B) Computer Science Department, University of Texas at Dallas, Richardson, TX, USA e-mail: [email protected]

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1 Introduction Wireless sensor networks (WSNs) have been a very active research area due to its wide applications in both military and civilian areas. These networks consist of sensor devices communicating with each other wirelessly. Due to physical limitations, these sensor devices are equipped with small batteries that have limited power. Therefore, energy consumption becomes one of the most crucial issues in this research area. A well-studied approach to achieve power efficiency is topology control. Specifically, the power usage at every sensor is adjusted to establish an energy efficient network, satisfying some desired features such as connectivity, low interference and fault tolerance. Concerning connectivity there are several interesting approaches regarding power consumption including minimizing the maximum-power utilized at any sensor nodes as well as minimizing the total power consumption of the whole network. While polynomial time algorithms are known for the former problem (Lloyd et al. 2005; Regina and Rosales-hain 2000), Kirousis et al. (2000) first proved the latter to be NP-hard and then proposed an approximation algorithm for this problem. Following this result, Lloyd et al. (2005) and Calinescu and Wan (2006) introduced different approximation algorithms which have better bounds. Along this line of research, considerable attention has also been given to the problem of minimizing the number of maximum power nodes. In the studies of Poojary et al. (2001) and Shah et al. (2004), applications of this problem have been analyzed. Subsequently, Lloyd et al. (2006) proposed an interesting result which is a reduction from minimizing the number of maximum-power nodes to minimizing the total power while preserving the approximation ratio. This implies that any α-approximation of the former is also an α-approximation of the latter. Recently, Jarray (2011) proposed an iterative exact solution using binary integer programming model for this problem. In this paper, we address the issue of fault tolerance in the dual power model. We investigate the problem of minimizing the number of maximum-power nodes in such a way that results in an edge-connected or vertex-connected topology. Recall that in the dual power model, only two discrete power levels are available to assign to the nodes. It is straightforward that in this model minimizing the number of maximum-power nodes is equivalent to minimizing total power. The connectivity problem in the dual power model was first studied by Rong et al. (2004). The authors provided an NP-completeness proof and proposed a 2approximation algorithm. The approximation ratio was later improved to 1.75 by Chen et al. (2005). While these studies focused only on the asymmetric version, Lloyd et al. (2006) in their study proved the NP-hardness of the symmetric version and gave an approximation algorithm with the ratio of 1.67. Addressing fault tolerance, Park et al. (2006) was the first to study the 2-edge connectivity and bi-connectivity problems. The authors proposed a 6-approximation for the former and a 5-approximation for the latter. Wang et al. (2008) proposed another algorithm to improve the ratio to 4 for both problems. They later improved the ratio to 3.67 Wang et al. (2010) by utilizing the algorithm of Lloyd et al. (2006). In this paper, we first prove the APX-completeness for the bi-connectivity problem in the metric model. Not only does this result imply the NP-completeness of this

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problem, but it also provides a lower bound for the approximation factor which is significant. We then address a more general problem, namely the dual power assignment problem for k-edge connectivity. We prove that this problem is NP-complete even in the (planar) geometric case. By formulating the problem as an integer linear programming problem, we design a 2-approximation algorithm using the iterated rounding method proposed by Jain (1998). We also study the performance of the 3.67-approximation algorithm in Wang et al. (2010) and our new heuristic through simulation. The rest of this paper is organized as follows. Section 2 provides the definitions and explanations used in this paper. Section 3 contains the APX-completeness proof. Section 4 is devoted to the NP completeness result, and Sect. 5 deals with the 2approximation algorithm. Section 6 discusses our heuristic and compares its performance against the approximation algorithm in Wang et al. (2010) through simulation. Finally, Sect. 7 contains some concluding remarks. 2 Network models Consider a set V of transceivers (nodes) in the plane. Each node u is assigned a power level denoted by p(u). The signal transmitted by node u can only received by a node v if the distance between u and v, denoted by d(u, v), is ≤ p(u). We only consider the bidirectional case (Wang et al. 2010) in which a communication edge (u, v) exists between two nodes u and v if and only if p(u) ≥ d(u, v) and p(v) ≥ d(u, v). Thus, the set V of nodes in the plane together with the power assignment p that assigns power levels to the nodes define a geometric (also known as intersection) graph G p = (V, E p ). In this paper we are specifically concerned with the dual power model as described in Wang et al. (2010). In this model, we use only two transmission power levels denoted by ph (high power) and pl (low power) where ( ph > pl ). A node is called high- (low-) power node if it is assigned high (low) power level ph ( pl ). Similarly, we define E h = {e(u, v)| ph ≥ d(u, v) > pl } and El = {e(u, v)|d(u, v) ≤ pl } as sets of high-power and low-power edges, respectively. The objective of our problem is to construct a power assignment so that the induced network is fault tolerant while minimizing the number of high-power nodes, thereby minimizing the total power consumption in the network. In this paper we consider the following two models of fault tolerance. 2.1 Bi-connectivity dual power assignment model This model was first studied in Park et al. (2006), aiming to construct a biconnected graph in which removing any node does not disconnect it. Denoted BDP, the optimization problem of this model can be formally defined as follows. Instance: Given a set of N nodes V = {v1 , v2 , . . . , v N } on the plane, a distance function d : V × V −→ R + and a set of two power levels P = { pl , ph } at which a node can transmit, and a positive integer Q ≤ N . Question: Is there a power assignment to each node so that it induces a symmetric bi-connected graph and the number of nodes assigned power level ph is bounded by Q?

