Chance-Constrained Programming Models and

Chance-Constrained Programming Models and Approximation Algorithms for General Stochastic Bottleneck Spanning Tree Problems Jue Wang Department of Industrial and Operations Engineering University of Michigan, Ann Arbor, MI 48109

Siqian Shen Department of Industrial and Operations Engineering University of Michigan, Ann Arbor, MI 48109, [email protected]

Murat Kurt Department of Industrial and Systems Engineering University at Buffalo (State University of New York)

Abstract: This paper considers a balance-constrained stochastic bottleneck spanning tree problem (BCSBSTP) in which edge weights are independently distributed but may follow arbitrary random distributions. The problem minimizes a scalar and seeks a spanning tree, of which the maximum edge weight is bounded by the scalar for a given certain probability, and meanwhile the minimum edge weight is lower bounded in another chance constraint. The paper formulates the BCSBSTP as a mixed-integer nonlinear program, and develops two mixed-integer linear programming approximations by using special ordered set of type one (SOS1) and special ordered set of type two (SOS2) variables. By relaxing the chance constraint on the minimum edge weight, the BCSBSTP is simplified as a stochastic bottleneck spanning tree problem, for which a bisection algorithm is developed to approximate optimal solutions in polynomial time. Based on the properties derived in the development of the bisection algorithm, we show that the BCSBSTP is NP-Complete. We demonstrate the computational results by testing all models and algorithms on a diverse set of graph instances with edge weights that are independently distributed. Keywords: Stochastic bottleneck spanning tree; chance-constrained programming; special ordered set; bisection algorithm; NP-Completeness

1.

Introduction

The Minimum Spanning Tree (MST) problem is one of the most well-studied network flow problems, and efficient algorithms can be found for tackling problems where arc costs are deterministic (Ahuja et al. 1993, Kruskal 1956, Prim 1957). The MST is closely related to problems of designing telecommunication systems. For example, consider wireless sensor networks used for environmental monitoring and data gathering, which are often maintained as minimum spanning trees because sensor nodes are energy constrained (Cheng et al. 2003, Tan and Körpeo˘glu 2003). The MST also arises in many other applications of designing pipelines or transmission lines, electronic circuits, connecting islands, leased-line telephone networks, laying cables in resident communities, etc. 1

In practice, network models excluding balance requirements may produce impractical solutions. Especially when elements of the associated network involve uncertainty, the necessity of such requirements becomes extremely important for ensuring solution robustness. Extended from the MST, a deterministic bottleneck spanning tree problem minimizes the maximum edge weight in a spanning tree. When network elements are uncertain, a Stochastic Bottleneck Spanning Tree problem (SBSTP) imposes a chance constraint (e.g., Charnes et al. 1958, Miller and Wagner 1965, Prékopa 1970) on the maximum edge weight, and minimizes a target upper bound in the chance constraint. The SBSTP is applied to address problems of link failure, message delay, and to guarantee a quality of service for building communication networks (e.g., Akyildiz et al. 2002, Seapahn et al. 2001). The SBSTP was first studied by Ishii and Nishida (1983) for edge weights that follow identically independently normal distributions. Ishii and Shiode (1995) considered a generalized SBSTP problem that aimed to determine an optimal reliability level associated with the chance constraint given a cost function measuring the selected reliability. They developed polynomial-time algorithms for two special functions of the reliability. Their algorithms involved a major step which relied on determining a set of breakpoints, as intersection points of cumulative probability functions of weights of edges emanating from a node. However, determining these intersection points can be extremely difficult even for the same type of cumulative probability functions. This paper starts with a balance-constrained stochastic bottleneck spanning tree problem (BCSBSTP), which is extended from the SBSTP by imposing an additional chance constraint on the minimum edge weight in the same spanning tree. Edge weights are independently distributed random variables defined on a common sample space, and are allowed to be characterized by any probability distributions. The paper formulates the problem as a mixed-integer nonlinear program, and approximates the reformulation with special ordered set of type one (SOS1) and special ordered set of type two (SOS2) variables. Moreover, we develop a polynomial-time algorithm for approximating solutions of general SBSTP with independently distributed edge weights. An exact optimal solution can be obtained when edge weights have the same distribution type and are allowed to vary one parameter that defines the distribution. We also show that the BCSBSTP is NP-Complete by transforming an NP-Complete subset sum (Garey and Johnson 1979) into a special instance of the problem. It is worth noting that the Sample Average Approximation (SAA) approach is a common Monte Carlo simulation-based method for bounding optimal objective values of general stochastic programs (see, e.g., Mak et al. 1999, Norkin et al. 1998). Shapiro and Homem-de-Mello (2000) applied the SAA for solving stochastic linear programs, and Kleywegt et al. (2002) extended the approach to stochastic discrete optimization problems. In particular, Luedtke and Ahmed (2008) used the SAA to approximate optimization problems with chance constraints. Their paper defined appropriate numbers of samples and scenarios to guarantee that an approximate solution yielded a lower bound or a feasible solution to the original problem with a certain confidence level. As a result, a finite number of scenarios was generated to represent the original random distribution, and the problem was solved as an equivalent deterministic integer program. 2

Both BCSBSTP and SBSTP discussed in this paper can be solved by applying the SAA and transforming the probabilistic models into mixed-integer programs. However, the sample size needed to ensure a good-quality solution is in general very large (Luedtke and Ahmed 2008). Moreover, one needs to optimize a deterministic intractable integer program for every scenario of a sample. In this paper, we use SOS1-and SOS2-based approximations for solving general BCSBSTP and derive a parametric MST-based bisection algorithm for approximating general SBSTP. The remainder of the paper is organized as follows. Section 2 describes the BCSBSTP and formulates it as a deterministic model with nonlinear terms and binary integer variables. Section 3 proposes SOS1- and SOS2-based approximations and investigates algorithms for solving both approximations. In Section 4, we elaborate details of a polynomial-time bisection algorithm for approximating SBSTP solutions. Section 5 employs properties used in the development of the bisection algorithm, and shows that the BCSBSTP is NP-Complete. Section 6 presents the computational results of the SBSTP and BCSBSTP with different settings. Section 7 concludes the paper and proposes future research directions.

2.

Problem Description and Formulations

Let G = (V, E) be an undirected connected graph with V = {v1 , ..., v|V | } and E = {e1 , ..., e|E| }. A spanning tree T = (V, ET ) of G is a connected acyclic spanning subgraph of G such that ET ⊆ E X and |ET | = |V | − 1. The MST and the bottleneck spanning tree problem minimize wj and j: ej ∈ ET

max wj , respectively, where wj is the weight of edge ej . Note that every minimum spanning

j: ej ∈ ET

tree is also a bottleneck spanning tree when edge weights are deterministic.

2.1

The BCSBSTP in a Probabilistic Form

In this paper, we assume that each edge ej has a random weight wj , characterized by a given continuous probability distribution independent from distributions of other edge weights. Denote T (G) as the set of all spanning trees of graph G, and α, β and κ are input parameters with α, β ∈ (0, 1). Let P be a probability measure defined on a σ-algebra based on a common sample space Ω, where all random variables wj are defined. We formulate the BCSBSTP as Q := min `:P max wj ≤ ` ≥ α, P min wj ≥ κ ≥ β , (1) T ∈ T (G)

j:ej ∈ ET

j:ej ∈ ET

where variable ` is a scalar and represents the α-quantile of the maximum edge weight in ET for some spanning tree T . Let Fj (·) be the cumulative probability function of weight wj , i.e. Fj (r) = P(wj ≤ r). Given order statistics of the independently distributed random weights (see, e.g., Bapat and Beg 1989), we have Y X P max wj ≤ ` ≥ α ⇔ Fj (`) ≥ α ⇔ log Fj (`) ≥ log α, and (2) j: ej ∈ ET

j: ej ∈ ET

j: ej ∈ ET

3

min

P

j: ej ∈ ET

wj ≥ κ ≥ β ⇔

Y

X

1 − Fj (κ) ≥ β ⇔

j: ej ∈ ET

log 1 − Fj (κ) ≥ log β.

