Determining r-Robustness of Arbitrary Digraphs Using Zero-One Linear Integer Programming

10/02/2018 ∙ by James Usevitch, et al. ∙ University of Michigan 0

There has been an increase in the use of resilient control algorithms based on the graph theoretic properties of r- and (r,s)-robustness. These algorithms guarantee consensus of normally behaving agents in the presence of a bounded number of arbitrarily misbehaving agents if the values of the integers r and s are sufficiently high. However, determining the largest integer r for which an arbitrary digraph is r-robust is highly nontrivial. This paper introduces a novel method for calculating this value using zero-one integer programming. The method only requires knowledge of the graph Laplacian matrix, and can be formulated with affine objective and constraints, except for the integer constraint. Integer programming methods such as branch-and-bound can allow both lower and upper bounds on r to be iteratively tightened. Simulations suggest the proposed method demonstrates greater efficiency than prior algorithms.

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I Introduction

Consensus on shared information is fundamental to the operation of multi-agent systems. In context of mobile agents, it enables formation control, agent rendezvous, sensor fusion, and many more objectives. Although a vast literature of algorithms for consensus exist, many are unable to tolerate the presence of adversarial attacks or faults. Recent years have seen an increase of attention on resilient algorithms that are able to operate despite such misbehavior. Many of these algorithms have been inspired by work such as [1], which is one of the seminal papers on consensus in the presence of adversaries; [2, 3, 4] which outline discrete- and continuous-time algorithms along with necessary and sufficient conditions for scalar consensus in the presence of Byzantine adversaries; and [5, 6, 7, 8]

, which outline algorithms for multi-agent vector consensus of asynchronous systems in the presence of Byzantine adversaries. Some of the most recent results that draw upon these works include resilient state estimation

[9], resilient rendezvous of mobile agents [10, 11], resilient output synchronization [12], resilient simultaneous arrival of interceptors [13], resilient distributed optimization [14, 15], reliable broadcast [16, 3], and resilient multi-hop communication [17].

Many of these results are based upon the graph theoretical properties known as -robustness and -robustness [2, 3]. These notions were defined after it was shown that traditional graph theoretic metrics (e.g. connectivity) were insufficient to analyze the convergence properties of certain resilient algorithms based on purely local information [3]. The properties of - and -robustness have been used in sufficient conditions for several resilient consensus algorithms including the ARC-P [4], W-MSR [2], SW-MSR [18], and DP-MSR [19] algorithms. Given an upper bound on the global or local number of adversaries in the network, these resilient algorithms guarantee convergence of normally behaving agents’ states to a value within the convex hull of initial states if the integers and are sufficiently large.

A key challenge in implementing these resilient algorithms is that determining the - and -robustness of arbitrary digraphs is an NP-hard problem in general [20, 21]. The first algorithmic analysis of determining the values of and for arbitrary digraphs was given in [20]. The algorithms in this work employ an exhaustive search to determine the maximum values of and for a given digraph, and have exponential complexity w.r.t. the number of nodes in the network. Subsequent work has focused on methods to circumvent this difficulty, including graph construction methods which increase the graph size while preserving initial values of and [2, 22], demonstrating the behavior of as a function of particular graph properties [21, 23, 24], lower bounding with the isoperimetric constant and algebraic connectivity of undirected graphs [25]

, and even using machine learning to correlate characteristics of certain graphs to the values of

and [26]. Finding more efficient ways of determining the exact robustness of digraphs however is still an open problem.

In this paper, we introduce a novel method for determining the maximum value of for which an arbitrary digraph is -robust by solving a zero-one linear integer programming (IP) problem. The problem only requires knowledge of the graph Laplacian matrix and can be formulated with affine objective and constraints, with the exception of the integer constraint. To the best of our knowledge, this is the first time the problem has been formulated in this way. This contribution provides several advantages. First, these results open the door for the extensive literature on zero-one integer programming to be applied to the -robustness determination problem. In particular, applying branch-and-bound algorithms to the problem can allow for lower and upper bounds on a digraph’s -robustness to be iteratively tightened. Prior algorithms are only able to tighten the upper bound on the maximum robustness for a given digraph. Second, this formulation enables commercially available solvers such as Gurobi or MATLAB’s intlinprog to be used to find the maximum robustness of any digraph. Finally, experimental results using this new formulation suggest a reduction in computation time as compared to the centralized algorithm proposed in [20].

This paper is organized as follows: notation and relevant definitions are introduced in Section II. The problem formulation is given in Section III. Our main result of formulating the -robustness determination problem as a zero-one linear integer programming problem is given in Section IV. Simulations are presented in Section V, and we present conclusions and directions for future work in Section VI.

