Trichotomy for the reconfiguration problem of integer linear systems

11/07/2019 ∙ by Kei Kimura, et al. ∙ 0

In this paper, we consider the reconfiguration problem of integer linear systems. In this problem, we are given an integer linear system I and two feasible solutions s and t of I, and then asked to transform s to t by changing a value of only one variable at a time, while maintaining a feasible solution of I throughout. Z(I) for I is the complexity index introduced by Kimura and Makino (Discrete Applied Mathematics 200:67–78, 2016), which is defined by the sign pattern of the input matrix. We analyze the complexity of the reconfiguration problem of integer linear systems based on the complexity index Z(I) of given I. We then show that the problem is (i) solvable in constant time if Z(I) is less than one, (ii) weakly coNP-complete and pseudo-polynomially solvable if Z(I) is exactly one, and (iii) PSPACE-complete if Z(I) is greater than one. Since the complexity indices of Horn and two-variable-par-inequality integer linear systems are at most one, our results imply that the reconfiguration of these systems are in coNP and pseudo-polynomially solvable. Moreover, this is the first result that reveals coNP-completeness for a reconfiguration problem, to the best of our knowledge.

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1 Introduction

Reconfiguration problem

In reconfiguration problem we are asked to transform the current configuration into a desired one by step-by-step operations. Formally, in this problem, we are given two feasible solutions of a combinatorial problem, then we find the transformation between them, such that all intermediate results are also feasible, and each step conforms to an adjacency relation defined on feasible solutions. The reconfiguration problem investigates the properties of solution spaces of combinatorial problems, and has a deep relationship to the optimization variants of them. After Ito et al. [12] introduced this reconfiguration framework, many researchers applied this framework to a variety of combinatorial problems, including not only graph problems such as independent set, vertex cover, and coloring, but also set cover, knapsack problem, and general integer programming problem. For recent surveys, see [9, 20]. The reconfiguration problem has many possible applications, particularly on ongoing services, e.g., maintenance of power stations and computer networks.

The reconfiguration framework has been also applied to the Boolean satisfiability problem (SAT), which is a central problem in computer science. In the reconfiguration problem of SAT, we are given a Boolean formula and two feasible solutions (i.e., satisfying assignments) and of , and asked to transform to by changing only one variable from true to false or from false to true at a time, while maintaining a feasible solution of throughout. This problem is shown to have dichotomy property in terms of constraint language [8]. Namely, it is shown that the reconfiguration problem of SAT is in P if the constraints satisfy a certain property and otherwise PSPACE-complete, where the property distinguishing these complexities is later corrected by Schwerdtfeger [22]. We note that the reconfiguration problem is called -connectivity problem in [8, 22].

Integer linear systems and complexity index

In this paper, we focus on the reconfiguration problem for integer linear system (ILS for short). In ILS, we are given a matrix

, a vector

, and a positive integer . A feasible solution is an integer vector satisfying , where . We denote by

an instance of ILS. ILS can formulate many combinatorial optimization problems and is a fundamental problem studied in many fields such as mathematical programming and theoretical computer science. The

feasibility problem of ILS asks if there exists a feasible solution of a given instance of ILS. The feasibility problem of ILS has been intensively studied, especially compared to the reconfiguration counter part. The feasibility problem is strongly NP-hard in general, but several (semi-)tractable subclasses are known to exist. For example, the problem can be solved in polynomial time, if is bounded by some constant [18], or if is totally unimodular [11]. Moreover, it can be solved in pseudo-polynomial time if (i) is bounded by some constant [21], (ii) it is a Horn system (i.e., each row of contains at most one positive element) [7, 19], or (iii) it is a two-variable-per-inequality (TVPI) system (i.e., each row of contains at most two nonzero elements) [10, 1]. It is also known that the problem is weakly NP-hard, even if is bounded by some constant or the system is Horn and TVPI (also known as monotone quadratic) [16].

In this paper, we investigate the reconfiguration problem of ILS through the complexity index for ILS introduced in [15]. The complexity index extends a complexity index for SAT introduced in [4]

, and classifies the complexity of the feasibility problem of ILS in terms of the sign structure of the input matrix. For an ILS

, the complexity index of

is the optimal value of the following linear programming problem (LP) with variables

.