(3)

j: ej ∈ ET

We transform Problem Q into the following equivalent nonlinear optimization program:     X 0 Q := min `: log Fj (`) ≥ log α ,  T ∈T 0 (G) 

(4)

j:ej ∈ ET

where   T 0 (G) = T ∈ T (G) : 

X j:ej ∈ ET

  log 1 − Fj (κ) ≥ log β 

(5)

represents the set of spanning trees that satisfy the balance constraint in the BCSBSTP. When κ is sufficiently small, T 0 (G) = T (G), and the BCSBSTP is reduced to the SBSTP. We discuss the feasibility of the SBSTP and the BCSBSTP as follows. For nonempty set T (G), it is obvious that the SBSTP is always feasible, because ` can be set sufficiently large to allow Q Fj (`) ≥ α for some tree solution T ∈ T (G). The feasibility of the BCSBSTP depends on j: ej ∈ ET

the value of κ. Define Fj−1 (·) as the inverse cumulative distribution function of edge weight wj , ∀ej ∈ E. Let o n 1 κ = min Fj−1 (1 − β |V |−1 ) . (6) j:ej ∈ E

Given nonempty set T (G), the BCSBSTP is feasible for a sufficiently small κ ∈ (−∞, κ ]. The 1 1 reasons are elaborated as follows. If κ ≤ κ = min Fj−1 (1−β |V |−1 ), then κ ≤ Fj−1 (1−β |V |−1 ), ∀ej ∈ j:ej ∈ E 1 1 Q −1 E. Hence, Fj (κ) ≤ Fj Fj (1 − β |V |−1 ) = 1 − β |V |−1 , ∀ej ∈ E, and thus [1 − Fj (κ)] ≥ j: ej ∈ ET

β, ∀ET ⊆ E, ET = |V | − 1. Similar arguments are used later in Section 3.3.1 to compute upper and lower bounds of the optimal `.

2.2

A Mixed-Integer Nonlinear Reformulation

We define binary decision variables xj , where ( 1 if edge ej ∈ ET , xj = 0 otherwise. Also, define binary decision variables δjk , where

δjk =

   1  

0

if vk ∈ V can be reached from some source node v1 ∈ V through a path containing edge ej ∈ E, otherwise.

4

Let Evi+ = {(vi , vj ) ∈ E : ∀vj ∈ V } and Evi− = {(vj , vi ) ∈ E : ∀vj ∈ V }, representing sets of outgoing and incoming arcs of node vi , respectively. We transform Q0 into a mixed-integer nonlinear program (MINLP): min: s.t.

` X

xj log Fj (`) ≥ log α

(7a)

xj log 1 − Fj (κ) ≥ log β

(7b)

xj = |V | − 1

(7c)

j:ej ∈ E

X j:ej ∈ E

X j:ej ∈ E

X

δjk −

j:ej ∈ Ev

(7d)

δjk = 0 ∀k = v2 , . . . , v|V | , ∀vr ∈ V − {v1 , k}

(7e)

1−

δjk −

j:ej ∈ Ev

X j:ej ∈ Ev

r−

r+

δjk ≤ xj

δjk = 1 ∀k = v2 , . . . , v|V |

j:ej ∈ Ev

1+

X

X

∀ej ∈ E, ∀k = v2 , . . . , v|V |

(7f)

xj ∈ {0, 1}

∀ej ∈ E

(7g)

δjk ∈ {0, 1}

∀ej ∈ E, ∀k = v2 , . . . , v|V |

(7h)

where (7a) and (7b) are equivalent to the two chance constraints in Problem Q. Constraint (7c) enforces that each spanning tree solution only contains |V | − 1 edges, and (7d)–(7e) ensure a path from a source node v1 to every other destination node k ∈ V − {v1 }. That is, (7d) guarantees that all nodes are connected, by sending one unit of flow from v1 to each node in V − {v1 }, and (7e) ensures that each node vr ∈ V − {v1 , k} is a transshipment node with respect to a path from node v1 to a destination node k, ∀k ∈ V − {v1 }. If edge ej ∈ E is not in set ET of a tree T , then it cannot be used to form the path enforced by (7f). Constraints (7g) and (7h) ensure that x- and δ-variables are binary valued, respectively. With ` being a variable, the nonlinear terms xj log Fj (`) in (7a) cause major difficulties for efficiently solving Formulation (7) to optimality. Next in Sections 3, we describe two approximations to address the problem.

3.

SOS1- and SOS2-based Approximations

We approximate log Fj (`) by using SOS1 and SOS2 variables defined on a sequence of discrete sample points from function log Fj (·). In general, SOS1 is used when only one of several variables can take on a nonzero value, and SOS2 is used when at most two adjacent variables can take on nonzero values. We refer the interested readers to Beale and Tomlin (1970) for applications of special ordered sets to nonconvex problems, and Sridhar et al. (2012) for applying SOS2 variables to approximate nonlinear functions in production planning problems with increasing byproduct. Dentcheva et al. (2002) employed SOS approximations for bounding optimal solutions to probabilistically constrained integer programs, and illustrated the methods on an example of a vehicle 5

routing problem. Also, we refer toBeale and Forrest (1976), Forrest et al. (1974) and Tomlin (1988) for other implementations of SOS-based approximations. General ideas include reformulating (7) by replacing function log Fj (`) with functions defined on extra binary variables, and linearizing nonlinear terms by using McCormick Inequalities (McCormick 1976). Given ` and ` as respective estimated lower and upper bounds for the maximum edge weight in a spanning tree, we generate a sampling sequence as {`1 , . . . , `n }, where ` = `1 < · · · < `n = `. The corresponding function values are calculated as {log Fj (`1 ), . . . , log Fj (`n )} for each edge ej ∈ E. Sections 3.1 and 3.2 provide formulations of the SOS1 and SOS2 approximations.

3.1

An SOS1-Based Approximation

We approximate functions log Fj (`), ∀ej ∈ E via discrete points {(`1 , log Fj (`1 )), . . . , (`n , log Fj (`n ))}. For each k = 1, . . . , n, define a binary variable zk , such that zk = 1 if ` equals to `k , and zk = 0 otherwise. We approximate Formulation (7) by min:

n X

zk `k

(8a)

k=1

s.t.

(7b)–(7h) ! n X X xj zk log Fj (`k ) ≥ log α j:ej ∈ E n X

(8b)

k=1

zk = 1

(8c)

k=1

zk ∈ {0, 1}

∀k = 1, . . . , n,

(8d)

where (8b) approximates (7a), (8c) enforces that only one of `k ∀k = 1, . . . , n equals to `, and (8d) ensures binary values of zk variables. McCormick Inequalities are used to linearize terms zk xj in Formulation (8). We define decision variables okj to replace bilinear terms zk xj , ∀k = 1, . . . , n, ej ∈ E, and rewrite Formulation (8) as min:

n X

zk `k

k=1

s.t.

(7b)–(7h), (8c), (8d) n X X okj log Fj (`k ) ≥ log α

(9a)

j:ej ∈ E k=1

okj ≤ zk

∀k = 1, . . . , n, ∀ej ∈ E

(9b)

okj ≤ xj

∀k = 1, . . . , n, ∀ej ∈ E

(9c)

okj ≥ zk + xj − 1 ∀k = 1, . . . , n, ∀ej ∈ E

(9d)

okj ≥ 0

(9e)

∀k = 1, . . . , n, ∀ej ∈ E,

where given zk ∈ {0, 1}, ∀k = 1, . . . , n, (9b)–(9e) are the McCormick constraints for linearizing okj , for all k = 1, . . . , n and ej ∈ E. 6

3.2

An SOS2-Based Approximation

Alternatively, we approximate functions log Fj (`) by a piecewise linear continuous function defined over a set of intervals [`k , `k+1 ] for all k = 1, . . . , n − 1. Given ` ∈ [`k , `k+1 ], we express ` as a convex combination of `k and `k+1 , i.e., ` = ρk `k + ρk+1 `k+1 , ρk + ρk+1 = 1, ρk , ρk+1 ≥ 0. Moreover, log Fj (`) ≈ ρk log Fj (`k ) + ρk+1 log Fj (`k+1 )

∀ej ∈ E.

To identify the interval to which ` belongs, define binary variables yk such that yk = 1 Pn if ` ∈ [`k , `k+1 ], and yk = 0 otherwise. By replacing ` with k=1 ρk `k , and log Fj (`) with Pn k=1 ρk log Fj (`k ) for all ej ∈ E, an SOS2-based approximation of Formulation (7) reads n X

min:

ρ k `k

k=1

s.t.

(7b)–(7h) ! n X X xj ρk log Fj (`k ) ≥ log α j:ej ∈ E n X k=1 n−1 X

(10a)

k=1

ρk = 1

(10b)

yk = 1

(10c)

k=1

ρ1 ≤ y1

(10d)

ρi ≤ yi + yi−1

∀i = 2, . . . , n − 1

ρn ≤ yn−1 yk ∈ {0, 1} ρk ≥ 0

(10e) (10f)

∀k = 1, . . . , n

∀k = 1, . . . , n,

(10g) (10h)

where (10a) approximates (7a). Constraints (10b)–(10f) are SOS2 constraints such that (i) the sum of all ρk is one (10b), (ii) ` locates in only one interval (10c), and (iii) if ` ∈ [`k , `k+1 ], then only ρk and ρk+1 are positive (10d)–(10f). Similar to (9b)–(9e), bilinear terms ρk xj , ∀k = 1, . . . , n, ej ∈ E are replaced with qkj ≡ ρk xj . We reformulate Formulation (10) as min:

n X

z k `k

k=1

s.t.