Ii Notation

The sets of real numbers and integers are denoted and , respectively. The sets of nonnegative real numbers and integers are denoted and , respectively. denotes an -dimensional vector space over the field , represents an dimensional vector with nonnegative integer vectors, and represents a binary vector of length . Scalars are denoted in normal text (e.g. ) while vectors are denoted in bold (e.g. ). The notation denotes the th entry of vector . The inequality symbol denotes a componentwise inequality between vectors; i.e. for , . An -dimensional vector of ones is denoted , and an -dimensional vector of zeros is denoted . In both cases the subscript will be omitted when the size of the vector is clear from the context. The union, intersection, and set complement operations are denoted by , and , respectively. The cardinality of a set is denoted as , and the empty set is denoted . The infinity norm on is denoted . The notation is sometimes used in this paper to denote the binomial coefficient with . Given a set , the power set of is denoted .

A directed graph (digraph) is denoted as , where is the set of indexed nodes and is the edge set. A directed edge is denoted , with , meaning that agent can receive information from agent . The set of in-neighbors for an agent is denoted . The minimum in-degree of a digraph is denoted . In this paper we consider simple digraphs of nodes, meaning digraphs without self loops and without redundant edges (i.e. if the directed edge , then it is the only directed edge from to ). Occasionally, will be used to denote an undirected graph where . The graph Laplacian for a digraph (or undirected graph) is defined as follows, with denoting the entry in the th row and th column:

(1)

Iii Problem Formulation

We begin with the definitions of -reachability and -robustness:

Definition 1 ([2]).

Let and be a digraph. A nonempty subset is -reachable if such that .

Definition 2 ([2]).

Let . A nonempty, nontrivial digraph on nodes is -robust if for every pair of nonempty, disjoint subsets of , at least one of the subsets is -reachable. By convention, the empty graph is 0-robust and the trivial graph is 1-robust.

Note that if a set is -reachable, it is -reachable for any . Similarly, if a graph is -robust it is also -robust for any .

This paper addresses the following problem:

Problem 1.

Given an arbitrary digraph , determine the maximum integer for which is -robust.

Remark 1.

We denote the maximum integer for which a given digraph is -robust as .

It should be clear from Definition 2 that determining the maximal -robustness involves checking the reachability of pairs of nonempty, disjoint subsets in a graph. Let the set be defined as

(2)

The set therefore contains all possible pairs of nonempty, disjoint subsets of . It was shown in [20] that .111Since , the total number of unique nonempty, disjoint subsets is , denoted as in [20].

Iii-a Alternate Formulation of Maximum -Robustness

In our first result, we derive an equivalent way of expressing the maximum robustness of a digraph . Given an arbitrary digraph and a subset , we define the reachability function as follows:

(3)

In other words, returns the maximum integer for which the set is reachable. The following Lemma presents an explicit formulation which yields :

Lemma 1.

Let be an arbitrary nonempty, simple digraph with . The following holds:

(4)
subject to
Proof.

For brevity, define the function

(5)

Note that or . Let be a minimizer of the right hand side (RHS) of (4). Then . Therefore either or . Therefore the graph is -robust, implying .

To show , we prove by contradiction. By definition, is -robust (Remark 1). Suppose . This implies and . Since such that and , this implies is not -robust. This contradicts the fact that is -robust by definition. Therefore, .

Remark 2.

Using the definition of in (2), the constraints on the RHS of (4) can be made implicit [27, section 4.1.3] as follows:

(6)

Iv -Robustness Determination as Linear Integer Program

The next step in the analysis is to demonstrate how the expression

(7)

can be calculated as a function of the graph Laplacian matrix. Recall that , and define the indicator vector as follows: for any ,

(8)

In other words the th entry of is 1 if the node with index is a member of the set , and zero otherwise. It is straightforward to verify that is a bijection. Therefore given , the set is defined by and .

Next, the following Lemma is needed for our results:

Lemma 2.

Let be an arbitrary nonempty, simple digraph, let be the Laplacian matrix of , and let . Then the following holds for all :

(9)

where is the th row of .

Proof.

The term is shortened to for brevity. We have

(10)

Since , and , we have . Next, . Therefore if ,

If , then implying . ∎

In words, given the indicator vector of set and any node , the number of in-neighbors of which are outside of is equal to this indicator vector multiplied by the th row of the Laplacian matrix . Our next Lemma demonstrates that recovering the maximum nonnegative value of yields the value of :

Lemma 3.

Let be an arbitrary nonempty, simple digraph, let be the Laplacian matrix of , and let . Then the following holds:

(11)
Proof.

The term is shortened to for brevity. Define . By Lemma 2,

(12)

being simple implies . Therefore ,

(13)

Furthermore, ,

(14)

Therefore and , . Using this fact and the definition of yields

being nonempty implies . Using the definition of in (3), we have

With these results in hand, (7) can finally be expressed directly as a function of .

