Minimal Reachability Problems

# Minimal Reachability Problems

V. Tzoumas, A. Jadbabaie, G. J. Pappas All authors are with the Department of Electrical and Systems Engineering, University of Pennsylvania, Philadelphia, PA 19104-6228 USA (email: {vtzoumas, pappasg, jadbabai}@seas.upenn.edu).This work was supported in part by TerraSwarm, one of six centers of STARnet, a Semiconductor Research Corporation program sponsored by MARCO and DARPA, in part by AFOSR Complex Networks Program and in part by ARO MURI W911NF-12-1-0509.
###### Abstract

In this paper, we address a collection of state space reachability problems, for linear time-invariant systems, using a minimal number of actuators. In particular, we design a zero-one diagonal input matrix , with a minimal number of non-zero entries, so that a specified state vector is reachable from a given initial state. Moreover, we design a so that a system can be steered either into a given subset, or sufficiently close to a desired state. This work extends the results of [1] and [2], where a zero-one diagonal or column matrix is constructed so that the involved system is controllable. Specifically, we prove that the first two of our aforementioned problems are NP-hard; these results hold for a zero-one column matrix as well. Then, we provide efficient algorithms for their general solution, along with their worst case approximation guarantees. Finally, we illustrate their performance over large random networks.

## I Introduction

Power grids, transportation systems, brain neural circuits and social networks are just a few of the complex dynamical systems that have drawn the attention of control scientists, [3, 4, 5, 6], since their vast size, and interconnectivity, necessitate novel control techniques with regard to:

1. tasks that are collective [7], e.g., reaching consensus in a system of autonomous interacting vehicles [8];

2. new cost constraints, e.g., with respect to the number of used actuators and the level of the input and communication power [9].

In this paper, we consider a set of minimal state reachability problems, for linear time-invariant systems, where the term ‘minimal’ captures our objective to use the least number of actuators towards the involved control tasks. Specifically, we design a zero-one diagonal input matrix , with a minimal number of non-zero entries, so that one of the following (collective) tasks are met: i) the resultant system can be steered into a subset, or ii) to a state, or iii) sufficiently close to a state. Therefore, our work relaxes the objective of [1] and [2], where a zero-one diagonal or column matrix is constructed, with a minimal number of non-zero entries, so that the designed system is controllable.

This is an important distinction whenever we are interested only in the feasibility of a state transfer, as in power grids [3]; transportation systems [4]; complex neural circuits [5]; infection processes over large-scale social networks [10] (e.g., from the infectious state to the state where all the network nodes are healthy): Consider for example the system in Fig. 1 and assume the transfer from the initial state zero to , where the first entry corresponds to the final state of node ‘0’, the second to that of ‘1’, and so forth; if we impose controllability in the design of , we get a with non-zero elements: ; that is, states through must be actuated so that this system is controllable. On the other hand, if we impose only state reachability, we get a with only one non-zero element, independently of ; e.g., a solution is , where only state is actuated. Thereby, whenever we are interested in the feasibility of a state transfer and in a with a small number of non-zero elements, the objective of state reachability should not be substituted with that of controllability: under controllability the number of used actuators could grow linearly with , while under state reachability it could be one for all . Similar comments carry through with respect to the rest of our objectives.

At the same time, the task to design a sparsest zero-one diagonal matrix is combinatorial, and, as a result, it may be computationally hard in the worst case. Indeed, we prove that the first two of our aforementioned problems are NP-hard — our proofs hold for a zero-one column matrix as well. Therefore, we then provide efficient algorithms for their general solution, along with their worst case approximation guarantees; to this end, we use an approximation algorithm that we provide for our third problem, where a sparse zero-one diagonal matrix is designed so that a system can be steered -close to a desired state.

These hardness results proceed by reduction to the minimum hitting set problem (MHS), which is NP-hard [11]. In particular, we prove that the problem of state reachability, using a minimal number of actuators, is NP-hard, by reducing it to the controllability problem introduced in [1], which is at least as hard as the MHS. Moreover, we prove that the problem of steering a system into a subset is NP-hard by directly reducing it to the MHS.