(7b)–(7h), (10b)–(10h), n X X qkj log Fj (`k ) ≥ log α j:ej ∈ E k=1

7

(11a)

qkj ≤ ρk

∀k = 1, . . . , n, ∀ej ∈ E

(11b)

qkj ≤ xj

∀k = 1, . . . , n, ∀ej ∈ E

(11c)

qkj ≥ ρk + xj − 1

∀k = 1, . . . , n, ∀ej ∈ E

(11d)

qkj ≥ 0 ∀k = 1, . . . , n, ∀ej ∈ E.

3.3

(11e)

Solving SOS1- and SOS2-based Approximations

We now describe an algorithm for solving SOS1- and SOS2-based approximations, i.e., Formulations (9) and (11). For a given graph G(V, E) with parameter α ∈ (0, 1), we first calculate initial lower and upper bounds as `0 and `0 . Then, we equally divide [`0 , `0 ], and generate a sampling sequence {`01 , . . . , `0n } for some pre-determined n as the number of break points. Next, compute log Fj (`k ), ∀k = 1, . . . , n, ∀ej ∈ E and solve Formulation (9) (or (11)) to obtain an optimal objective ∗ ∗ value `0 . Suppose that `0 locates in [`0k0 , `0k0 +1 ] for some k 0 = 1, . . . , n−1. Set [`1 , `1 ] = [`0k0 , `0k0 +1 ] as the new bounds. We repeat the procedures and re-solve Formulation (9) (or (11)) to obtain a ∗ ∗ ∗ new objective value `t at iteration t. The process is terminated when |`t−1 − `t | ≤ ∆, where ∆ is a pre-specified error tolerance. 3.3.1

Computing valid `0 and `0 .

For both Formulations (9) and (11), let n o 1 Fj−1 (α |V |−1 ) j:ej ∈ E n o 1 = max Fj−1 (α |V |−1 ) ,

`0 = `0

min

(12a) (12b)

j:ej ∈ E

Next we show the validity of the two bounds in (12a) and (12b) by demonstrating that any optimal objective value `∗ to Problem Q will locate between the two values. Proposition 1 is applicable to cases where edge weights are independently distributed and follow general continuous distributions. Proposition 1. Let `∗ and T ∗ be the optimal objective value and a corresponding spanning tree to Problem Q. Then Y Fj (`∗ ) = α, (13) j: ej ∈ ET ∗

for any continuous cumulative distribution function Fj (·) of edge weight wj , ∀ej ∈ E. Q Proof. Suppose that Fj (`∗ ) > α. Given continuous Fj (·) of edge weight wj , ∀ej ∈ E, j: ej ∈ ET ∗ Q because Fj (·) is continuous and nondecreasing, there exists a `− < `∗ such that j: ej ∈ ET ∗

Y j: ej ∈ ET ∗

Fj (`∗ ) >

Y

Fj (`− ) ≥ α,

j: ej ∈ ET ∗

contradicting that `∗ is the optimal objective to Problem Q. This completes the proof.

8

Theorem 1. Let `∗ and T ∗ be the optimal objective value and a corresponding spanning tree to 1 1 Q Fj (`∗ ) = α. Problem Q. Then `∗ ≤ max Fj−1 (α |V |−1 ), `∗ ≥ min Fj−1 (α |V |−1 ), and j:ej ∈ E

j:ej ∈ E

j: ej ∈ ET ∗

1

1

Proof. We first show that max Fj−1 (α |V |−1 ) is a valid upper bound. let ` =

max Fj−1 (α |V |−1 ),

j:ej ∈ E

then ` ≥ Fj−1 (α

1 |V |−1

j:ej ∈ ET ∗

), ∀ej ∈ ET ∗ . Therefore,

1 1 Fj (`) ≥ Fj Fj−1 (α |V |−1 ) = α |V |−1 , ∀ej ∈ E, and

Y

Fj (`) ≥ α,

j: ej ∈ ET ∗ 1

implying that ` ≥ `∗ . Moreover, max Fj−1 (α |V |−1 ) ≥ j:ej ∈ E

1

max Fj−1 (α |V |−1 ) = `, given ET ∗ ⊆ E

j:ej ∈ ET ∗ 1

for any given spanning tree T ∗ . Meanwhile, min Fj−1 (α |V |−1 ) is a valid lower bound. Let ` = j:ej ∈ E

1

min

j:ej ∈ E

Fj−1 (α |V |−1 ),

then ` ≤

1

Fj−1 (α |V |−1 ),

∀ej ∈ ET . As a result,

1 1 Fj (`) ≤ Fj Fj−1 (α |V |−1 ) = α |V |−1 , ∀ej ∈ E, and thus

Y

Fj (`) ≤ α,

j: ej ∈ ET ∗ 1

implying that ` ≤ `∗ . We complete the proof by deriving min Fj−1 (α |V |−1 ) ≤ ` given ET ∗ ⊆ E for any given spanning tree T ∗ . 3.3.2

j:ej ∈ E

1

min

j:ej ∈ ET ∗

Fj−1 (α |V |−1 ) =

Solving the approximations.

Algorithm 1 demonstrates a bisection method for optimizing either the SOS1 or SOS2 approximation of Problem Q. Algorithm 1 A bisection algorithm for solving Formulation (9) (or (11)). 1: Input a connected undirected graph G(V, E), number of intervals n, probability level α and error tolerance ∆. 2: Set the current iteration n t : = 0.1 o n o 1 −1 −1 t 3: Compute ` : = min Fj (α |V |−1 ) and `t : = max Fj (α |V |−1 ) . j:ej ∈E

4: 5: 6: 7: 8:

9: 10:

j:ej ∈E

repeat Generate an equally distributed sequence {`t1 , . . . , `tn } in between interval [`t , `t ]. Compute log Fj (`tk ) ∀ej ∈ E, k = 1, . . . , n. ∗ Solve Formulation (9) (or (11)) and record the current optimal objective value `t . ∗ For (9), if `t = `tkt , set `t+1 : = `tkt −1 and `t+1 : = `tkt +1 . ∗ For (11), if `t ∈ [`tkt , `tkt +1 ], set `t+1 : = `tkt and `t+1 : = `tkt +1 . Set t : = t + 1. ∗ ∗ until |`t−1 − `t | ≤ ∆

9

Example 1. Figure 1 shows a network G = (V, E) with |V | = 6, |E| = 9 and α = 0.95. We relax κ as −∞ in this example, and demonstrate Algorithm 1 on an SBSTP instance. Each edge weight in the network follows an exponential distribution such that wj ∼ Exp(λj ), j = 1, . . . , 9. The number alongside each edge represents λj .

Figure 1: The topology of graph G(V, E) and parameter λj , ∀ej ∈ E.

Table 1: Demonstrating Algorithm 1 on Example 1. Iteration t

SOS1

SOS2

∗

`t

∗

ETt

`1

`2

`3

`4

`5

`6

0

0.458

0.825

1.192

1.559

1.926

2.292

1.192

(1-3) (2-5) (3-5) (4-6) (5-6)

1

0.825

0.972

1.119

1.265

1.412

1.559

1.119

(1-3) (2-5) (3-5) (4-6) (5-6)

2

0.972

1.031

1.089

1.148

1.207

1.265

1.031

(1-3) (2-5) (3-5) (4-6) (5-6)

3

0.972

0.995

1.019

1.042

1.066

1.089

1.042

(1-3) (2-5) (3-5) (4-6) (5-6)

4

1.019

1.028

1.038

1.047

1.056

1.066

1.028

(1-3) (2-5) (3-5) (4-6) (5-6)

5

1.019

1.023

1.026

1.030

1.034

1.038

1.023

(1-3) (2-5) (3-5) (4-6) (5-6)

`6 2.292 1.192 1.045

t∗

Iteration t 0 1 2

`1 0.458 0.825 0.972

`2 0.825 0.899 0.987

`3 1.192 0.972 1.001

`4 1.559 1.045 1.016

`5 1.926 1.119 1.031

` 1.077 1.021 1.019

∗

ETt (1-3) (2-5) (3-5) (4-6) (5-6) (1-3) (2-5) (3-5) (4-6) (5-6) (1-3) (2-5) (3-5) (4-6) (5-6)

To implement Algorithm 1, we set ∆ = 0.01, and n = 6. Table 1 presents a sequence of sample ∗ points, the current optimal objective value `t , and a spanning tree solution in each iteration t. For SOS1, the circled, underlined, and overlined numbers in each iteration t indicate the current ∗ optimal objective value `t , lower and upper bounds for iteration t + 1, respectively. For SOS2, the underlined and overlined numbers in each iteration t respectively indicate lower and upper bounds used for iteration t + 1. n o √ Taking SOS1 as an example, we first calculate initial lower and upper bounds, `0 = min Fj−1 ( 5 0.95) = j:ej ∈E

10

−ln(1 −

√ 5

0.95)/10 = 0.458 and `0 = max

n o √ √ Fj−1 ( 5 0.95) = −ln(1 − 5 0.95)/2 = 2.292. Af-

j:ej ∈E 0 {`1 , . . . , `06 }

ter generating a sampling sequence in between interval [0.458, 2.292] and computing ∗ ∗ 0 0 {log Fj (`1 ), . . . , log Fj (`6 )}, ∀ej ∈ E, we solve Formulation (9) to obtain `0 = 1.192. As `0 = `03 , we let `02 and `04 be the new lower and upper bounds in iteration 1. The process terminates at ∗ ∗ iteration 5 with |`5 − `4 | = |1.023 − 1.028| < ∆ = 0.01. The optimal objective value `∗ is 1.023, and the corresponding spanning tree solution for this example is: {(1-3) (2-5) (3-5) (4-6) (5-6)}.