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2.2 k-Edge connectivity dual power assignment model Recall that a k-edge connected graph is a graph in which removing any k − 1 edges does not disconnect it. The problem of minimizing the number of high-power nodes to obtain a k-edge-connected graph, denoted k E D P, can be formally defined as follows. Instance: Given a set of N nodes V = {v1 , v2 , . . . , v N } on a plane, a distance function d : V × V −→ R + , a set of two power levels P = { pl , ph } at which a node can transmit, and a positive integer Q ≤ N . Question: Is there a power assignment to each node so that it induces a symmetric k-edge-connected graph and the number of nodes assigned power level ph is bounded by Q? 3 The APX hardness of metric BDP In this section, we prove the APX-hardness for the BDP problem in the metric model (MBDP), where the distance function satisfies the triangular inequality. Theorem 1 MBDP is APX-hard. Proof This proof can be done by constructing an L-reduction from the Minimum kSet Cover (MkSC) problem which has been shown to be APX-complete (Duh and F ürer 1997). This problem is a variation of the well-known Minimum Set Cover problem in which the cardinality of all sets are bounded by a fixed constant k. Let I S be an instance of MkSC consisting of a collection S of subsets of a finite set E of elements where the cardinality of each set in S is ≤ k. (Note that k is a fixed constant). Let n and m denote the cardinalities of E and S, respectively. A solution to I S is a subset S  ⊆ S such that every element e ∈ E is in at least one set A ∈ S  . Letting pl and ph be two real numbers where ph  pl , we design the instance I M of MBDP from I S as follow: 1. For each element in the MkSC instance I S we construct an element gadget which is a group of 4 nodes forming a square shape. The distance between two neighboring nodes is pl . Let us call these nodes the element nodes in the MBDP instance I M . 2. Each set in I S is represented by 2 nodes which are at distance pl from each other. Let us call these nodes set nodes in I M (Nodes A, A , B, B  , C, C  in Fig. 1). Every two set nodes of I M representing a set in I S are separated such that one node belongs to group I (nodes A, B, C) and the other belongs to group II (nodes A , B  , C  ). 3. An element node of an element gadget is defined to have distance ph to a set node of one group (I or II) if this element is in that set of the instance I S ; another element node is also defined to be at distance of ph to the corresponding set node in the other group (II or I). This means that among the 4 element nodes in an element gadget, exactly one element node is at distance ph to a group of set nodes in group I whereas another element node of the same gadget is at distance ph to the corresponding group of set nodes in II if this element appears in these sets in the instance I S of MkSC.

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Fig. 1 Example of an MBDP instance constructed from an MBKSC instance