Then, we first provide an efficient approximation algorithm so that a system can be steered -close to a desired state. This algorithm returns a with a number of non-zero elements up to a multiplicative factor of from any optimal solution. Therefore, it allows the designer to select the level of approximation , with respect to the trade-off between the reachability error and the number of used actuators (recall that the number of non-zero elements of coincides with the number of used actuators). Afterwards, we use this algorithm to provide efficient approximation algorithms for the rest of our reachability problems as well.

In addition to [1] and [2], other relevant studies to this paper are [12, 13, 14] and [15], where their authors consider the design of a sparse input matrix so that an input energy objective is minimized. Moreover, [16] and [17] address the sparse design of the closed loop linear system, with respect to its feedback gain, as well as, a set of sensor placement problems. Other recent works that study sensor placement problems are the [18] and [19].

Furthermore, [20] considers the decidability of a set of problems related to ours; for example, it asks whether the problem of deciding if there exists a control that can drive a given system from an initial state to a desired one is decidable or not. The main difference between this set of problems and ours is that they consider the feasibility of state transfer given a fixed system, whereas we design a system so that the feasibility of a state transfer is guaranteed.

The remainder of this paper is organized as follows. The formulation and model for our reachability problems are set forth in Section II, where the corresponding integer optimization programs are stated. In Section III-A, we prove the intractability of these problems and, then, in Section III-B, we provide efficient algorithms for their general solution, along with their worst case approximation guarantees. Finally, in Section IV, we illustrate our analytical findings, using an instance of the network in Fig. 1, and afterwards, we test the efficiency of the proposed algorithms over large random networks that are commonly used to model real-world networked systems. Section V concludes the paper.

## Ii Problem Formulation

##### Notation

We denote the set of natural numbers as , the set of real numbers as , and we let for all . Also, given a set , we denote as its cardinality. Matrices are represented by capital letters and vectors by lower-case letters. For a matrix , is its transpose and is its element located at the th row and th column. Moreover, we denote as the identity matrix; its dimension is inferred from the context. Additionally, for , we let denote an diagonal matrix such that for all . The rest of our notation is introduced when needed.

### Ii-a Model

Consider a linear system of states, , whose evolution is described by

 ˙x(t)=Ax(t)+Bu(t),t>t0, (1)

where is fixed, , , and is the input vector. The matrices and are of appropriate dimension. Without loss of generality, ; in general, whenever the -th column of is zero, is ignored. Moreover, we denote (1) as the duple and refer to the states as nodes , respectively; finally, we denote their collection as .

In what follows, is fixed and the following structure is assumed on :

###### Assumption 1

is a diagonal zero-one matrix: , where .

Therefore, if , state is actuated, and if , is not and is ignored. That is, the number of non-zero elements of coincides with the number of actuators (inputs) that are implemented for the control of system (1).

In this paper, we design so that satisfies a control objective among the following presented in the next section.

### Ii-B Minimal Reachability Problems

We introduce two control objectives, the state and subset reachability, which we use to define the design problems of this paper. In particular, consider , , and fixed:

###### Objective 1 (State Reachability)

The state is reachable by at time if and only if there exists input defined over such that .

A parallel notion to the state reachability is the state feasibility:

###### Definition 1 (State Feasibility)

The transfer from to by , denoted as , is feasible if and only if is reachable by at time .

We now present our second objective:

###### Objective 2 (Subset Reachability)

The subset is reachable by at time if and only if there exist and input defined over such that is reachable.

The corresponding definition of subset feasibility parallels that of state feasibility and it is omitted.

Evidently, Objective 2 generalizes Objective 1: According to it, targets from a subset, instead of a single state. Nevertheless, subset reachability of does not imply that all states are reachable. Similarly, although may not be reachable by , can be; thus, Objective 1 is not a special case of Objective 2. Overall, Objectives 1 and 2 define the two separate design problems that follow.

###### Problem 1 (Minimal State Reachability)

Given and , design a with the smallest number of non-zero elements so that the state transfer is feasible.

Note that Problem 1 is always feasible, since for any , is controllable.