4.

Solving General SBSTP via a Bisection Algorithm

In this section, we relax the lower-bound chance constraint on the minimum weight in the BCSBSTP, and analyze the SBSTP. As demonstrated in Example 1, one can use Formulation (7) to solve SBSTP by letting κ = −∞ and relaxing (7b). Thus, both SOS1 and SOS2 approximations are still applicable. Here we propose a bisection algorithm for approximating general SBSTP with independently distributed edge weights. Consider a parametric maximum spanning tree formulation as X Q0 (r) := max log Fj (r) for r ∈ R, (14) T ∈ T (G)

j: ej ∈ ET

which is a nondecreasing continuous function of r shown in Ishii and Nishida (1983), and `∗ ∈ R is optimal if and only if Q0 (`∗ ) = log α. Although their result is derived for graphs with normally distributed edge weights, it holds for all graphs with any type of continuous probability distributions that are independent. (The result is also not affected by whether T 0 (G) = T (G), and thus in next section we use the result in the NP-Completeness analysis of the BCSBSTP.) For self-contained purpose, we present the following results. Property 1. (Ishii and Nishida 1983) Q0 (r) is a continuous function nondecreasing in r. Theorem 2. (Ishii and Nishida 1983) Let `∗ be the optimal objective of Problem (4). (i) Q0 (r) > log α ⇔ `∗ < r, (ii) Q0 (r) = log α ⇔ `∗ = r, (iii) Q0 (r) < log α ⇔ `∗ > r.

4.1

4.1.1

An Approximation Algorithm for SBSTP with General Weight Distributions An Overview.

Our algorithm consists of two parts: • Part (i): for any given set of |V |−1 edges and parameter α, we derive upper and lower bounds for seeking a value r∗ , such that the summation of log Fj (r∗ ) for all edge ej among the |V | − 1 edges equals to log α. • Part (ii): apply procedures in Part (i) to two sets of |V | − 1 edges that are specially selected, to calculate an upper and a lower bound for searching the optimal objective value. 11

4.1.2

Solution Properties and Valid Bounds.

We derive some basic structural properties of the SBSTP which we make use of later to develop a polynomial-time approximation algorithm. The properties mainly rely on the fact that any spanning tree of graph G consists of |V | − 1 edges. Recall that Fj−1 (·) is the inverse cumulative probability function of edge weight wj , for all ej ∈ E. For a subset of edges S ⊆ E with |S| = |V | − 1, define X • hS (r) = log Fj (r) for all r ∈ R, j: ej ∈ S 1

1

• bmin (S, α) = min Fj−1 (α |V |−1 ), and bmax (S, α) = max Fj−1 (α |V |−1 ), j: ej ∈ S

j: ej ∈ S

n o 1 • Sα := ej ∈ E : bmin (S, α) ≤ Fj−1 (α |V |−1 ) ≤ bmax (S, α) , • For a set of edges S 0 with S 0 ⊇ S: n o 1 1 −1 −1 |V |−1 ) : h |V |−1 ) ≤ log α F (α , F (α S j j j: ej ∈ S 0 n o 1 1 US 0 (S, α) = min 0 Fj−1 (α |V |−1 ) : hS Fj−1 (α |V |−1 ) ≥ log α . LS 0 (S, α) = max

j: ej ∈ S

Let p0 (S, α) be the value for which hS p0 (S, α) = log α, i.e., the desired value r∗ , such that the sum of log Fj (r∗ ) for all edge ej ∈ S equals to log α. Proposition 2. LSα (S, α) ≤ p0 (S, α) ≤ USα (S, α), for all S ⊆ E with |S| = |V | − 1.

(15)

Proof. For any two edges ek and ek0 ∈ Sα , without loss of generality, we suppose 1 1 |V |−1 ) ≥ log α. hS Fk−1 (α |V |−1 ) ≤ log α, and hS Fk−1 0 (α

Because hS (r) is a nondecreasing continuous function of r, this implies that there must exist a point 1 1 min (S, α) |V |−1 ) such that h (r 0 ) = log α. Next, consider the values F r0 ∈ Fk−1 (α |V |−1 ), Fk−1 0 (α j b S and Fj bmax (S, α) for every edge j ∈ S. Because Fj (r) is nondecreasing in r, by definitions of bmin (S, α) and bmax (S, α): 1 1 log α, log Fj bmin (S, α) ≤ log α, ∀ej ∈ S. (16) |V | − 1 |V | − 1 Given |S| = |V | − 1, (16) implies hS bmin (S, α) ≤ log α ≤ hS bmax (S, α) . By definition, we have bmax (S, α) ≥ USα (S, α) and bmin (S, α) ≤ LSα (S, α), which guarantee the existence and validity of bounds in (15). We complete the proof. log Fj bmax (S, α) ≥

For a given S ⊆ E with |S| = |V |−1, Proposition 2 provides lower and upper bounds of p0 (S, α) for any given (|V | − 1)-cardinality set S ⊆ E and a fixed α. Because hS (r) is nondecreasing in r, these bounds allow us to numerically approximate p0 (S, α) by a bisection type of interval elimination method. 12

1

To obtain initial bounds (denoted by L1 and U1 ), we first calculate Fj−1 (α |V |−1 ) for all ej ∈ E. For some pre-selected (|V |−1)-cardinality set S, we identify edges in Sα and let L1 = LSα (S, α) and U1 = USα (S, α). If the middle point of interval [L1 , U1 ], denoted by m(S, α), satisfies hS m(S, α) = log α, then p0 (S, α) = m(S, α). Otherwise, we either update L1 or U1 to the current middle point, and repeat the previous steps. Denote 1 as a sufficiently small number such that we stop when U1 − L1 < 1 . Algorithm 2 elaborates the details. Algorithm 2 A bisection approach for calculating p0 (S, α) for a given set S and α. 1: Input an edge set S ⊆ E with |S| = |V | − 1, values of α, and 1 2: Initialize L1 = LSα (S, α), U1 = USα (S, α) 3: while U1 − L1 > 1 do 4: Set m(S, α) = (L1 + U1 )/2 and compute hS m(S, α) 5: if hS m(S, α) = log α then 6: exit the while loop 7: else 8: if hS m(S, α) > log α then 9: set U1 = m(S, α) 10: else 11: set L1 = m(S, α) 12: end if 13: end if 14: end while 15: return p0 (S, α) = m(S, α) α α and Emin as edge sets with the largest and the smallest |V | − 1 values of Consider Emax 1 −1 α | = |E α | = |V | − 1. Proposition 3 provides tighter bounds |V |−1 ), respectively, where |Emax Fj (α min of the optimal SBSTP objective `∗ compared with the bounds given by Theorem 1.

Proposition 3. α α p0 (Emin , α) ≤ `∗ ≤ p0 (Emax , α). α , α) and p0 (E α , α). Because F (r) is nondecreasing in r, the Proof. Consider points p0 (Emin j max α α definitions of Emin and Emax imply α α hE 0 (r) ≤ log α ∀r ≤ p0 (Emin , α), and hE 0 (r) ≥ log α ∀r ≥ p0 (Emax , α),

(17)

for all E 0 ⊆ E with |E 0 | = |V |−1, which are also true for all spanning trees. The result immediately follows from Property 1 and Theorem 2. 4.1.3

Algorithm Details.