(1)

where for a real number , its sign is defined as

(2)

Since LP (1) depends only on the sign pattern of , the index captures the sign structure of ILSes. For , we denote by the family of ILSes with . For the feasibility problem of ILS, the following trichotomy result is shown in [15]: (i) is solvable in linear time for any , (ii) is weakly NP-complete and pseudo-polynomially solvable, and (iii) is strongly NP-complete for any ; see also Table 1 in the next subsection. It should be noted that ILS(1) includes Horn and TVPI ILSes, well-studied subclasses of ILSes. In fact, for a Horn ILS , is a feasible solution to LP (1), and thus the optimal value of LP (1) is at most one. For a TVPI ILS , is a feasible solution to LP (1), and thus the optimal value of LP (1) is at most one. Therefore, these ILSes are included in ILS(1). Horn and TVPI ILSes arise in, e.g., program verification and scheduling, respectively, and many algorithms have been devised to solve the feasibility problems of these subclasses [1, 7, 10, 19]. On the other hand, ILS(1) can be decomposed to Horn and TVPI ILSes in a certain way [15]; see also Section 3. It should be also noted that we can recognize which class a given ILS belongs to in linear time, without solving LP (1[15]. This is useful for practice, since if we recognized in linear time that the index of a given ILS is, say, less than one, then we could use the linear time algorithm to solve the feasibility problem.

Main results of the paper

In this paper, we consider the reconfiguration problem of ILS. Namely, we are given an ILS and two feasible solutions and of , and then asked to transform to by changing a value of only one variable at a time, while maintaining a feasible solution of throughout. We analyze the complexity of this problem using the complexity index described in the previous subsection and show the following three results: the reconfiguration problem of ILS is

(i) always yes if the complexity index is less than one

(ii) weakly coNP-complete and pseudo-polynomially solvable if the complexity index

is

     exactly one

As mentioned in the previous subsection, both Horn and TVPI ILSes are contained in ILS(1) [15]. Therefore, the reconfiguration problem of these ILSes are both in coNP and pseudo-polynomially solvable from this result. Furthermore, SAT can be formulated as ILS with the constant-size numerical inputs, by representing each clause as and setting . Thus, the reconfiguration problem of SAT with index at most 1 is polynomially solvable from this result.

(iii) PSPACE-complete if the complexity index is greater than one

We show that the reconfiguration problem is PSPACE-complete even for SAT. Combining this result with result (ii), we obtain a complexity dichotomy for the reconfiguration problem of SAT in terms of the complexity index. We compare this dichotomy result with the dichotomy result in [8, 22] in the next subsection.

From the above results, we obtain a complexity trichotomy for the reconfiguration problem of ILS; see also Table 1.

We also analyze how far two feasible solutions can be, namely, the maximum of the minimum number of value changes between two feasible solutions. This can be cast as the analysis of the diameter of the solution graph of ILS, where the solution graph is defined as follows: the vertex set is the set of feasible solutions and two vertices are adjacent if their hamming distance (i.e., the number of components having different values) is one. We then show that the diameter of the solution graph of ILS is , , and if the complexity index is respectively less than one, equal to one, and greater than one; see Table 1.

feasibility [15] reconfiguration diameter
P (linear time) (always yes)
weakly NP-complete weakly coNP-complete
pseudo-polynomially solvable pseudo-polynomially solvable
strongly NP-complete PSPACE-complete
Table 1: Results for integer linear systems (results of this paper are in bold). Here, is the number of variables and is the upper bound of the values of the variables.

In Table 1, a problem is pseudo-polynomially solvable if it is solvable in polynomial time in the numeric value of the input. Moreover, a problem is weakly NP-complete (resp., weakly coNP-complete) if it is NP-complete (resp., coNP-complete) in the usual sense, and strongly NP-complete if it is NP-complete even when all of its numerical parameters are bounded by a polynomial in the size of the input.