4. There is a distinguished node (node S) called the top node, which is at distance pl to every set nodes in both group I and group II. 5. All other distances follow from symmetry or are induced by the shortest path metric. Lemma 1 The MkSC instance I S with n elements and m sets has a set cover S O L S of size |S O L S | = η if and only if the corresponding MBDP instance I M with 4n +2m +1 nodes has a power assignment S O L M to all nodes that yields a biconnected graph and the total number of nodes assigned the high power level ph denoted by |S O L M | is |S O L M | = (2η + 2n). That is |S O L M | = 2|S O L S | + 2n. Proof For the “Only If” direction suppose that I S has solution S O L S of size η. We construct a power assignment S O L M for I M as follows. Assign the high power level ph to all pair of set nodes in I M that represent of sets in S O L S . For a group of 4 element nodes representing an element, assign the high power level ph to every two element nodes that have distance ph to the set nodes in group I or group II. Assign the low power level pl to all remaining element nodes and set nodes. Also assign the low power level pl to the top node. Obviously, this assignment yields a biconnected graph since all neighboring element nodes of an gadget are connected to each other and two of them are connected to a set node in group I and another set node in group II. The remaining set nodes in one group are connected to their corresponding set nodes in the other group. They are also connected to the top node that connects to every set node in both groups. The total number of high power nodes is 2η + 2n: each element gadget has two element nodes with power level ph and each set in S O L S has a set node in group I and a corresponding set node in group II using the high power level ph . For the “If” direction suppose that I M has a power assignment S O L M that yields a biconnected graph and the total number of high power nodes is λ. W.l.o.g. assume that S O L M is minimal in the sense that no high power node can be assigned the low power level pl without destroying the biconnectivity property. First note that in order for the resulting graph to be biconnected there must be exactly 2 element nodes in

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each element gadget that have the high power level ph to connect to some set nodes in groups I and II. Thus, there are a total of 2n element nodes that are assigned the high power level ph . This leaves λ − 2n set nodes in groups I and II to have the high power level in order to connect to element nodes. Due to the minimality of S O L M and the requirement that the resulting graph be biconnected, there must be an equal number of set nodes in group I and group II to be assigned the high power level ph to connect to element nodes also having the high power level ph . Thus, λ = 2n + 2η for some η. Now, w.l.o.g. consider group I and select the set nodes in I S which are represented by set nodes in group I having the high power level ph to be included in the set cover for I S . Clearly, this selection of sets is a set cover for I S of size η. This concludes the proof of the lemma.

Let O P TS denote the size an optimal solution to the MkSC instance I S and O P TM be the number of high power nodes in an optimal solution to MBDP instance I M . Observe that in I S there are n elements and each set in the collection S has size ≤ k. This implies that O P TS ≥ n/k. Now, noting that |S O L M | = 2|S O L S | + 2n we have O P TM = 2O P TS + 2n ≤ 2O P TS + 2k O P TS ≤ 2 · (k + 1)O P TS In addition, we also have: O P TM − 2n |S O L M | − 2n − 2 2 1 = (|S O L M | − O P TM ) 2

|S O L S | − O P TS =

Therefore, the above reduction from MkSC to MBDP is an L-reduction. Hence, the theorem follows.

Theorem 2 MBDP is APX-complete. Proof There exists an algorithm with a 3.67 approximation ratio presented in Wang et al. (2010). This result together with the APX-hardness result in the preceding theorem show that MBDP is APX-complete.

4 NP completeness of k E D P In this section, we prove the NP-completeness of the k E D P problem for geometric graphs. To obtain this result, it is sufficient to show the NP-completeness for a specific value of k. The following theorem is a result for the case k = 2.

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Theorem 3 2E D P is NP-complete. Proof Given a power assignment we can verify in polynomial time that the resulting graph is 2-edge-connected by removing any edge and check for connectivity, and that the number of nodes that are assigned power level ph is ≤ Q. To show that 2E D P is NP-hard, we construct a polynomial-time reduction from the Planar 2 Exact 3-SAT (Pl-2EX3SAT) problem. The Pl-2EX3SAT problem is the problem of determining if a CNF formula in which each clause has exactly 3 literals has a satisfying assignment such that exactly two literals per clause have the value TRUE. Note that if we negate the literals in an instance of Pl-1EX3SAT, which was proven to be NP-complete in Dyer and Frieze (1986), we have an instance of Pl2EX3SAT. Hence, the Pl-2EX3SAT problem is also NP-complete. From an instance of Pl-2EX3SAT we construct an instance of 2E D P as follow: Let F be an instance of Pl-2EX3SAT. Each clause in F is represented by a node called clause node in the instance of 2E D P. Let k be the maximum number of occurrences of a variable and its negation in F. For each variable x we create a variable gadget with 2 ∗ k literal nodes x and ˜ x (see Fig. 2). Between every pair of a variable node and its negation in the gadget, we add one extra node (node p in Fig. 2). For convenience, let us call a variable node and its negation variable nodes, and the new extra node variable-control node. The variable and variable-control nodes are connected to nodes (o’s and r ’s) placed on an inner cycle inside the variable gadget. These nodes (o’s and r ’s) are called variable-connect nodes. (In Fig. 2, nodes x’s and ˜ x’s are variable nodes, nodes p’s are variable-control nodes, and nodes o’s and r ’s are variable-connect nodes.) Each variable gadget uses 2 variable nodes to connect with other variable gadgets. Consecutive variable gadgets are connected by edges that link all variable gadgets together (see Fig. 3). To connect a variable node to a clause node, only variable nodes with degree 3 are used, and each such variable node is used to connect to at most one clause node if the variable or its negation appears in a clause in F. Thus, the degree of every node is bounded by 4.