According to Proposition 3, we compute lower and upper bounds of the optimal SBSTP objective α , α) and U = p0 (E α , α). Similar to Algorithm 2, we approximate `∗ by value as L2 = p0 (Emin 2 max 13

using Algorithm 3, where X ∗ represents an optimal spanning tree solution. Algorithm 3 A bisection approach for solving `∗ of general SBSTP. 1: Input a graph G = (V, E) with random edge weights wj , functions Fj , α, and 2 . α , α), U = p0 (E α , α) by applying Algorithm 2. 2: Initiate L2 = p0 (Emin 2 max 3: while U2 − L2 > 2 do ˆ defined in (14). 4: Set `ˆ = (L2 + U2 )/2 and solve Q0 (`) ˆ be the corresponding maximum spanning tree solution. 5: Let X 0 ˆ = log α then 6: if Q (`) 7: exit the while loop 8: else ˆ > log α then 9: if Q0 (`) 10: set U2 = `ˆ 11: else 12: set L2 = `ˆ 13: end if 14: end if 15: end while ˆ ˆ `) 16: return (X ∗ , `∗ ) = (X, ˆ For a fixed `, ˆ solve Q0 (`) ˆ In Algorithm 3, we compute the middle point of interval [L2 , U2 ] as `. ˆ in Formulation (14) and compute the corresponding maximum spanning tree by setting log Fj (`) ˆ if as the cost associated with each arc ej ∈ E. According to Property 1 and Theore 2, `∗ = `, ˆ = log α. Otherwise, we either set L2 = `ˆ or U2 = `, ˆ and repeat the foregoing steps. Denote Q0 (`) 2 as a sufficiently small number such that we stop the iterations until U2 − L2 < 2 . 4.1.4

Analysis of the Approximation and Complexity.

α , α) and p0 (E α , α) as initial lower In Algorithm 3, we first use Algorithm 2 to calculate p0 (Emin max and upper bounds of `∗ . Therefore, if we consecutively implement Algorithms 2 and 3 with precision levels 1 and 2 , respectively, the solution provided by Algorithm 3 will be optimal by a precision at most 1 + 2 . Algorithm 3 terminates in

O(log2

α α ) × O(Maximum Spanning Tree) (p0 (Emax , α) − p0 (Emin , α) −1 2

steps. α , α) ≥ p0 (E α , α), Algorithm 3 can α (E α , α) ≤ p0 (E α , α) and UE α Given that LEmin (Emax max min min max be directly implemented without applying Algorithm 2. Instead, we initialize upper and lower bounds as α α α (E α L2 = LEmin (Emax , α), (18) min , α), and U2 = UEmax yielding an optimal solution with a precision at most 2 . Therefore, there are two possible schemes to implement Algorithm 3 and the performance of each scheme depends on specific SBSTP instances. 14

4.2

A Special Case of the SBSTP

We analyze a special case of the SBSTP where each wj follows the same type of probability distribution but we allow one parameter d in the distribution to vary. For example, assume that each edge weight follows normal distribution with the same mean value, but different standard deviation. We denote the cumulative probability function of an edge weight by F (d, `), i.e., F (dj , `) = P(wj ≤ `), implying that the distribution of each edge weight wj is characterized by parameter dj . We also assume that F (d, `) is monotone in d for all ` ∈ R. This assumption can be satisfied by most of the well-known probability distributions such as uniform, normal, log-normal, exponential, gamma and beta.

Figure 2: Illustrating the definition and property of F (d, `) by two example distributions. Figure 2 illustrates the definition and monotonic change of F (d, `) that is affected by the change of parameter d for all ` ∈ R. We use two examples as an exponential distribution Exp(λ) with variable λ, and a uniform distribution U (a, b) with fixed a = 0 and variable b. The horizontal axis reflects continuous values of `, and the vertical axis provides the values of F (d, `) where d corresponds to λ in Exp(λ), and corresponds to b in U (a, b). Functions F (d, `) of the two distributions are increasing in λ, and decreasing in b for all `. Corollary 1 proposes an approach to find optimal tree solutions by solving a parametric maximum or minimum spanning tree depending on the monotonicity of parameter d. Corollary 1. Denote 0

T ∈ arg max

  X

T ∈T 0 (G)  j:ej ∈ET

dj

 

T 0 ∈ arg min

,

  X

T ∈T 0 (G)  j:ej ∈ET



0 (i) If F (d, r) is nondecreasing in d, then (X ∗ , `∗ ) = T , p0 (ET 0 , α) , 15

dj

  

.

(ii) If F (d, r) is nonincreasing in d, then (X ∗ , `∗ ) = T 0 , p0 (ET 0 , α) . Proof. We provide the proof of case (i) and omit the other which is similar. By definition, ET 0 is the set of edges with the largest |V | − 1 values of parameter d in the feasible set T 0 (G) that do not form a cycle. Because F (d, `) is nondecreasing in d, it must contain the set of edges with the 0 largest |V | − 1 values of function F (d, `) for any ` ∈ R. Therefore, T must be the optimal solution to Problem (14) for all ` ∈ R. As a result, it is sufficient to seek the optimal objective value for all edges in ET 0 . Given p0 (S, α) with S = ET 0 , the result directly follows from Theorem 2. According to Corollary 1, under special conditions imposed on the cumulative probability functions of edge weights, when κ = −∞ and T 0 (G) = T (G), an exact optimal tree solution of the SBSTP can be obtained by solving an ordinary maximum or minimum spanning tree problem. The corresponding optimal objective value `∗ can be computed by only applying Algorithm 2. Note that Corollary 1 also holds for general BCSBSTP (i.e., Problem Q in (4)). Corollary 2 derives tighter bounds for (4) under the assumption that all edge weights follow the same type of distributions. Corollary 2. Suppose F (d, `) is monotone in d. Let κ = −∞ and,      X   X  dj , T ∈ arg min dj . T ∈ arg max   T ∈T (G)  T ∈T (G)  j:ej ∈ET

j:ej ∈ET

Define E ∗ = ET ( E ∗ = ET ) if F (d, `) is nondecreasing (nonincreasing) in d. Then, α α max p0 (Emin , α), LEα∗ (E ∗ , α) ≤ `∗ ≤ min p0 (Emax , α), UEα∗ (E ∗ , α) . Proof. Because κ = −∞, we have T (G) = T 0 (G), which implies that `∗ = p0 (E ∗ , α) according to Corollary 1. According to Proposition 2, it further implies LEα∗ (E ∗ , α) ≤ `∗ ≤ UEα∗ (E ∗ , α). By applying Proposition 3, we complete the proof.

5.

NP-Completeness of the BCSBSTP

The general BCSBSTP can be shown NP-Complete by applying Corollary 1. Theorem 3. BCSBSTP is NP-Complete. Proof. First, given a BCSBSTP instance, one can verify its solution feasibility in polynomial steps, and thus BCSBSTP belongs to NP. We prove the result by reducing the subset sum problem (Garey and Johnson 1979) to an instance of the BCSBSTP, where each edge weight is exponentially distributed with different expectations. Denote m = |E| as the number of arcs in a general graph G(V, E). Let K ∈ Z+ and {λi }m i=1 ˜ ˜ ˜ ˜ be a set of nonnegative integers. G = (V , E) is an undirected graph with V = {1, ..., 2m + 1} and ˜ = {e1,2 , e1,3 , e2,3 , ..., e2m−1,2m , e2m−1,2m+1 , e2m,2m+1 }, where ei,j is the edge connecting nodes i E and j. Let Wi,j be the weight of edge ei,j , where ( λ(i+1)/2 if j = i + 2, Wi,j = 0 otherwise; 16

˜ = (V˜ , E) ˜ and the designated weights of all edges. Figure 3: Graph G as shown in Figure 3. (The value next to a node denotes the index of that node, and the value next to an edge denotes the associated weight of that edge.) Consider the following two decision problems. • Q1 (subset sum): Given {λi }m i=1 , does there exist a nonempty subset X ⊆ {1, ..., m} such X λi = K? that i∈X

˜ = (V˜ , E), ˜ does there exist a spanning tree T ∈ T (G) ˜ such that • Q2 : Given the graph G X Wi,j = K? (i,j)∈ET

Obviously, the answer to question Q1 is “YES” if the answer to question Q2 is “YES”. Next, we will show that the answer to question Q2 is “YES” if the answer to question Q1 is “YES”. Suppose X λi = K for some X ⊆ {1, ..., m}. Let I(i, j) be an indicator variable which attains the value of i∈X

1 if we include edge ei,j in a spanning tree. The assignment ( 1 if (i + 1)/2 ∈ X, and j = i + 2 or j = i + 1, I(i, j) = 0 otherwise; ˜ with weight K. Therefore, Q1 is equivalent to Q2 , and the constructs a spanning tree of graph G latter problem is also NP-Complete. Consider the optimization version of Q2 as    X  X Qo2 := arg max Wi,j : Wi,j ≤ K .  ˜  T ∈T (G) (i,j)∈ET

(i,j)∈ET

Now, choose an arbitrary β ∈ (0, 1) and let κ = log(1/β)/K. We have X

Wi,j

(i,j)∈ET

    X ˜ ≤ K ⇔ exp −κ Wi,j ≥ β for all T ∈ T (G).  