Our pseudo-polynomial solvability is based on the decomposition of any ILS in ILS(1) into Horn and TVPI ILSes introduced in [15]. In fact, we first show that the reconfiguration problems of these two ILSes are pseudo-polynomially solvable. For Horn ILS, this is done by extending the greedy algorithm for Horn SAT [8], using the fact that the set of solutions of any Horn ILS is closed under a minimum operation (see Section 2 for details). On the other hand, for TVPI ILS, extending the algorithm for 2-SAT in [8] is not straightforward. This is because the majority operation on the Boolean domain (i.e., the SAT case) is uniquely determined, whereas there are many majority operations on non-Boolean domains (i.e., the ILS case), and properties of the set of solutions depend on a majority operation under which it is closed. We reveal that the solution sets of any TVPI ILS are closed under a median operation. Using this closedness property, we can induce a partial order on the set of solutions and devise an algorithm that changes values of variables according to the partial order. For our coNP-completeness result, we use the reduction by Lagarias [16] that shows the weak NP-hardness of the feasibility problem of monotone quadratic ILSes.

For the case of , we show that the solution graph of ILS is always connected. Therefore, any instance of the reconfiguration problem is a yes instance. We show this by reformulating the structural result for ILS with index less than one in [15].

For the case of , we show that the reconfiguration problem is PSPACE-complete even for the SAT problem. As mentioned above, ILS can formulate SAT by representing each clause as and setting . Through this formulation, we can also define complexity index for SAT, and this index actually coincides with the complexity index for SAT problem introduced by Boros et al. [4]. Using the structural expression introduced in [8], we show that the reconfiguration problem is PSPACE-complete for SAT with index greater than one.

Finally, we obtain some positive results complementing the hardness results for ILS(1). An integer linear system is called unit if for positive integers and . We show that the reconfiguration problem of unit ILS(1) is solvable in polynomial time. Note that unit ILS() is PSPACE-complete for , since it contains SAT(). Therefore, we obtain a dichotomy result for unit ILS. Interestingly, the diameter of the solution graph of a system in unit ILS(1) is still and thus the length of the shortest path between two feasible solutions can be exponential in the input size, where we note that is a part of the input and its input size is . Hence, we cannot output an actual path in any polynomial time algorithm for unit ILS(1). Therefore, we devise an algorithm that repeats a certain sequence of value changes implicitly exponential time for these systems. We also show that the reconfiguration problem of ILS(1) is solvable in polynomial time if the number of variables is a fixed constant.

Related work

Our results imply that the reconfiguration problem for SAT has a dichotomy property in terms of the complexity index. Namely, the problem is in P if and otherwise (i.e., if ) PSPACE-complete. This result is incomparable to the dichotomy result in [8, 22], that is, the set of instances in the polynomially solvable class in [8, 22] differs from that with . This is because the results in [8, 22] concern the restriction on the constraint language, namely constraints used to build a problem instance. On the other hand, our result (or the complexity index) focuses on the combination of constraints. For example, consider any one-inequality ILS . Then it is in general within the PSPACE realm in terms of constraint language. However, the complexity index is zero for the ILS, implying that the ILS lies in the P realm in our result. On the other hand, consider an affine equation in the field of order two, where is an addition modulo two. Then the equation lies in the P realm in the result in [8, 22]. The equation can be formulated as an ILS by

(3)

and , and its complexity index is defined by the following LP.

(4)

By a simple calculation, one can verify that the optimal value of this LP is . Hence, the ILS lies in the PSPACE realm in our result. Therefore, the set of instances in the polynomially solvable class in [8, 22] differs from that with . It should be noted that Horn and 2-SAT lie in the P realm in both the results.