Lemma 4.

Let be an arbitrary nonempty, simple digraph, and let be the Laplacian matrix of . Let such that . Then the following holds:

(15)
Proof.

For brevity, denote and . By Lemma 3, and . Therefore

(16)

Because and are nonnegative, the left hand side (LHS) of (16) satisfies

(17)

Consider the RHS of (16). being nonempty implies . Also, (3) implies and . Therefore

(18)

The result follows from (16), (17), and (18).

From Lemma 1 and Lemma 4 we can immediately conclude that satisfies

(19)
subject to

However, the formulation can be simplified further by changing the optimization variables to be binary vectors rather than subsets of .

Lemma 5.

Consider an arbitrary nonempty, simple digraph . Let and define as

(20)

Define the function as

(21)

where is the image of under . Then both of the following statements hold:

  1. The mapping is a bijection

Proof.

To prove 1), we first prove . Choose any , . . By similar reasoning, . Next, and imply and . Therefore and . Finally, implies that and . Therefore and . This implies .

Next we prove by contradiction. Suppose such that . Consider sets and . If , then , which contradicts the assumption. Therefore . This implies either or or . We consider all three cases. For the first case, , which contradicts the assumption. For the second case, , which contradicts the assumption. For the third case, such that and . Therefore , implying . This implies , which contradicts the assumption. By these arguments , which implies . Therefore .

Lastly, we prove 2). implies is surjective. To show is injective, we first define an equivalence relation on as if and only if and . For any and , implies . Therefore and . Since is a bijection, and . This implies . Therefore is injective.

Note that is the feasible set of the right hand side of (19). Since is a bijection, instead of optimizing over the feasible set we can instead use binary vectors as variables and optimize over the feasible set , which is the image of under . We now present the main result of this paper.

Theorem 1.

Let be an arbitrary nonempty, simple digraph and let be the Laplacian matrix of . The maximum -robustness of , denoted , is obtained by solving the following minimization problem:

(22)
subject to
Proof.

Lemma 4 and Theorem 4 imply that

subject to

As per Remark 2, the definition of can be used to make the constraints implicit. This along with the definition of from (21) yields

(23)

Since is a bijection by Lemma 5, (23) is equivalent to

Making the constraints on the domain explicit, we obtain

(24)
subject to

Note that since are binary vectors, and if and only if .

This minimization problem can actually be reformulated to an linear integer programming problem (IP) where the objective and all constraint functions are affine except for the integer constraint. This is shown in the following corollary:

Corollary 1.

Let be an arbitrary nonempty, simple digraph, and let be the Laplacian matrix of . The maximum -robustness of , denoted , is obtained by solving the following linear integer program:

(25)
subject to
Proof.

First, it can be demonstrated [27] that the formulation is equivalent to

subject to

Reformulating the objective of the RHS of (22) in this way yields the objective and first two constraints of (25):

(26)
subject to

The variables and from (22) are combined into the variable in (25); i.e. . The third and fourth constraints of (25) restrict . The last three constraints of (25) restrict and are simply a reformulation of the last two constraints in (22).

Iv-a Discussion

Since -robustness is equivalent to -robustness,222See section VII-B of [2]. the solution to the optimization problem in Theorem 1 determines the maximum for which the graph is -robust. As per [2, 20], in general the parameter has higher precedence than when ordering a set of graphs by robustness. In addition, -robustness implies -robustness [28],333E.g. a graph which is -robust for a nonnegative integer is therefore -robust. and so a certain degree of -robustness can be inferred from knowing the maximum value of .

When comparing the optimization problem in Theorem 1 with the algorithms in [20], it is important to note that those in [20] determine both of the parameters and for which graphs are -robust. Since the method in Theorem 1 only determines the largest for which a digraph is -robust (with fixed at 1), it cannot be directly compared to Algorithm 3.2, in [20]. However, by replacing each initialization in with the initialization , a modified algorithm is obtained which only determines -robustness and can be directly compared with the method in Theorem 1. This modified algorithm is presented as Algorithm 1. In addition, the initialization condition in

incorrectly classifies some rooted outbranchings as 0-robust (those with in-degree of the root being 0), when they are actually 1-robust (see Lemma 7 of

[2]). We have revised the initialization of accordingly. The reader is referred to [20] for the definition of the function Robustholds. In short, the function returns the boolean true if the number of -reachable nodes from and is at least , and false otherwise.

(Different than Alg. 3.2 in [20])
comment: is the min. in-degree of nodes in
for each  to  
       comment: is the set of unique subsets of
      for each  
            for each  
                 
comment: is set of partitions of into
and
                 
                 if ( false) and  then
                                         
                 while  false and  
                       while  false and