(19)

(i,j)∈ET

˜ = (V˜ , E) ˜ in Figure 3. For all edges ei,j ∈ E, let wi,j denote the weight of edge ei,j Consider G that is exponentially distributed with rate Wi,j . Given that min wi,j is exponentially distributed (i,j)∈ET X ˜ with rate Wi,j for all T ∈ T (G), (19) implies (i,j)∈ET

X (i,j)∈ET

Wi,j ≤ K ⇔ P

min wi,j ≥ κ

(i,j)∈ET

17

˜ ≥ β for all T ∈ T (G).

(20)

˜ and (20) Based on the definition of T 0 (G)       X   X X Wi,j : Wij ≤ K = arg max Wi,j . arg max   ˜  ˜  T ∈T 0 (G) T ∈T (G) (i,j)∈ET

(i,j)∈ET

(21)

(i,j)∈ET

As a cumulative probability function of an exponential random variable is nondecreasing in its rate λ (i.e., condition (i) in Corollary 1), we have      X    X arg max Wi,j = arg max `: log Fj (`) ≥ log α . (22)   ˜  ˜  T ∈T 0 (G) T ∈T 0 (G) j:ej ∈ET

(i,j)∈ET

Equations (21) and (22) establish the equivalence between BCSBSTP (i.e., Problem Q) and Proble Qo2 . Thus, we complete the proof.

6.

Computational Results

We compare computational results of the SOS1, SOS2 approximations, and Algorithm 3 proposed in Section 4.2 for solving SBSTP. In general, the SOS1 is much faster than the SOS2 and also yields relatively good quality solutions. We then apply the SOS1 approach to solve the BCSBSTP instances with diverse network topologies and different κ values. The rest of the section is organized as follows. Section 6.1 sets up the design of computations. The results of all algorithms for the SBSTP are compared in Section 6.2. In Section 6.3, we employ SOS1 for solving BCSBSTP and provide illustrations of how the values of κ affect the optimal objectives and spanning tree solutions.

6.1

Computational Design and Setup

We vary {|V |} × {D} over {10, 20, 30} × {10%, 20%, 30%, 50%}, where D ≈ 2|E|/(|V | × (|V | − 1)) refers to the density of a graph. A random number pij is generated based on a uniform distribution over interval [0, 1] for each edge (vi , vj ), vi , vj ∈ V and i < j. If pij < 2/(|V | × (|V | − 1)), we add edge (vi , vj ) into the current graph and update the number of edges as |E| = |E|+1. We repeat this process until achieving a desired density of D. We ensure that the graph is connected by randomly adding an edge (vi , vj ) into set E if there is no path between any node pair of vi and vj in the graph. Table 2 shows five types of distributions we tested where all weights wj , ∀ej ∈ E are independently distributed. For each edge ej , parameters kj , λj , bj and σj are the same by rounding up a number randomly generated from uniform distribution U ∼ (0, 10]. All models and algorithms use C++, and CPLEX 12.2 via ILOG Concert Technology is implemented for solving integer-programming models. The computations are performed on a HP Workstation Z210 Windows 7 machine with Intel(R) Xeon(R) CPU 3.20 GHz, and 8GB memory. In all tables, Columns Time, Itrns, |V|, D and n present average CPU seconds, average number of iterations, number of nodes in the network, graph density and number of sampling sequence 18

Table 2: Distribution types and parameter settings for characterizing random edge weights. Type

1

2

3

4

5

Distribution Setting

Chi-Squared wj ∼ χ2 (kj )

Exponential wj ∼ Exp(λj )

Normal1 wj ∼ N (10, (3.5σj )2 )

Normal2 wj ∼ N (1, (0.35σj )2 )

Uniform wj ∼ U(0, bj )

points, respectively. The error tolerance is set as 0.01 for all algorithms implemented. For each parameter combination, ten instances are tested and average results are reported. The CPU time is reported in seconds.

6.2

Results of Solving the SBSTP

6.2.1

Cases when all edge weights follow the same distribution type.

Table 3 reports results of Algorithm 1 for solving SBSTP instances with edge weights following the same distribution type. For both SOS1 and SOS2, we test n = 6 and 12 as the number of sample points used for approximating log Fj (`), ∀ej ∈ E. Table 3: SBSTP instances with edge weights following the same distribution type solved by SOS1 and SOS2. Type

|V |

10 1

20 30 10

2

20 30 10

3

20 30 10

4

20 30 10

5

20 30

n

D = 10% Time(s) Itrns

SOS1 D = 20% D = 30% Time(s) Itrns Time(s) Itrns



SOS2 D = 20% D = 30% Time(s) Itrns Time(s) Itrns


6 12 6 12 6 12

0.03 0.16 0.14 0.62 2.38 10.95

1.2 3.2 1.0 2.8 1.4 4.0

0.27 0.45 2.89 5.80 73.41 136.10

2.7 3.0 3.9 4.4 4.7 3.7

0.29 0.62 8.72 10.58 239.67 199.75

3.5 4.5 5.1 3.5 5.8 4.0

0.57 0.91 29.20 70.30 1440.99 2105.81

4.4 4.1 4.9 4.2 4.4 5.0

0.17 0.15 2.50 1.92 51.54 75.01

3.9 3.0 3.9 2.9 3.8 3.0

0.53 0.57 20.22 26.27 1790.87 2741.46

3.3 3.0 3.5 3.0 3.4 3.0

0.52 0.63 102.94 300.23 -

3.7 2.9 3.9 3.0 -

2.95 4.33 1782.24 4209.69 -

3.9 3.0 4.0 3.0 -

6 12 6 12 6 12

0.14 0.12 0.47 0.93 7.38 11.91

4.1 3.1 3.5 3.4 3.6 3.6

0.19 0.18 3.00 3.31 108.23 43.81

3.5 3.5 4.3 3.7 5.0 4.0

0.37 0.26 6.44 5.12 267.15 124.74

4.2 3.1 4.6 3.5 5.1 3.6

0.70 0.58 20.04 14.86 1331.64 548.08

4.5 3.6 5.8 2.8 6.2 2.8

0.22 0.10 2.68 1.83 22.89 122.34

2.8 2.2 2.8 2.6 2.9 2.8

0.22 0.16 20.34 18.04 -

2.9 2.4 2.8 2.7 -

0.46 0.47 -

2.9 2.3 -

3.35 2.93 -

3.3 2.7 -

6 12 6 12 6 12

0.13 0.10 0.70 0.63 7.84 8.44

3.9 2.6 3.8 2.8 4.1 3.0

0.23 0.23 2.30 2.96 131.15 138.64

3.9 3.0 3.2 3.1 2.9 2.9

0.26 0.29 5.38 4.30 191.36 200.52

3.4 2.7 3.6 2.7 4.3 3.2

0.54 0.35 10.64 13.75 637.26 689.21

3.4 2.3 3.2 3.0 3.6 2.7

0.30 0.12 1.02 0.66 26.13 29.55

2.8 2.0 3.0 2.0 2.7 2.0

0.53 0.20 13.28 14.93 1724.53 1187.70

3.0 2.0 2.7 2.2 2.7 2.2

0.56 0.33 68.84 54.56 -

2.8 2.0 3.0 2.3 -

2.03 1.74 542.41 2676.82 -

2.9 2.0 2.7 2.6 -

6 12 6 12 6 12

0.13 0.21 0.93 0.93 9.82 12.01

5.4 4.3 4.6 3.6 4.7 4.2

0.27 0.37 2.24 4.00 138.69 209.45

4.9 4.8 3.5 4.1 3.5 3.7

0.28 0.42 5.68 6.30 189.34 152.14

3.9 3.9 4.7 3.8 5.4 4.6

0.72 0.58 11.61 9.66 674.76 564.97

4.3 3.6 3.5 4.1 4.3 4.4

0.16 0.22 0.94 1.27 32.27 44.74

3.5 2.9 3.6 3.0 3.7 2.9

0.37 0.44 17.83 19.26 1766.86 2238.67

3.4 3.0 3.8 2.9 3.3 2.9

0.45 0.49 73.04 104.84 -

3.4 2.8 3.9 2.9 -

2.02 2.51 762.31 1606.54 -

3.8 3.0 4.0 3.0 -

6 12 6 12 6 12

0.08 0.07 0.13 0.17 0.98 2.42

1.6 1.3 1.0 1.7 1.4 3.4

0.09 0.13 0.85 1.04 24.43 25.91

1.8 1.9 2.9 3.9 4.1 4.1

0.10 0.10 2.23 1.96 58.72 40.39

2.2 2.4 5.3 4.5 6.1 4.7

0.27 0.29 5.82 4.42 167.43 71.83

4.1 4.0 6.1 4.5 5.7 4.2

0.14 0.09 0.27 0.27 5.03 5.38

3.0 2.1 3.0 2.2 3.1 2.7

0.18 0.21 2.21 4.76 165.66 197.66

2.7 2.2 3.0 2.7 3.1 2.6

0.15 0.16 9.69 13.30 771.38 137.10

3.0 2.3 3.5 2.8 3.5 3.0

0.46 0.67 51.60 185.61 -

3.1 2.8 3.6 3.0 -

“–” represents that more than five instances cannot be solved within the time limit.

