On the Tightness of Bounds for Transients of Weak CSR Expansions and Periodicity Transients of Critical Rows and Columns of Tropical Matrix Powers
Glenn Merlet, Thomas Nowak, Sergei Sergeev
TL;DR
This paper investigates the bounds on transients in max-plus algebra matrices, characterizing matrices that attain these bounds and extending classical graph theory bounds to the weighted case.
Contribution
It characterizes matrices that attain specific bounds on the weak CSR threshold, generalizing Wielandt and Dulmage-Mendelsohn bounds to weighted digraphs in max-plus algebra.
Findings
Characterization of matrices attaining bounds on $T_1$
Extension of classical bounds to weighted digraphs
Insights into periodicity transients of critical matrix components
Abstract
We study the transients of matrices in max-plus algebra. Our approach is based on the weak CSR expansion. Using this expansion, the transient can be expressed by , where is the weak CSR threshold and is the time after which the purely pseudoperiodic CSR terms start to dominate in the expansion. Various bounds have been derived for and , naturally leading to the question which matrices, if any, attain these bounds. In the present paper we characterize the matrices attaining two particular bounds on , which are generalizations of the bounds of Wielandt and Dulmage-Mendelsohn on the indices of non-weighted digraphs. This also leads to a characterization of tightness for the same bounds on the transients of critical rows and columns. The characterizations themselves are generalizations of those for the non-weighted case.
Peer Reviews
No public reviews on file for this paper yet. If you reviewed it on a platform where reviews are public (OpenReview, ICLR, NeurIPS, ICML), you can paste yours below so the community can read it here.
Videos
No videos yet. Explain this paper in a talk, walkthrough, or lecture? Add one.
On the Tightness of Bounds for Transients of Weak CSR Expansions and Periodicity Transients of
Critical Rows and Columns of Tropical Matrix Powers222This work was partially supported by ANR Perturbations grant (ANR-10-BLAN 0106). The work of S. Sergeev was also supported by EPSRC grant EP/P019676/1.
Glenn Merlet
Thomas Nowak333T. Nowak was with the Ăcole Normale SupĂ©rieure, Paris, France, when this work was initiated.
SergeÄ Sergeev
Aix Marseille Univ, CNRS, Centrale Marseille, I2M, Marseille, France
Université Paris-Saclay, CNRS, Orsay, France
University of Birmingham, School of Mathematics, Edgbaston B15 2TT, UK.
Abstract
We study the transients of matrices in max-plus algebra. Our approach is based on the weak CSR expansion. Using this expansion, the transient can be expressed by , where is the weak CSR threshold and is the time after which the purely pseudoperiodic CSR terms start to dominate in the expansion. Various bounds have been derived for and , naturally leading to the question which matrices, if any, attain these bounds.
In the present paper we characterize the matrices attaining two particular bounds on , which are generalizations of the bounds of Wielandt and Dulmage-Mendelsohn on the indices of non-weighted digraphs. This also leads to a characterization of tightness for the same bounds on the transients of critical rows and columns. The characterizations themselves are generalizations of those for the non-weighted case.
keywords:
Max-plus, matrix powers, transient, periodicity, digraphs. AMS Classification: 15A18, 15A23, 90B35
1 Introduction
Max-plus algebra is a version of linear algebra developed over the max-plus semiring, which is the set equipped with the operations (additive) and (multiplicative). This semiring has a zero , neutral with respect to , and a unity , neutral with respect to . The multiplicative operation is invertible, that is, for each there exists an element such that .
These arithmetical operations are extended to matrices and vectors in the usual way. Matrix addition is defined by for two matrices and of equal dimensions, and matrix multiplication by for two matrices and of compatible dimensions. Here we are interested in tropical matrix powers:
[TABLE]
assuming that , the max-plus identity matrix, in which all diagonal entries are equal to and all off-diagonal entries are equal to .
As we see, matrix powers are easy to define for natural . As for negative , the problem is that the set of max-plus matrices for which inverse exists is very scarce (see, e.g.,[6] Theorem 1.1.3 for a complete description). However, we will make use of invertible max-plus diagonal matrices: matrices in which are real for all and when . The inverse of , denoted by , is also a diagonal matrix with diagonal entries equal to for all , so that we have .
In what follows, the multiplication sign will be always omitted in the case of matrix multiplication, but always kept in the case of multiplication by scalars. In particular, we write and for .