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Fig. 3 The chain of variable gadgets contains a series of single and variable gadget

We now use Valiant’s result (Valiant 1981) to embed a planar graph with maximum degree 4 into the Euclidean plane: A planar graph with maximum degree 4 can be embedded in the plane using O(|V |) area in such a way that its vertices are at integer coordinates and its edges are drawn so that they are made up of line segments of form x = i or y = j, for integers i and j. After the embedding, we modify the edges of the graph already embedded in the plane to create an instance of 2E D P as follows. Let d be a unit distance in the plane. We define two distances d2 = d/10 and d1 = d2 /10. 1. On the edge connecting a variable node with a clause node, starting at distance d2 from the variable node, we add consecutive nodes at distance d1 , i.e., adjacent nodes are at distance d1 apart. We call these nodes auxiliary nodes. We now place additional nodes to the auxiliary nodes to form bridges to ensure 2edge connectivity in certain parts of the graph. Consider the three edges connecting a clause node with variable nodes. Being embedded in the plane, two of these edges are collinear and the other is perpendicular. √ On each of the collinear edges, one set of new nodes are added at distance (d1 / 2) from each consecutive pair of auxiliary nodes to create isosceles right triangles (see Fig. 4). On the perpendicular edge from the variable node to a clause node, we add a similar set of new nodes to the auxiliary nodes. However, the last new node is added at distance d1 from the auxiliary nodes (see node u in Fig. 4). (This adds the additional property of planarity in the resulting graph.) Let us call all these newly added nodes bridge nodes, and the last auxiliary nodes at distance d2 from a variable node variableinterface nodes (see nodes t’s).

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Fig. 5 Placing bridge nodes from a variable node to a variable-connect node

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r r Fig. 6 Placing node between a variable-control node and two neighboring variable nodes connected by collinear edges

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2. On the edge connecting a variable node to a variable-connect node (nodes o’s), we add auxiliary nodes such that consecutive nodes√are at distance d1 apart. We also add bridge nodes as done above at distance (d1 / 2) from each pair of consecutive auxiliary nodes to create an isosceles right triangles forming bridges as before (see Fig. 5). 3. On the two edges connecting a variable-control node (nodes p’s) to both of its neighboring variable nodes, starting from distance d2 from both variable nodes, we add auxiliary nodes at distance d1 from each other. The first added nodes are also called variable-interface nodes (nodes √ k’s and t’s in Figs. 6, 7). As before, we also add bridge nodes at distance (d1 / 2) from each pair of consecutive auxiliary nodes to create isosceles right triangles forming bridges (Figs. 6, 7). 4. On the remaining edges we place nodes at distance d1 from each other. We now have a set of nodes on the plane. To complete the construction of the instance of 2E D P problem, we define the power levels pl , ph , and the positive integer Q as follows: pl = d1 , ph = d2 , Q = n ∗ (3 ∗ k ∗ u) + 2 ∗ m where n and m are the numbers of variables and clauses in F, respectively.