19

In Table 3, the CPU time increases as the number of nodes or graph density increases. However, no clear relation can be drawn between the CPU time and the number of sample points used. This is because increasing n will better approximate function log Fj (`) and potentially reduce the number of iterations needed. Meanwhile, each iteration takes long CPU time for computing Formulation (9) or (11), which contains more binary variables due to the increase of n. Furthermore, the CPU time of Type 3 and Type 4 distributions does not have significant difference, indicating that scalings of distribution parameters have little impact on the computation. For both SOS1 and SOS2, when a network is small and sparse, all five types of probability distributions perform very well. As the graph size and density grow, instances of Uniform distributions take the least CPU time, followed by Normal, Exponential and Chi-Squared distributions. We explain the reason in Figure 4 by showing the log cumulative distribution functions. In general, if a log cumulative distribution function is closer to a linear function, SOS1 and SOS2 piecewise linear functions better approximate log Fj (·), ∀ej ∈ E.

Figure 4: Log cumulative distribution functions of all distribution types in Table 2. We also compute the same instances by using the bisection algorithm. Note that distributions in Table 2 satisfy requirements for the special case of SBSTP discussed in Section 4.2, i.e., a distribution has one parameter not fixed, and its cumulative distribution function is monotone in changing the parameter. According to Corollary 1, an optimal spanning tree solution can be obtained by solving an ordinary maximum or minimum spanning tree problem. We then apply 20

Algorithm 2 to compute the optimal objective value associated with the optimal tree solution. Cumulative distribution functions of Exponential and Normal distributions are nondecreasing in λ and σ, respectively, and the function of Uniform distribution is nonincreasing in parameter b. Therefore, we seek parametric maximum spanning trees for Type 2 and Type 3, and a parametric minimum spanning tree for Type 5. Table 4 demonstrates the results, for which we only need to find a maximum or minimum spanning tree, and then apply a bisection algorithm (i.e., Algorithm 2) to compute the objective value. Table 4: CPU seconds of SBSTP instances with the same distribution type edge weights solved by Algorithm 2. |V |

D = 10%

10 20 30

0.15 0.57 3.44

Type 2 D = 20% D = 30% 0.22 1.58 7.51

0.31 2.39 11.51

D = 50%

D = 10%

0.51 4.40 24.68

0.07 0.33 2.50

Type 3 D = 20% D = 30% 0.12 1.07 6.88

0.15 1.85 13.05

D = 50%

D = 10%

0.27 3.16 17.74

0.14 0.54 3.46

Type 5 D = 20% D = 30% 0.24 1.58 6.93

0.37 2.40 10.72

D = 50% 0.59 4.73 28.02

In Table 4, it comes as no surprise that computational time is significantly shorter than the previous one of SOS1 and SOS2. Distribution types do not affect the CPU time, because the bisection approach does not involve steps of approximating log Fj (·), ∀ej ∈ E. 6.2.2

Cases when all edge weights follow general diverse distribution types.

We test SOS1, SOS2 and Algorithm 3 to solve general SBSTP instances where edge weights wj , ∀ej ∈ E are independently distributed and can randomly choose one distribution among ChiSquared, Exponential, Normal and Uniform. Table 5 specifies different distributions that we mix and randomly assign to edge weights for generating SBSTP instances. Table 5: Distribution choices for generating random SBSTP instances. No.

1

2

3

4

5

6


Normal wj ∼ N (10, 1)

Normal wj ∼ N (10, 1.5)


Exponential wj ∼ Exp(0.4)



No.

7

8

9

10

11

12


Uniform wj ∼ U(0, 10)



Chi-Squared wj ∼ χ2 (2)



No.

13

14

15

16

17

18








We generate instances in Group 1 by mixing distributions No. 1–12, and create instances in Group 2 by mixing No. 1–6 and No. 10–15. Group 2 aims to examine how replacing Uniform with Normal may change the computational results. We further create Group 3 instances by mixing No. 1–15 and Group 4 instances by mixing No. 1–18. This is to test the influence of increasing the diversity of edge weight distributions. Table 6 reports results of applying all three approaches on Group 1 instances, as well as SOS1

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and Algorithm 3 on Groups 2, 3, and 4 instances. For SOS1 and SOS2, we choose n = 6 as the number of sample points used in Algorithm 1. Table 6: SBSTP instances with general distributed edge weights solved by SOS1, SOS2, and Algorithm 3. D=10% Time(s) Itrns

D=20% Time(s) Itrns

D=30% Time(s) Itrns

D=50% Time(s) Itrns

6.3 6.2 6.6

1.55 15.73 1218.86

6.4 6.5 6.3

1.95 43.73 1576.86

6.6 6.5 6.8

0.32 17.89 3246.80

2.9 3.1 2.9

0.78 147.61 -

3.0 3.3 -

2.55 3987.24 -

3.3 3.5 -

4.9 11.1 14.2

0.54 12.88 109.54

8.1 13.3 15.3

1.41 27.40 163.69

13.8 14.5 15.0

2.20 61.64 203.66

14.4 14.1 16.7

0.27 1.22 10.80

5.7 6.5 6.7

0.48 5.77 91.00

5.9 6.8 6.4

0.77 11.45 297.53

6.4 6.5 6.6

1.12 85.85 1172.75

7.4 7.1 7.7

10 20 30

0.13 1.90 24.26

4.0 11.5 14.3

0.44 11.30 72.62

7.9 13.3 14.2

1.22 22.51 93.17

13.5 14.2 13.5

1.98 40.21 150.82

12.9 13.8 11.9

SOS1 : Group 3 n=6

10 20 30

0.23 1.22 11.09

5.9 6.2 6.2

0.46 5.22 91.51

5.3 6.3 6.2

0.73 7.69 217.81

6.1 6.5 6.7

1.24 18.11 881.47

7.6 7.4 7.5

Algo3 : Group 3

10 20 30

0.16 2.66 22.74

4.9 11.6 13.2

0.50 13.31 61.24

8.3 13.3 13.4

0.80 16.47 124.85

11.3 14.0 16.6

2.50 32.67 189.87

13.5 13.8 15.4

SOS1 : Group 4 n=6

10 20 30

0.24 1.11 9.28

5.2 5.4 5.8

0.69 5.36 151.74

5.1 5.8 5.7

0.56 13.08 185.78

5.0 6.2 5.8

1.14 20.12 3376.32

5.5 5.6 5.7

Algo3 : Group 4

10 20 30

0.16 2.48 21.06

4.9 11.4 12.9

0.43 10.05 61.97

7.8 13.6 14.8

1.16 16.33 85.10

13.1 14.3 16.0

2.21 38.97 656.38

13.9 16.1 15.3

Instance

|V |

SOS1 : Group 1 n=6

10 20 30

0.38 2.22 16.23

5.8 6.7 6.7

0.95 7.30 118.38

SOS2 : Group 1 n=6

10 20 30

0.16 2.11 56.08

3.1 3.1 3.5

Algo3 : Group 1

10 20 30

0.18 2.01 34.49

SOS1 : Group 2 n=6

10 20 30

Algo3 : Group 2

“–” represents that more than five instances cannot be solved within the time limit.

In Table 6, CPU time increases as the number of nodes or graph density increases. For Group 1, SOS2 takes much longer CPU time than SOS1 and Algorithm 3. In general, instances in Group 2 and Group 3 take less CPU time than Group 1. Parameter {|V |, D} = {30, 50%} has the longest CPU time in Group 4 among all groups using either SOS1 or Algorithm 3. But for other graph configurations, Group 4 performs as good as Group 2 and Group 3 by using SOS1. High diversity of edge weight distributions reduces CPU time when a graph small and sparse. For large and dense graphs, the CPU time increases dramatically because more edge weights have the same distribution type. In general, Algorithm 3 takes shorter CPU time than SOS1 when graph instances are large and dense. At every iteration, Algorithm 3 solves a maximum spanning tree which is computationally 22

easier than Formulation (9) solved by SOS1, and hence the former has significantly shorter CPU time per iteration than the latter. However, the number of iterations taken by Algorithm 3 is more than SOS1 given that the former could bisect over a long interval. When a graph is sparse, both algorithms have short CPU time per iteration, but Algorithm 3 requires more iterations. When a graph is dense, per iteration CPU time of SOS1 increases while the number of iterations in Algorithm 3 does not change significantly, leading to longer CPU time of SOS1.