Therefore, the objective of Problem 1 relaxes that of [1, 2] where is designed with the smallest number of non-zero elements so that the resultant is controllable.

###### Problem 2 (Minimal Subset Reachability)

Given , and , design a with the smallest number of non-zero elements so that the subset is reachable from at time .

We refer to Problem 2 as minimal subset reachability as well. As with Problem 1, Problem 2 is always feasible, since for any , is controllable.

Evidently, the ‘minimal’ term in the definition of Problems 1 and 2 captures our objective to design a sparsest111A matrix is sparse if it has a small number of non-zero elements compared to each dimension. .

Finally, all of our results carry through if we consider the output of (1), where is fixed and of appropriate dimension, instead of . In particular, denote as the column space of and consider the following objectives:

###### Objective 3 (Output Reachability)

The output state is reachable by at time if and only if there exists input defined over such that .

Naturally, Objectives 1 and 3 coincide for . Thereby, a generalized version of Problem 1, where a sparsest is designed so that an output transfer is feasible, is due. Similar comments apply with respect to the objective below.

###### Objective 4 (Output Subset Reachability)

The is reachable by at time if and only if there exist and input defined over such that is reachable.

In what follows, we continue with the original Problems 1 and 2.

## Iii Main Results

In the first part of this section, III-A, we prove that Problems 1 and 2 are NP-hard. The proofs proceed by reduction to the minimum hitting set problem (MHS), which is NP-hard [11], and is defined as follows:

###### Definition 2 (Minimum Hitting Set Problem)

Given a finite set and a collection of non-empty subsets of , find a smallest cardinality that has a non-empty intersection with each set in .

In particular, we prove that Problem 1 is NP-hard providing an instance that reduces to the controllability problem introduced in [1], which is at least as hard as the MHS; as a result, we conclude that Problem 1 is as well. Moreover, we prove that Problem 2 is NP-hard by directly reducing it to the MHS.

In the second part of this section, III-B, since Problems 1 and 2 are NP-hard, we provide efficient approximation algorithms for their general solution. Towards this direction, we first generalize Definition 1 as follows:

###### Definition 3 (ϵ-close feasibility)

The transfer by is -feasible if and only if there exists reachable by at time such that , where denotes the euclidean norm.

For , Definitions 1 and 3 coincide.

We use Definition 3 to relax the objective Problem 1, by replacing the feasibility of with that of -close feasibility — from a real-world application perspective, and for small , this is a weak modification: the convergence of a system exactly to a desired is usually infeasible, e.g., due to external disturbances. We then provide for this problem a polynomial time approximation algorithm, Algorithm 1, that returns a with sparsity222The sparsity of a matrix is the number of its non-zero elements. up to a multiplicative factor of from any optimal solution of the original Problem 1.

Next, to address Problem 1 with respect to Objective 1, we prove that for all , where is positive and sufficiently small, Definitions 1 and 3 still coincide; hence, we implement a bisection-type execution of Algorithm 1, Algorithm 2, that quickly converges to an and, as a result, returns a that makes the exact transfer feasible.

Finally, we provide an approximation algorithm for Problem 2 when is finite, by observing that in this case can be approximated as a finite union of euclidean balls in . Specifically, let be their centres and their corresponding radii. Moreover, without loss of generality, assume . Then, by executing Algorithm 1 for and selecting the sparsest solution among all , we return an approximate solution to Problem 2 with Algorithm’s 1 worst case guarantees.

### Iii-a Intractability of the Minimal Reachability Problems

We prove that Problems 1 and 2 are NP-hard. The proofs proceed with respect to the decision version of Problems 1 and 2 and that of MHS. The latter is defined as follows:

###### Definition 4 (k-hitting set)

Given a finite set and a collection of non-empty subsets of , find an of cardinality at most that has a non-empty intersection with each set in .

Without loss of generality, we assume that every element of appears in at least one set in and all set in are non-empty.

The decision versions of Problems 1 and 2 are defined in Sections III-A1 and III-A2, where we present their NP-hardness, respectively.