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~x

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p d1

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r As this construction is clearly computable in polynomial time, we proceed to show the correctness of the reduction. For the “only if” direction, suppose that the Pl2EX3SAT instance F is satisfiable. Let φ be a Boolean assignment that satisfies F such that each clause in F has exactly 2 literals with the value TRUE. We assign the power level ph to the nodes as follows: 1. If a variable x is assigned value TRUE, then the variable node x in the 2E D P instance is assigned power level ph ; otherwise the variable node ˜ x is assigned the power level ph . 2. The variable-interface nodes which “interface” with the variable nodes having power level ph are assigned power level ph . 3. All remaining nodes are assigned power level p1 . Thus, in every variable gadget for variable x either k variable nodes x or their negations ˜ x are assigned power level ph . The variable-interface nodes of each of these variable nodes within the gadget are also assigned ph . Hence, in each gadget there are 3 ∗ k nodes with power level ph . Similarly, for the three variable nodes connected to a clause node, two of them have power level ph , and therefore two variable-interface nodes are assigned the power level ph . Thus, the total number of nodes assigned power level ph is n ∗ (3 ∗ k ∗ u) + 2 ∗ m which is ≤ Q. Now let us examine the graph resulting from this power assignment and show that it is 2-edge-connected. First consider the edges that link all variable gadgets together in a circular fashion. Obviously, at least two nodes/edges must be removed for the graph to be disconnected. Also, due to the bridges, each of the connections within a variable gadget requires at least 2 edges to be removed for the graph to be disconnected. Finally, for the connections from variable nodes to a clause nodes, at least 2 nodes have to be removed for the graph to be disconnected since there are exactly two literals with value TRUE in a clause. For the “if” direction suppose that for the 2E D P instance there is a power assignment that results in a 2-edge-connected geometric graph such that the total number of nodes assigned power level ph is ≤ Q. Without loss of generality assume that the number of nodes with power level ph is minimum. We show how to construct a Boolean assignment that satisfies F. We have these observations:

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1. Each clause node must be connected to at least 2 different variable nodes for the graph to be 2-edge-connected. This requires that at least 2 variable-interface nodes of each clause have power level ph . While one variable-interface node with power level ph does not make the graph 2-edge-connected, three variable-interface nodes with power level ph would violate the fact that number of nodes with power level ph is mimimum since one of the three variable-interface nodes with ph can be assigned power level pl , and 2-edge-connectivity is preserved. 2. For each variable gadget, at least 2 ∗ k nodes on the edges from the variablecontrol nodes to the variable nodes must have power level ph for the graph be 2-edge-connected. 3. For the minimum number of 3 ∗ k nodes in each variable gadget to have power level ph , exactly k variable nodes x or their negations ˜ x must have the power level ph to connect to the variable-control nodes. From the above observations, we can assign a variable in F the value TRUE if its variable node in the 2E D P instance has the power level ph . Observation 1 ensures that each clause will have exactly 2 variables with value TRUE, whereas Observations 2 and 3 guarantee that the Boolean assignment is consistent. This concludes the proof of Theorem 3.

Remarks The reader should note that the proof of Theorem 3 also holds for vertex connectivity. Thus, the dual power assignment optimization problem for k-vertex connectivity is also NP-complete for geometric graphs. Furthermore, the reduction in the proof yields a geometric graph that is planar, i.e., no edge crosses another. Hence the result applies to planar geometric graphs as well. 5 2-Approximation algorithm for k E D P In this section, we first construct a linear programming formulation for k E D P and then utilize the approach in Jain (1998) to design for it a 2-approximation algorithm. 5.1 Formulation Let δ E (S) := {e(u, v) ∈ E|u ∈ S ⊕ v ∈ S} where S ⊂ V (E). In other words, δ E (S) is the set of edges in E with exactly one endpoint in S. In addition, we use E(v) to denote a subset of E containing all edges incident to v and V (E) to denote a set of all vertices incident with edges in E. Since the objective of the k E D P problem is to minimize the number of high power nodes, all low power edges can be used freely without increasing the value of the objective function. So the main issue is to efficiently pick appropriate high power edges to obtain a k-edge connected graph. For every high power edge e we use variable xe to express whether this edge is used xe = 1 or not xe = 0. We also need to formulate the requirement of k-edge connectivity. Recall that a k-edge connected graph is a graph in which every cut has at least k edges crossing it. So this requirement can be formulated as a restriction on every cut of the induced graph. Now the k E D P problem can be formulated as follows.

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IP1 min s.t.



max xe

e∈E h (v) v∈V 

xe ≥ k − |δ El (S)| for all S ⊆ V

e∈δ E h (S)

xe ∈ {0, 1}

for all e ∈ E

The system of inequalities in the above formulation means that any feasible solution which is a selection x of xe , e ∈ E h , satisfies that requirement that for every cut in the induced graph, there are at least k edges crossing this cut. The relaxation of IP1 is: LP2  max xe min e∈E h (v) v∈V  s.t. xe ≥ k − |δ El (S)| for all S ⊆ V e∈δ E h (S)

0 ≤ xe ≤ 1

for all e ∈ E

Although the objective function is not linear, LP2 is equivalent to the following linear programming formulation.  yv min v∈V

s.t.xe ≤ yv yv ≤



xe

for all v ∈ V and e ∈ E h (v) for all v ∈ V

 e∈E h (v)

xe ≥ k − |δ El (S)| for all S ⊆ V

e∈δ E h (S)