In the literature, reconfiguration problems of many combinatorial problems are investigated [9, 20]. For many polynomially solvable problems such as minimum matroid basis and matching problem, the corresponding reconfiguration problems are shown to be polynomially solvable [12]. One may wonder if any reconfiguration for NP-hard problems tends to be PSPACE-complete and conversely, any reconfiguration for the problems in P is shown to be polynomially solvable. However, there are many exceptions. For example, the reconfiguration of the three-coloring problem is solvable in polynomial time [14], while the feasibility problem is known to be NP-complete. Conversely, the reconfiguration of the shortest path problem is PSPACE-complete [3], while the feasibility problem is trivially in P. While there exist results for specific problems, the general framework is still open that delineates a line between easy and hard reconfiguration problems. In this paper, we obtain a trichotomy result for ILS, a large class of combinatorial optimization problems, and we hope the result sheds some light on the general reconfiguration framework. Furthermore, we obtain the first coNP-completeness result for reconfiguration problems as far as we know. Thus, our result gives a new insight to the complexity of reconfiguration problems.

The complexity of a reconfiguration problem is, so far, strongly tied to the diameter of the solution graph. Namely, a reconfiguration problem tend to be in P if the diameter of the solution graph is polynomially bounded in the input size, and PSPACE-complete if the diameter can be exponential. In fact, polynomially solvable reconfiguration problems with exponential diameters have been only known for trivial problems such as Tower of Hanoi (which is always yes), to the best of our knowledge. Our result for unit ILS(1) provides a first example that the reconfiguration problem is in P even if the diameter of the solution graph can be exponential in the input size and the reconfiguration problem is not always yes. This is of independent interest.

The rest of the paper is organized as follows. Section 2 formally defines the reconfiguration problem of ILS and observes useful properties. Section 3 and 4 consider the case where , which is the most technically involved part. Sections 5 and 6 present our results for and , respectively. Finally, we conclude the paper in Section 7.

2 Preliminaries

We assume that the reader is familiar with the standard graph theoretic terminology as contained, e.g., in [2].

2.1 Definitions

We first define the integer linear systems (ILSes). In an ILS, we are given a matrix , a vector , and a positive integer , where and denote positive integers and denotes the set of rational numbers. We denote an ILS by . A feasible solution of is an integer vector satisfying , where . The feasibility problem of ILS is the problem of finding a feasible solution of a given ILS. Note that the bounds on variables, i.e., the domain , allow us to analyze the problem in more details, and also ensure that the solution graph defined below is finite.

For an integer , a subset is called an (-ary) relation on .

Definition 2.1 (Solution graph).

For a relation , we define the solution graph as follows: and , where is the Hamming distance of and . For an ILS , we denote by the solution graph of the set of the feasible solutions of , that is, with .

We call a path from to an - path. If there exists an - path, we say is reachable from . Using this definition, we can treat the reconfiguration problem of ILS as following: in the reconfiguration problem of ILS, we are given an ILS and two feasible solutions and of , and then we are asked whether is reachable from or not.

2.2 Basic observations

In this subsection, we give three lemmas which play important roles in this paper.

The following lemma is an extension of Lemma 4.1 in [8] to a multiple-valued version, and used throughout the paper. We say that an -ary relation is closed under a -ary operation if for every , the tuple is in . We denote this tuple by . An operation is called idempotent if holds for any .

Lemma 2.1.

If a relation is closed under an idempotent operation , then for each connected component in , is closed under .

Proof.

The proof goes along the same line as the proof of Lemma 4.1 in [8], since the proof of Lemma 4.1 only uses the idempotency of . However, we describe the proof for completeness of this paper.

Let be a relation which is closed under an idempotent operation . Consider vectors that all belong to the same connected component of . In the rest of this proof, we show that also belongs to the connected component. To this end, we show that there exists a path between and on .

We first prove that for any integer , , and in the same connected component of , there exists a path from to for any . Let be an - path. For every , the tuples and belong to the same component of , because they differ in at most one variable (the variable in which and are different). Thus and belong to the same component.

Therefore, there exist paths from to , from to and from to . Thus there exists a path between and on . ∎

We next see that we can replace any column of with its opposite vector without changing the reachability. Observe first that the feasibility of an integer linear system does not change if we replace a variable with a new variable . Namely, the feasibility of is equivalent to that of , where is obtained from by replacing the -th column with . Moreover, the reconfiguration problem of can be reduced to that of as the following lemma shows.

Lemma 2.2.

Let be an ILS and be solutions of . Then, for any , is reachable from in if and only if is reachable from in , where and is obtained from by replacing the -th column with .