#### Iii-A1 Intractability of Problem 1

We prove that the decision version of Problem 1 reduces to the -hitting set and, as a result, that Problem 1 is NP-hard.

This version of Problem 1 is defined by replacing the feasibility objective with that of -feasibility:

###### Definition 5 (k-feasibility)

The transfer is -feasible if and only if there exists -sparse333A matrix is -sparse if it has non-zero elements. such that is feasible by .

To present our instance of the decision Problem 1 that reduces to the -hitting set problem, let and , with respect to Definition 4, and define such that if the -th set contains the element and zero otherwise.

###### Lemma 1

For , denote as the matrix of all-ones and set , , where444 is invertible since it strictly diagonally dominant.

 V1=⎡⎢⎣2Im×m0m×pem×1Φ(m+1)Ip×p0p×101×m01×p1⎤⎥⎦,

and , as well as, . For any , is -feasible if and only if has a -hitting set.

Therefore, with Lemma 1 we provide an instance of Problem 1 that is -feasible if and only if any instance of , (that is, also the hardest ones with respect to the hitting set problem), has a -hitting set. Hence (cf. [11]):

###### Theorem 1

Problem 1 is NP-hard.

Thereby, the generalized version of Problem 1, with respect to Objective 3, is NP-hard as well (for the above instance where we additionally set ).

We illustrate the proof Lemma 1: The instance of and the initial and final condition are constructed so that the is -feasible if and only if there exists -sparse such that is controllable; on the other hand, the latter holds if and only if has a -hitting set [1]. Thereby, the theorem follows. Additionally, due to the controllability properties of linear time-invariant systems [21], it holds for any .

However, the proof of Lemma 1 suggests that the sparse reachability of a system is hard merely because its sparse controllability is. To show the contrary, we generalize Lemma 1 by constructing an and a so that is -feasible if and only if has a -hitting set, while the resultant system is not controllable.

###### Lemma 2

For , denote as the matrix of all-ones and set , , where

 V2=⎡⎢ ⎢ ⎢ ⎢⎣2Im×m0m×pem×10m×1Φ(m+1)Ip×p0p×10p×101×m01×p1001×m01×p01⎤⎥ ⎥ ⎥ ⎥⎦,

and , as well as, . For any , the is -feasible if and only if has a -hitting set.

With this instance, we prove that is -feasible if and only if a sub-system of is -controllable, a fact that is equivalent to having a -hitting set [1]. On the other hand, remains uncontrollable. Therefore, the NP-hardness of Problem 1 emanates from this class of instances as well, where state reachability is achieved without implying controllability to the resultant system.

Lemma 1 extends to the case where is a column zero-one vector as well. Furthermore, in Theorem 1 the assumption is without loss of generality, since we consider the linear dynamics (1) [21]. Finally, Lemmas 1 and 2 extend to the case where is a column zero-one vector as well. Furthermore, in both theorems, the assumption is without loss of generality, since we consider the linear dynamics (1) [21].

In the following paragraphs, we prove the NP-hardness of Problem 2.

#### Iii-A2 Intractability of Problem 2

We prove that the decision version of Problem 2 reduces to the -hitting set and, as a result, that Problem 2 is NP-hard.

This version of Problem 2 is defined by replacing the reachability objective with that of -reachability:

###### Definition 6 (k-reachability)

The subset is -reachable if and only if there exists -sparse such that is reachable by .

To present our instance of the decision Problem 2 that reduces to the -hitting set problem, let and , with respect to Definition 4, and define such that if the -th set contains the element and zero otherwise.

###### Lemma 3

Set and

 A=[0m×m0m×pΦ0p×p].

is -reachable if and only if has a -hitting set.

Therefore, with Lemma 3 we provide an instance of Problem 2 that is -feasible if and only if any instance of , (that is, also the hardest ones with respect to the hitting set problem), has a -hitting set. Hence (cf. [11]):

###### Theorem 2

Problem 2 is NP-hard.

Thereby, the generalized version of Problem 2, with respect to Objective 4, is NP-hard as well (for the above instance where we additionally set ).

Since Problems 1 and 2 are NP-hard, we need in the worst case to provide approximate algorithms for their solution; this is the subject of the next section.