0 ≤ yv ≤ 1 0 ≤ xe ≤ 1

for all v ∈ V for all e ∈ E

However, it is not straightforward that LP2 is solvable in polynomial time since the number of constraints is exponential. Recall that while running the Ellipsoid method, each iteration of this algorithm either checks if the center is feasible or returns a violated constraint. And it is known that finding a violated constraint for any infeasible solution can be done in polynomial time by utilizing the Max-flow Min-cut Theorem. The solution is not feasible if and only if there exists a pair of nodes whose min-cut corresponds to a violated inequality. So we assume that such a separation oracle for solving LP2 is given. 5.2 IR-APPROX algorithm In this subsection we describe a 2-approximation algorithm in which we first solve LP2 fractionally and then use the iterated rounding technique first introduced in Jain

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Fig. 8 Iterated Rounding

(1998) to convert a fractional solution of LP2 into an integral solution which can be shown to be a 2-approximate solution of IP1. Let x∗ be an optimal basic solution of LP2 found by the given separation oracle. Denoting F = {e|xe∗ ≥ 21 }, we round xe∗ for all e in F to 1 and solve the residual linear programming problem which can be formulated as follows. LP3  max xe min s.t.

v∈V \V (F) 

e∈E h (v)

xe ≥ k − |δ El ∪F (S)| for all S ⊆ V

e∈δ E h \F (S)

0 ≤ xe ≤ 1

for all e ∈ E h \ F

We iteratively apply this rounding method for the residual problem to construct F until the graph induced by El ∪ F is k-edge connected. The corresponding integral solution to IP1 can be obtained as follows.  1 e∈F xe = 0 e ∈ Eh \ F  We also have v∈V maxe∈E h (v) x e = |V (F)|. So for convenience, we call F an integral solution of IP1. The pseudo-code of this algorithm can be found in Fig. 8. 5.3 Performance analysis In this subsection, we will show that IR-APPROX produces a 2-approximate solution in polynomial time. First, let us recall the fundamental result of Jain (1998) for the iterated rounding technique. Lemma 2 In any basic solution to LP2 and LP3, there is one component whose value is at least 21 . From Lemma 2, it follows that at every iteration at least one edge is rounded, and hence, the number of iterations is bounded by |E h |. Thus, in order to prove that IR-

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APPROX is in P, it is sufficient to show how to find a basic solution to LP3 efficiently. It is clear that LP3 has a tight relation to LP2. Therefore, we can also construct a separation oracle for LP3 from the given one for LP2. The construction can be found in Jain (1998). We now derive the approximation ratio of 2. Let the optimal values of LP2 and LP3 be z ∗ and zr∗es , respectively. We have the following lemma and theorem which are similar to the ones in Jain (1998). Lemma 3 If Q is an integral solution of LP3 with value at most 2zr∗es , then Q ∪ F is an integral solution to LP2 with value at most 2z ∗ . Proof Clearly, Q ∪ F is a feasible integral solution to LP2. Notice that the projection of x∗ on V \ V (F) is a feasible solution to LP3, hence we have zr∗es ≤ z ∗ −

 v∈V (F)

max xe

e∈E h (v)

For any v ∈ V (F), we know that there exists e ∈ E h (v) such that xe ≥ 21 . Hence it follows that 2z ∗ ≥ 2zr∗es + |V (F)| By the assumption, we have |V (Q)| ≤ 2zr∗es . Therefore 2z ∗ ≥ |V (Q)| + |V (F)| ≥ |V (Q ∪ F)|

Theorem 4 IR- APPROX produces a 2-approximation. Proof The optimal solution to IP1 is also a feasible solution to LP2. Hence, we have 2opt ≥ 2z ∗ . So, applying Lemma 3 inductively we obtain the approximation ratio.

6 Heuristic and performance for kEDP Although the IR-APPROX algorithm theoretically produces a 2 approximation factor for the k-edge connectivity problem, its implementation may not be practical since it requires linear programming. Therefore, in this section we first review the approximation algorithm proposed by Wang et al. (2010), and then propose a new heuristic whose performance is much better in experiments. The rest of this section is organized as follows. The first two subsections describe the approximation algorithm in Wang et al. (2010) and our heuristic. The last subsection contains the experimental results that show a better performance of our heuristic.