Proof.

Assume that is reachable from in . Let an - path be . By replacing with for each , we obtain a path from to in . Hence, is reachable from in . The converse can be proven similarly. ∎

By inductively applying Lemma 2.2, we can replace any columns of by their opposite vectors without changing reachability by changing vector appropriately.

We also use the following lemma in Sections 3 and 4. A matrix is Horn if each row of has at most one positive element. An ILS is called Horn if the input matrix is Horn. The minimum operation is a binary operation that outputs the smaller value of the two inputs. For an ILS, a feasible solution is called a unique minimal solution of the ILS if it satisfies for all the feasible solutions of the ILS. Here, for two vectors and , holds if for all .

Lemma 2.3 (E.g., [19]).

The set of feasible solutions of a Horn ILS is closed under the minimum operation. Since any nonempty relation on closed under the minimum operation has a unique minimal solution, so does any feasible Horn ILS.

3 The general case of

In this section, we show that the reconfiguration problem of is weakly coNP-complete and pseudo-polynomially solvable.

3.1 Basic Properties

In this subsection, we summarize useful properties of ILS(1).

3.1.1 Horn integer linear systems

In this subsection, we treat Horn ILS. Recall that ILS is called Horn if each row of the input matrix has at most one positive element. Let be an Horn ILS. From Lemma 2.3 in Subsection 2.2, the set of feasible solutions of is closed under the minimum operation. Since the minimum operation is idempotent, each connected component of is also closed under the minimum operation by Lemma 2.1 in Subsection 2.2, It follows that there exists a unique minimal solution in each connected component by Lemma 2.3. We show that any vertex of is connected to the unique minimal solution in the same connected component via a monotone path. Here, a path is monotone if holds, where we recall that for two vectors and , we have if and only if holds for all .

Lemma 3.1.

For a Horn system , each vertex in is connected via a monotone path to the unique minimal solution in the same connected component.

Proof.

Let be a Horn ILS. Let be an arbitrary feasible solution of and let be the unique minimal solution in the same component as on . Since they are in the same connected component, there exists an - path on . Let such a path be . Note that this path may not be a monotone path.

Now we show that we can construct a monotone - path. Let and for each , . Note that by minimality of , and for each , . In the rest of the proof, we show that for each , . Then we immediately have a monotone - path (if needed, we delete the redundant feasible solutions).

Let be arbitrary vectors with . We can easily have that . Using this property, we show that in the following. For , we have

since . For , we have and

(5)

where we use associativity of the minimum operation in the last equality. Since , we have . Then follows as desired. This completes the proof. ∎

3.1.2 Two-variable-per-inequality (TVPI) integer linear systems

In this subsection, we treat TVPI ILS, i.e., ILS where each row of the input matrix has at most two nonzero elements. We first show that the solution set of a TVPI ILS is closed under a median operation, where a ternary operation is the median operation on if it outputs the middle value of the three inputs. For example, we have and . The fact might be already known, however, the authors cannot find it in the literature.

Proposition 3.2.

The solution set of a TVPI ILS is closed under a median operation.

Proof.

Let be the median operation on . Note that holds for all .

For a TVPI inequality , let be solutions of the inequality, i.e., we have , , and . We show that holds, which proves the proposition.

Without loss of generality, we assume that . Hence, we have . Then it suffices to show that holds. We show this for all the sign patterns of and .

Case 1: and
In this case, we have

(6)

Case 2: and
In this case, it follows that

(7)

where the second last inequality follows from .

Case 3: and
In this case, we have

(8)

where the second last inequality follows from .

Case 4: and
In this case, it follows that

(9)

where the second last inequality follows from .

Therefore, we show that in all cases holds. This completes the proof. ∎

We also use the following observation to show our result. For , let be defined as for any , where is the median operation. Then is a semilattice operation, which is shown in the following lemma. Here, a binary operation is semilattice if it is (i) associative, (ii) commutative, and (iii) idempotent.

Lemma 3.3.

For , defined as above is a semilattice operation.

Proof.

We show that each axiom of semilattice operations holds for .