### Iii-B Approximation Algorithms for the Minimal Reachability Problems

We provide efficient approximation algorithms for the general solution of Problems 1 and 2. Recall that these problems aim for a sparse so that a transfer is feasible or a subset of the state space is reachable, respectively. At the same time, the sparsity of equals the number of actuators that we should implement in system (1) so to satisfy these goals. Therefore, the objective of these algorithms is the sparse control of system (1).

To implement an approximation algorithm for Problem 1, we use Definition 3 to relax Objective 1, by replacing the feasibility of with that of -close feasibility. We then provide Algorithm 1, that returns a with sparsity up to a multiplicative factor of from any optimal solution of the original Problem 1.

Next, to address Problem 1 with respect to Objective 1, we prove that for all , where is positive and sufficiently small, Definitions 1 and 3 still coincide; hence, we implement a bisection-type execution of Algorithm 1, Algorithm 2, that quickly converges to an and, as a result, returns a that makes the exact transfer feasible.

Finally, using Algorithm 1, we provide an approximation algorithm for Problem 2 as well.

#### Iii-B1 Approximation Algorithm for Problem 1

We develop the notation and tools that lead to an efficient approximation algorithm for Problem 1.

For and , we denote as the projection of onto and as its euclidean norm. Moreover, we denote as the set of columns of , as the -th unit vector and as the set of columns . For per Assumption 1, we set

 S(B)≡span[B|AB|…|An−1B].

Since the dynamics (1) are linear, is feasible if and only if is. Moreover, since these dynamics are also continuous and time-invariant, whenever is feasible for some , it is also for any [21]. Hence, we study directly , suppressing .

In particular, is feasible if and only if [21]. Therefore, is feasible if and only if : if , , while, if , , that is, 555 is the orthogonal complement of .. Similarly, is feasible if and only if : if , , while, if , .

Definition 3 is restated as follows:

###### Definition 7 (ϵ-close feasibility)

The is -close feasible by if and only if .

###### Remark 1

Since is orthogonal to , and, as a result, -close feasibility implies .

We provide the following greedy approximation algorithm for Problem 1 with respect to the relaxed feasibility objective of Definition 7. Its quality of approximation is quantified in Theorem 3.

###### Theorem 3

Given the transfer , denote as an optimal solution to Problem 1 and as the corresponding output of Algorithm 1. Then, is -close feasible by and

 n∑i=1Bii≤⌈ln(∥x(t1)−exp[A(t1−t0)]x(t0)∥2/ϵ)⌉n∑i=1B⋆ii.

That is, the polynomial time approximation Algorithm 1 returns a with sparsity up to a multiplicative factor of from any optimal solution of the original Problem 1, and makes the , or , -close feasible.

Next, to address Problem 1 with respect to Objective 1, we show that there exists , positive, such that for any , Definitions 1 and 3 coincide. Thereby, running Algorithm 1 with , results to a that makes the exact transfer feasible.

In particular, for , let ; that is, is the sub-matrix of that is also present in if and only if . Moreover, for , consider if and only if . Moreover, assume that is infeasible by , i.e., . Then, denote as the event where can become feasible by making one more element of one, that is, . It is,

 ϵ(A)=minS⊆[n]:Ξ(S) is true.(∥v∥2−∥v[S]∥2).

Therefore, is positive.

In general, is unknown in advance. Hence, we need to search for a sufficiently small value of so that . Since is lower and upper bounded by and , respectively, we achieve this by performing a binary search. In particular, we implement Algorithm 2, where we denote as the matrix that Algorithm 1 returns for given , and .

In the worst case, when we first enter the while loop, the if condition is not satisfied and, as a result, is set to a lower value. This process continues until the if condition is satisfied for the first time, from which point and on, the algorithm converges, up to the accuracy level , to ; specifically, , due to the mechanics of the bisection. Then, Algorithm 2 exits the while loop and the last if statement ensures that is set below so that is feasible.

The efficiency of Algorithm 2 for Problem 1 is summarized below.