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6.1 Wang et al.’s 3.67-approximation algorithm In the following, we will briefly describe the Candidate Sets Filtering Algorithm proposed by Wang et al. (2010) to solve the dual power assignment problem for 2edge connectivity. This 3.67-approximation algorithm consists of two steps. The first step is to utilize the 1.67-approximation algorithm proposed by Lloyd et al. (2006) to construct a rooted spanning tree. In the second step, the algorithm modifies this tree to obtain a 2-edge connected graph. This can be done by first using the low power edges to cover “bridges”, which are defined to be edges whose removal disconnects the graph, and then trying with the maximum-power edges. It is easy to check if low power edges could help to cover bridges in the first step. The main issue is how to pick good high power edges in the second step so as to minimize the number of high power nodes. The authors define a candidate set of high power edges for each node. Their algorithm traverses the tree in a depth-first manner to construct candidate sets and also filter them. Finally, these sets contain only “good” edges. It is sufficient to pick one in each set to obtain the desired graph. The authors also claimed an O(N 3 ) time complexity for this algorithm. The reader is referred to Wang et al. (2010) for further details. 6.2 Heuristic In this subsection, we provide a simple greedy heuristic. The basic idea of this heuristic comes from the following lemma, which follows from Proposition 3.1.3 in Diestel (2000). Recall that an H -path of a subgraph H of G is defined to be a simple path starting from a node s ∈ H , going through several nodes v ∈ G \ H and stopping at a node t∈H . Lemma 4 Given a 2-edge connected graph G. Starting from some 2-edge connected subgraph H of G, we can constructed a 2-edge connected spanning subgraph by successively adding H -paths to H . Our heuristic proceeds in a greedy fashion. It starts with a trivial 2-edge connected component H and successively selects in each iteration a proper H -path to expand the current subgraph H until it becomes a 2-edge-connected spanning subgraph of the original graph G. The major issue is the greedy property used to determine the proper H -path at each step. Since our objective is to minimize the number of high power nodes, it is intuitively reasonable to consider the number of high power nodes required in an H path as its cost. Following this approach we define the cost of an H -path T as the amount of additional (based on the current status) high power nodes required in order to construct the connection through all nodes in T . So at each greedy iteration, we simply pick a minimum-cost H -path to expand the current subgraph H . The heuristic is described in the pseudo-code in Fig. 9. The remainder of this subsection focuses on designing an algorithm to find a minimum-cost H -path. For a given node s ∈ H, ws (u, p) denotes the cost of a minimum-cost path from s to u, where u is assigned power level p ∈ {0, pl , ph }.

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Fig. 10 RELAXATION

Once all values ws (u, p) are computed, we simply pick the minimum-cost (s0 , u 0 , p0 ) = argmin{ws (u, p)|s ∈ H, u ∈ H \ {s}, p ∈ {0, pl , ph }} to construct the desired H -path. It is easy to compute ws (u, p) following the approach of Bellman-Ford algorithm (Cormen et al. 2001). We also use relaxation (Fig. 10) to progressively decrease an estimate ws (u, p) on the value of a minimum-cost path from s to each vertex u ∈ V until it becomes the actual minimum one. Figure 11 is the pseudo-code of the algorithm to find a minimum-cost H -path starting from a given source node s in H .

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The reader should note that an H -path may be a cycle, and we can compute minimum-cost H -cycles as follows. Given s ∈ H , for each edge (s, u), u ∈ / H, we find the minimum-cost path from u to s. Together with (s, u), this will form a minimum-cost cycle. Finally, we pick the best one among all cycles found. To compute the minimum-cost paths, we use πs (u, p) to keep track the predecessor of (u, p) in the path from s. This algorithm will set πs so that we can traverse from any node back to s to obtain the corresponding optimum path. It is straightforward to see that the Minimum- Cost- H- Path algorithm runs in time O(N 3 ). The I nitiali zation step which simply sets all values as infinity, requires O(N ) time while each of the O(V ) outer iterations checking over all edges takes O(N 2 ) steps. Thus, the time complexity of this procedure is O(N 3 ). This procedure is applied for each source node s in H . Therefore, each iteration in H- Path- Construction requires O(N 4 ) time. In addition, after every iteration, we expand H by at least one node. Hence, the number of iterations is bounded by the number of vertices. Thus, the heuristic has O(N 5 ) time complexity. 6.3 Simulation In our experiments, we only consider the geometric model. We randomly generate 100 nodes on an area of size of 1000 × 1000. We then find the minimum power level pmin that induces a 2-edge-connected graph. Eventually, ph is picked randomly from the interval ( pmin , dmax ], where dmax is the maximum distance between any pair of nodes, and pl is picked randomly from the interval (0, pmin ). We generate 100 cases in our simulation. For each case, we run the two algorithms and compare their performance. The results are shown in Table 1. As seen in this table, in 78 out of 100 cases, the results of our heuristic are 19.50 % better on average. In the 7 cases where our heuristic perTable 1 Simulation results