For associativity, we have to show that

(10)

The middle equality can be shown by checking all the possibility of the magnitude relations on and . For example, if holds, then we have and , and thus the equality holds. We leave the reader to check the equality for the other possibilities.

For commutativity, it follows that . Finally, for idempotency, we have . Hence, is a semilattice operation. ∎

From Lemma 3.3, we can construct a poset induced by for , where for any , if and only if .

From Lemma 2.1 in Subsection 2.2 and Proposition 3.2, each connected component of is closed under the median operation on for a TVPI system , since median operations are idempotent. For any feasible solution to , define as for any . Then, as in Lemma 2.1, we can show that each connected component of is closed under , i.e., for two feasible solutions to in the same connected component is also in the component. For any two vectors and , let hold if and only if holds for all . Similar to Lemma 3.1, we show that there exists a -monotone path from any vertex of to the -unique minimal solution in the same connected component, where a -monotone path and unique -minimality are defined analogously to a monotone path and unique minimality, respectively.

Lemma 3.4.

For a TVPI system and a feasible solution to , each vertex is connected to the unique -minimal solution in the same connected component via a -monotone path.

Proof.

The proof is similar to the one for Lemma 3.1.

Let be a TVPI integer linear system and be an arbitrary feasible solution to . Let be an arbitrary feasible solution of and let be the unique -minimal solution in the same component as on . Since they are in the same connected component, there exists an - path on . Let such a path be . Note that this path may not be a -monotone path.

Now we show that we can construct a -monotone - path. For each , , we define as follows:

(11)

By the same discussion on Lemma 3.1, we have a -monotone - path (if needed, we delete the redundant feasible solutions). This completes the proof. ∎

3.1.3 Decomposition of ILS(1)

We here recall that any instance of ILS(1) can be decomposed into Horn and TVPI systems. All the result in this subsection are from [15]. It is known that an instance of ILS(1) admits a -partition Let be a variable index set. A partition (and ) is called a -partition of , if it satisfies the following three conditions:

(a)

Each row of contains at most two nonzero elements with .

(b)

Each row of contains at most one positive element with .

(c)

If a row of contains a positive element with , then the elements with are all zeros.

For a -partition, let denote the set of rows of such that for all . Define . For a row and column index sets and , let denote the submatrix of whose row and column sets are and , respectively. Moreover, for a vector and , let denote the restriction of to . Then, we can decompose the integer linear system as follows:

(12)

where we note that by the definition of . Moreover, note that by the condition (b) of -partition, the system is Horn, i.e., each row of contains at most one positive element. Similarly, the elements of are nonpositive and each row of contains at most two nonzero elements, respectively by conditions (c) and (a) of -partition.

3.2 Pseudo-polynomial solvability

In this subsection, we show the following theorem.

Theorem 3.5.

The reconfiguration problem of is pseudo-polynomially solvable.

To show the theorem, we first consider two subclasses of ILS(1). Namely, we show that the reconfiguration problems of Horn and two-variable-per-inequality (TVPI) ILS are pseudo-polynomially solvable in Subsections 3.2.1 and 3.2.2, respectively. Then, using these results, we show Theorem 3.5 in Subsection 3.2.3.

3.2.1 Horn integer linear systems

In this subsection, we treat Horn ILS, i.e., ILS where each row of the input matrix has at most one positive element. To show that the reconfiguration problem of Horn ILS is pseudo-polynomially solvable, we use Lemma 3.1 in Subsection 3.1.1.

Proposition 3.6.

The reconfiguration problem of Horn ILS is pseudo-polynomially solvable.

Proof.

Let be a Horn ILS and let and be feasible solutions of . Let and be the unique minimal solution in the same component as and respectively. Clearly, and are connected if and only if holds. From Lemma 3.1, (resp., ) can be obtained by greedily following a smaller feasible solution from (resp., ). Note that the length of any monotone path is at most since a value of one component decreases in each step. Therefore, we can obtain (resp., ) in time polynomial in and , which implies that the reconfiguration problem of Horn ILS is pseudo-polynomially solvable. ∎

Proposition 3.7.