6.3

Results of Solving the BCSBSTP

According to Theorem 3, the BCSBSTP is NP-Complete. As the bisection algorithm is no longer applicable, we use SOS1 in this section to solve the BCSBSTP given its result reliability and significantly shorter CPU time compared with SOS2, demonstrated in the previous section. We set both α and β to be 0.95 in Problem Q and test different κ values to derive empirical observations and insights. For each combination of {|V |, D}, we test five instances of the BCSBSTP on graphs with edge weights following distributions in Group 1. Tables 7–10 report results of {|V |} × {D} = {20, 30} × {30%, 50%}, respectively. Table 7: Results of the BCSBSTP with Group 1 and {|V |, D} = {20, 30%}. Instance 1 2 3 4 5

κ=0 Time(s) Obj 11.15 13.87 14.90 17.85 14.90

12.00 12.14 10.62 11.94 11.94

κ = 0.025 Time(s) Obj 52.95 19.61 28.19 27.64 30.31

12.63 12.50 12.78 12.03 12.59

κ = 0.05 Time(s) Obj 25.49 16.55 28.99 22.71 79.50

13.80 12.68 14.32 12.49 13.54

κ = 0.075 Time(s) Obj 19.50 21.39 32.46

13.50 12.68 14.25

κ = 0.1 Time(s) Obj 13.18 25.46 14.40

13.87 13.59 14.69

κ = 0.125 Time(s) Obj 22.84 -

14.04 -

“–” represents that the instance is infeasible for the given κ.

Table 8: Results of the BCSBSTP with Group 1 and {|V |, D} = {20, 50%}. Instance 1 2 3 4 5

κ=0 Time(s) Obj 58.13 28.61 36.97 22.12 24.07

10.10 9.98 11.09 10.06 9.99

κ = 0.1 Time(s) Obj 79.23 78.91 39.38 42.65 59.64

13.38 13.14 12.74 13.38 13.14

κ = 0.2 Time(s) Obj 141.99 44.73 27.11 40.89 42.28

14.48 13.73 13.29 14.15 14.58

κ = 0.3 Time(s) Obj 50.53 26.08 33.01 22.45 -

14.83 14.02 13.90 14.55 -

κ = 0.4 Time(s) Obj 22.89 23.26 19.75 -

14.46 14.15 14.63 -

κ = 0.5 Time(s) Obj -

-


In Tables 7–10, as κ increases, the objective value in each instance invariably increases because previously feasible (and “good”) spanning tree solutions are no longer feasible for larger κ. We show that instances in Table 7, Table 8, and Table 10 become infeasible approximately when we increase κ to 0.125, 0.5, and 5.7, respectively. Comparing optimal solutions in Tables 7 and 10, we find that for Instance 1 in Table 10 at κ = 5.7, fifteen edge weights take on distribution choice No. 1, thirteen take on No. 2, and one takes on No. 3 for the optimal spanning tree, while for Instance

23


κ=0 Time(s) Obj 367.01 338.73 187.61 107.97 155.74

κ = 0.1 Time(s) Obj

11.32 10.82 11.95 11.27 10.47

272.77 268.06 299.49 169.04 541.68

14.22 14.11 13.85 13.99 13.87

κ = 0.2 Time(s) Obj 509.56 75.57 66.10 185.39 96.25


14.98 14.76 14.20 14.49 14.49

63.66 86.92 85.54 -


15.09 14.49 14.77 -

41.22 -


14.84 -

-

-



κ=0 Time(s) Obj


5390.76 266.14 423.11 314.53 296.01

248.82 222.15 551.46 507.08 1950.72

10.68 10.00 10.00 9.99 9.99

13.98 13.76 14.39 13.63 13.81

κ = 0.6 Time(s) Obj 100.98 87.99 151.62 383.79


13.99 14.04 13.95 13.93

38.16 28.32 65.60 208.07

14.20 14.27 14.18 14.11

κ = 5.7 Time(s) Obj* 42.90 22.64 57.19 145.38

κ=6 Time(s) Obj

14.20 14.27 14.18 14.11

-

-

“∗”: same objective values are obtained from κ = 0.9 to κ = 5.7. “–” represents that the instance is infeasible for the given κ.

1 in Table 7, edge weights take on a variety of No. 1, 2, 4, 7, 8, 9, 11, and 12. Table 11 presents the 5% quantile for No. 1–12 (i.e., Group 1) distributions introduced in Table 5. Table 11: 5% quantile for group 1 distributions specified in Table 5 Distribution No.

1

2

3

4

5

6

7

8

9

10

11

12

5% Quantile

8.36

7.53

6.71

0.13

0.10

0.09

0.50

0.60

0.70

0.10

0.35

0.71

The values in Table 11 set the upper bound of κ for each distribution to be valid when β = 0.95. For example, if a spanning tree solution contains an edge, of which the weight follows No. 5 distribution, then a κ value greater than 0.1 will for sure make any spanning tree solution infeasible. As a graph becomes dense, more feasible spanning trees are yielded. High graph density (i.e., D = 50%) in general decreases the optimal objective values, compared with D = 30%. That is, one has higher chance to obtain spanning tree solutions with lower objective values when more nodes are connected. Efficient Frontier of κ. Finally, we solve a BCSBSTP problem on the graph depicted in Figure 1 to illustrate an efficient frontier of the optimal objective `∗ versus κ. Recall that in Figure 1, all edge weights follow exponential distributions, with the number alongside each edge representing parameter λ. We use SOS1 to compute optimal solutions for different κ where we increase κ by 0.0005 each time. As a result, a total of 79 points are obtained and depicted in Figure 5 by varying κ from 0 to 0.0039. The horizontal and vertical axes represent values of κ and the corresponding optimal objective values of Problem Q, respectively. Table 12 presents frontier points and their corresponding optimal tree solutions. Increasing κ

24

Figure 5: Illustrating a κ-efficient frontier by solving the BCSBSTP on the graph in Figure 1 will change previous optimal tree solutions as infeasible solutions, which is in turn reflected by the increase in `∗ . Table 12: A summary of frontier solutions

7.

κ (10−3 )

`∗

1.45 1.75 1.80 1.85 2.10 2.30 2.40 3.00 3.90

1.020 1.104 1.522 1.588 1.603 1.604 1.845 1.882 2.065

ET∗ (1-3) (1-3) (1-3) (1-2) (1-3) (1-3) (1-3) (1-3) (1-2)

(2-5) (2-3) (2-3) (1-3) (2-3) (2-3) (2-3) (2-3) (2-3)

(3-5) (3-5) (3-5) (3-5) (3-5) (2-4) (2-4) (2-4) (2-4)

(4-6) (4-6) (4-5) (4-6) (4-5) (4-6) (4-5) (4-5) (4-5)

(5-6) (5-6) (5-6) (5-6) (4-6) (5-6) (5-6) (4-6) (4-6)

Conclusions

In this paper, we investigate (balance-constrained) stochastic bottleneck spanning tree problems (BCSBSTP or SBSTP). We propose three approaches for handling nonlinearity in the original optimization model: SOS1- and SOS2-based approximations (valid for both BCSBSTP and SBSTP) and a parametric maximum spanning tree-based bisection approximation that runs in polynomial time for approximating solutions of general SBSTP. For general SBSTP in which each edge weight may have different distributions, our computational results demonstrate that the bisection algorithm is significantly better than the SOS approximations when a graph is large and dense, and SOS1 is better when a graph is sparse. Compared with the bisection algorithm, the performance

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of SOS1 and SOS2 highly relies on the distribution type of edge weights. We show that general BCSBSTP is NP-Complete, and use SOS1 to derive an efficient frontier of the optimal objective values in the values of κ. The practical use of the research is limited by the assumed independence between edge weights. In future research, we will consider problems with generally distributed edge weights. One possibility is to use an SAA-based approach and to generate realizations of edge weights by following the Monte Carlo sampling approach, which does not require edge weight independence. Moreover, we aim to identify specific distributions making BCSBSTP instances tractable, and develop polynomial-time algorithms. Our future research will focus on improving the effectiveness of approximation algorithms, seeking tight bounds, and incorporating more complicated decision making factors, such as specific restrictions on spanning tree solutions.

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