(Over 100 cases)

Worse

Better

Cases

7

78

Maximum value

6.25 %

56.25 %

Average value

1.98 %

19.50 %

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forms worse, its results are only 1.98 % worse on average. While among the worse cases, the worst value is at most 6.25 %, there is a better case where our heuristic performs 56.25 % better. Thus, our heuristic performs much better than the approximation algorithm of Wang et al. overall. 7 Conclusions In this paper, we have studied fault tolerance issues on the metric and geometric networks. We assumed the dual power model which uses two energy levels to assign for every node and have shown that the problem MBDP is APX-complete. In addition, we have also proved that kEDP is NP-complete and introduced a 2-approximation algorithm using linear programming techniques. This approximation ratio beats the current record proposed by Wang et al. (2010). Finally, we proposed a new heuristic which performs better than the algorithm of Wang et al. in simulations. References Calinescu G, Wen PJ (2006) Range assignment for biconnectivity and k-edge connectivity in wireless ad hoc networks. Mob Netw Appl 11(2):121–128 Chen J-J, Lu H-I, Kuo T-W, Yan C-Y, Pang A-C (2005) Dual power assignment for network connectivity in wireless sensor networks. In: GLOBECOM ’05. IEEE, vol 6, pp 5–3642 Cormen TH, Stein C, Rivest RL, Leiserson CE (2001) Introduction to algorithms, 2nd edn. McGraw-Hill Higher Education, New York Diestel R (2000) Graph theory. Springer, Berlin Duh R-c, F ürer M (1997) Approximation of k-set cover by semi-local optimization. In: Proceedings of the twenty-ninth annual ACM symposium on theory of computing. ACM, New York, NY, USA, pp 256–264 Dyer M, Frieze A (1986) Planar 3DM is NP-complete. J Algorithms 7(2):174–184 Jain K (1998) A factor 2 approximation algorithm for the generalized steiner network problem. In: Proceedings of the 39th annual symposium on foundations of computer science, pp 448–457 Jarray F (2011) An iterative exact solution for the dual power management problem in wireless sensor network. J Math Model Algorithms 10(2):205–212 Kirousis LM, Kranakis E, Krizanc D, Pelc A (2000) Power consumption in packet radio networks. Theor Comput Sci 243(1–2):289–305 Lam NX, Nguyen TN, An M-K, Huynh DT (2011) Dual power assignment optimization for k-edge connectivity in WSNs. In: IEEE SECON 2011, pp 566–573 Lloyd EL, Liu R, Marathe MV, Ramanathan R, Ravi SS (2005) Algorithmic aspects of topology control problems for ad hoc networks. Mob Netw Appl 10(1–2):19–34 Lloyd EL, Liu R, Ravi SS (2006) Approximating the minimum number of maximum power users in ad hoc networks. Mob Netw Appl 11(2):129–142 Park M-A, Wang C, Willson J, Wu W, Farago A (2006) Fault- tolerant dual-power assignment in wireless sensor networks (Technical Report Nos. UTDCS -52-06). University of Texas at Dallas Computer Sciences Department Poojary N, Krishnamurthy S, Dao S (2001) Medium access control in a network of ad hoc mobile nodes with heterogeneous power capabilities. In: IEEE international conference on communications (ICC), vol 3, pp 872–877 Regina RR, Rosales-hain R (2000) Topology control of multihop wireless networks using transmit power adjustment. In: IEEE INFOCOM, vol 2, pp 404–413 Rong Y, Choi H, Choi H-A (2004) Dual power management for network connectivity in wireless sensor networks. In: International parallel and distributed processing symposium, p 225 Shah V, Krishnamurthy S, Poojary N (2004) Improving the mac layer performance in ad hoc networks of nodes with heterogeneous transmit power capabilities. In: 2004 IEEE international conference on communi- cations, vol 7, pp 3874–3880

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