###### Corollary 1

Given the transfer , denote as an optimal solution to Problem 1 and as the corresponding output of Algorithm 2. Then, is feasible by and

 n∑i=1Bii≤⌈ln(∥x(t1)−exp[A(t1−t0)]x(t0)∥2/ϵ)⌉n∑i=1B⋆ii.

where is the approximation level where Algorithm 2 had converged when terminated.

The results of this section apply to the generalized version of Problem 1 with respect to Objective 3 by replacing , and with , and , respectively (where is the output matrix of (1)). Similarly with regard to the approximation algorithm described below.

#### Iii-B2 Approximation Algorithm for Problem 2

We sketch the approximation algorithm for Problem 2 (for the case where is finite), since, then, its implementation is straightforward: Without loss of generality, assume , as the dynamics (1) are linear, and consider the problem of reaching a finite . Observe that can be approximated as a finite union of euclidean balls in . Specifically, let be their centres and their corresponding radii. Then, by executing Algorithm 1 for and, afterwards, selecting the sparsest solution among all , we return an approximate solution to Problem 2. As in Algorithm 1, two levels of approximation underlie here: First, we approximate with a sufficient number of balls, and, then, we approximate the sparsity of the optimal solution to Problem 2; the quality of the latter approximation is quantified in Theorem 3.

We illustrate our analytical findings, and test their performance, in the next section.

## Iv Examples and Discussions

We test the performance of Algorithm 2 over various systems, starting in Subsection IV-A with the networked system of Fig. 1 and following up in Subsection IV-B with Erdős-Rényi random networks. Extending the simulations of this section to the algorithm for Problem 2 is straightforward and, as a result, due to space limitations we omit this discussion.

### Iv-a Star Network

We illustrate the mechanics and efficiency of Algorithm 2 using the star network of Fig. 1, where and

 A=⎡⎢ ⎢ ⎢ ⎢ ⎢ ⎢⎣−111110−100000−100000−100000−1⎤⎥ ⎥ ⎥ ⎥ ⎥ ⎥⎦.

In particular, we run Algorithm 2 for the , and and for . The algorithm returned a equal to , and , respectively; indeed, is feasible by the minimum number of actuators if and only if either is actuated or one among is; is feasible by the minimum number of actuators if and only if and are actuated and, finally, is feasible by the minimum number of actuators if and only if and are actuated. Overall, Algorithm 2 operated optimally.

Evidently, this star network is controllable by the minimum number of actuators if and only if all are actuated. Therefore, whenever we are interested merely in the feasibility of a state transfer, it is cost-effective, with respect to the number of actuators that should be implemented, to design a that does not result to a controllable system as well.

### Iv-B Erdős-Rényi Random Networks

Erdős-Rényi random graphs are commonly used to model real-world networked systems [22]. According to this model, each edge is included in the generated graph with some probability independently of every other edge. We implemented this model for varying network sizes where the directed edge probabilities were set to . In particular, we first generated the binary adjacencies matrices for each network size so that each edge is present with probability and then we replaced every non-zero entry with an independent standard normal variable to generate a randomly weighted graph. The network size varied from to , with step .

For each network size, we run Algorithm 2 for a , where was randomly generated using MATLAB’s “randn” command; for all cases, the algorithm returned a -sparse . This is in accordance with the simulation results of [1], where similarly randomly generated networks were made controllable by actuating one or two states.

Extending the simulations of this section to the algorithm for Problem 2 is straightforward and, as a result, due to space limitations we omit this discussion.

## V Concluding Remarks

We addressed a collection of state (and output) space reachability problems for a linear system, under the additional objective of sparse control, i.e., the control using a minimal number of actuators. In particular, we proved that these problems are NP-hard and provided efficient approximation algorithms for their general solution, along with worst case approximation guarantees. Finally, we illustrated the efficiency of these algorithms with a set of simulations. Optimal behaviour was observed.

Moreover, any optimal control problem, e.g., the LQR. where an objective is optimized with respect to i) the input vector and ii) the sparsity of , subject to the system dynamics, as well as, an initial and final condition of the form and or , respectively, is NP-hard as well. This conclusion suggests a future direction: Which is an efficient approximation algorithm for such optimal control problems? A relevant result is [14], where the authors provide an efficient approximation algorithm for minimizing the input energy for a desired state transfer, subject to a -sparse and a controllable .