The diameter of each component of is for any Horn ILS .

Proof.

Let be a Horn ILS. From Lemma 3.1, any two vertices of in the same component are connected by two monotone paths via the unique minimal solution in the component. Since the length of any monotone path is at most , the diameter of each component of is at most .

We now show that the diameter of can be even for a monotone quadratic ILS, where an ILS is monotone quadratic if each inequality has at most one positive coefficient and at most one negative coefficient. Note that a monotone quadratic ILS is a Horn (and TVPI) ILS.

Example 3.1.

Consider the following monotone quadratic ILS.

(13)

The diameter of the solution graph of ILS (13) is . Indeed, consider a path from to . Then we can increase a value of a variable at most one in each step, since any two consecutive variables can differ by at most one. Therefore, the length of the path is at least . Thus, the diameter of the solution graph of system (13) is .

Combining the upper and lower bounds, we obtain that the diameter of each component of is for any Horn ILS . ∎

3.2.2 Two-variable-per-inequality (TVPI) integer linear systems

In this subsection, we treat TVPI ILS, i.e., ILS where each row of the input matrix has at most two nonzero elements. To show that the reconfiguration problem of TVPI ILS is pseudo-polynomially solvable, we use Lemma 3.4 in Subsection 3.1.2.

Proposition 3.8.

The reconfiguration problem of TVPI ILS is pseudo-polynomially solvable.

Proof.

The proof goes along the same line as that of Lemma 3.6. Let be a TVPI integer linear system and be solutions of . Let be the unique -minimal solution in the same component as . Then and are connected if and only if holds. From Lemma 3.4, can be obtained by greedily following a smaller feasible solution (in terms of ) from . Note that the length of any -monotone path is at most since a value of one component decreases (in terms of ) in each step. Therefore, we can obtain in time polynomial in and . This implies that the reconfiguration problem is pseudo-polynomially solvable. ∎

Proposition 3.9.

The diameter of each component of is for any TVPI integer linear system .

Proof.

Since any two vertices of in the same component are connected by a path of length at most , the diameter of each component of is .

Moreover, since ILS (13) is a TVPI system, the diameter of each component of can be .

Combining the upper and lower bounds, we obtain that the diameter of each component of is for any TVPI ILS . ∎

3.2.3 General case of

We now show Theorem 3.5, that is, the reconfiguration problem of ILS(1) is pseudo-polynomially solvable. We describe our algorithm to solve the reconfiguration problem of ILS(1) in Algorithm 1; see Subsection 3.1.3 for notation. The algorithm first solves the reconfiguration problem on the Horn system and then solves the reconfiguration problem on a certain TVPI system.

1:  compute a -partition of and let
2:  if  and are not connected in  then
3:     output “NO” and halt
4:  else
5:     compute the unique minimal solution in the same component as
6:      and in and let .
7:  end if
8:  if  and are connected in  then
9:     output “YES” and halt
10:  else
11:     output “NO” and halt
12:  end if
Algorithm 1 Solving the reconfiguration problem of with
Lemma 3.10.

Algorithm 1 solves the reconfiguration problem of ILS(1) in time polynomial in , and .

Proof.

We show that and are connected in if and only if Algorithm 1 outputs “YES”.

We first prove the if direction. Assume that Algorithm 1 outputs “YES”. Then we can construct an - path in using monotone paths from to and from to , and a -monotone path from to , which exist since the algorithm outputs “YES” and from Lemmas 3.1 and 3.4. Since we use monotone paths, vectors from and are all solutions of , since the elements of are nonpositive. Indeed, for in , we have by monotonicity and thus holds since is a nonpositive matrix. Therefore, we have

(14)

where the second inequality holds since is a solution to . Since is a solution to , this implies that is a solution to . Similarly, any vector in is a solution to . Moreover, vectors in are solutions of by the definition of in the algorithm. Therefore, we obtain an - path in , implying that and are connected in .

We next prove the only-if direction. We assume that and are connected in . Let be an - path in . Then clearly the restriction of to variable indices is an - path in . Moreover, the restriction of to variable indices is an - path in