Finally, due to Lemmas 1 and 3, and since for the hitting set problem it is NP-hard to find a set whose cardinality is within a factor of from the optimal set [23], it is an open problem to find for Problem 1 an approximation algorithm that achieves an approximation factor, or to prove that this is the case for Algorithm 2.

## Appendix A Proofs of the Main Results

### A-a Lemma 1

{proof}

Denote as the -th row of . It is proved in [1] that has a -hitting set if and only if is -controllable (that is, is controllable for being -sparse). Therefore, we prove that is -feasible at time by if and only if is -controllable.

If is -feasible at time , then

 χ=t1∫t0eA(t1−τ)Bu(τ)dτ,

for some input defined over . Let such that and observe that all the entries of are non-negative. Then,

 en×1≤(1+ϵ)V1Bt1∫t0u(τ)dτ.

Set . Therefore, : Assume that there exists such that . Then, ; contradiction. As a result, for all , , which implies, from the PBH theorem, that is -controllable.

Conversely, if is -controllable, then is -feasible at any time by , that is, also for .

### A-B Lemma 2

Due to space limitations, this proof is omitted; it can be found in the full version of this paper, located at the authors websites.

### A-C Lemma 3

{proof}

Let ,

Assume that is a hitting set of cardinality at most for . For all , set . Then, there exists , , i.e., is -reachable, since by writing as

 B=[B(1)m×m0m×p0p×mB(2)p×p],

then

 [B|AB]=[B(1)m×m0m×p000p×mB(2)p×pΦB(1)m×m0].

Conversely, assume that is -reachable. That is, there exists , and consider : Choose an such that and the smallest such that : Set and . It remains true that there exists (possibly different than ), , i.e., that is -reachable. Proceeding likewise for all such that , we construct a -sparse matrix , (while becomes zero). Then, the set is a -hitting set for .

### A-D Theorem 3

{proof}

We denote as a set of columns of such that and the cardinality of is minimum. Also, we denote as the zero-one diagonal matrix such that if and only if . That is, is a sparsest matrix such that is feasible.

For any ,

 v[span{S∪i∈I(#)Ci}]=v.

As successively runs over all the elements of , decreases from to . Thereby, there is some for which the dimension decreases by at least ; otherwise, the total decrease is strictly less that , contradiction. Thus, denoting as the previous indices of in the succession,

 ∥v∥2−∥v[span{(S∪i∈I(#)∖i′Ci)∪Ci′}]∥2≤ ∥v∥2−∥v[span{S∪i∈I(#)∖i′Ci}]∥2 −∥v∥2−∥v[span{S}]∥2|I(#)|.

Furthermore, from Lemma 8.1 in [24]

 ∥v[span{S∪(Ci′∖S)}]∥2−∥v[span{S}]∥2≥ ∥v[span{(S∪i∈I(#)∖i′Ci)∪(Ci′∖(S∪i∈I(#)∖i′Ci))}]∥2 −∥v[span{S∪i∈I(#)∖i′Ci}]∥2,

and since and ,

 ∥v∥2−∥v[span{S∪Ci′}]∥2≤ (2) (1−1|I(#)|)(∥v∥2−∥v[span{S}]∥2). (3)

At Algorithm 1, consider that the while loop has been executed for times, and let denote the corresponding constructed matrix. By the inequality in (2)-(3), there is an such that the next time that the while loop will be executed

 ∥v∥2−∥v[S(Bk+1)]∥2≤ (1−1|I(#)|)(∥v∥2−∥v[S(Bk)]∥2).

Thus,

 ∥v∥2−∥v[S(Bk+1)]∥2≤…≤ (1−1|I(#)|)k∥v∥2≤e−k/|I(#)|∥v∥2.

Thereby, after steps (with being equal to the number of the non-zero elements of ),

 ∥v∥2−∥v[S(B¯k)]∥2≤ϵ,

and, as a result, is -close feasible.

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