The fundamental result on tropical matrix powers [9] states that if is irreducible then there exist a real and integers and such that
[TABLE]
The smallest nonnegative for which (2) holds is called the *transient * of matrix , denoted by . The transient can be shown to be independent of the choice of and . In fact, is the largest mean cycle weight in the weighted digraph described by . Bounds on transients have been studied by many authors, e.g., Hartmann and Arguelles [13], Bouillard and Gaujal [4] Soto y Koelemeijer [22], Akian et al. [1], Charron-Bost et al. [7], and the authors [14]. The time behavior and complexity of several real systems can be reduced to determining the transient of max-plus matrices. These applications include communication networks [3], cyclic scheduling [10], link-reversal algorithms [8], network synchronizers [12], as well as transportation and manufacturing systems [2].
An analog of the ultimate pseudoperiodicity of irreducible max-plus matrices for real-valued non-negative matrices is provided by the Perron-Frobenius theorem. The largest mean cycle weight is analogous to the spectral radius of non-negative matrices. The spectral radius of stochastic matrices, i.e., non-negative matrices whose row sums are all , is equal to . The sequence of powers of an irreducible stochastic matrix is known to converge to a rank- stochastic matrix and the rate of convergence is known to depend on the matrixâs spectral gap, i.e., the difference between its largest and its second largest eigenvalue. Similarly, the transient of an irreducible max-plus matrix depends on the difference between the largest and the second-largest mean cycle weight.
If all matrix entries in are restricted to or then the tropical matrix algebra becomes Boolean matrix algebra (i.e., linear algebra over the Boolean semiring), and the associated digraph becomes unweighted. Powers of Boolean matrices have been thoroughly studied in combinatorics (see, e.g., [5]), and various bounds on their transient of periodicity, called index or exponent in these cases, have been obtained. One well-known application is the Frobenius coin problem, which can be seen as calculating the transient of a specifically constructed graph. For general connected graphs, Wielandt [23] proved the bound and Dulmage and Mendelsohn [11] proved , where is the girth of the graph.
Note that the same problem can be considered locally: for each pair , given and that work for (2), find the minimal such that
[TABLE]
Denoting that minimal by , we can also consider and . These quantities are called the transient of the th column of and the transient of the th row of , respectively. The bounds for such transients can be much lower than those for , as it was shown by [15] in an important special case of critical rows and columns, i.e., in the case where the index of the row or column corresponds to a critical node (see Definition 2.7 below).
The eventual periodicity was reformulated and generalized by Sergeev and Schneider [19, 20] via the concept of CSR expansions. They observed that in the case of eventual periodicity of there is a big enough for which we have
[TABLE]
where and are constructed from (see Definition 2.11 below) in such a way that is a purely pseudoperiodic sequence (i.e. for any and some and ). The smallest for which (4) holds is equal to .
In [14], we used this approach to unify and improve the known bounds on . To this aim, we introduced the so-called weak CSR expansions. Observe that, given any , there exists a big enough for which
[TABLE]
Here , , and are defined as in (4) and is a matrix obtained from by setting some entries to . The smallest number for which (5) holds is called the weak CSR expansion threshold and denoted by . In this paper we consider only the case , where all entries with an index corresponding to a critical node are set to (see Definition 2.11 below). In this case will be abbreviated to .
For irreducible Boolean matrices, and coincide. For reducible ones, they coincide, as soon as . This allows to hope for extensions of bounds on exponents of graph to in the framework of max-plus algebra, In particular, Wielandt and Dulmage-Mendelsohn bounds were extended in Theorem 4.1 of [14].
However, no information was given in [14] on the question of which classes of matrices attain these bounds for . For the index of digraphs, those results are well-known. In particular, the digraph attaining the Wielandt bound is unique up to renumbering the nodes, and the digraphs attaining the bound of Dulmage-Mendelsohn were studied by Shao [21]. The main results of the present paper are Theorem 3.1, which characterizes all matrices (or all weighted digraphs) such that attains the Dulmage-Mendelsohn bound, and Theorem 3.7, which characterizes those attaining the Wielandt bound. Unfortunately, we have not been able to characterize the matrices that reach the bounds for other choices of studied in [14].
On the other hand, in [15] we had proved that the same bounds (and others) also apply to the transient of a critical row or column of matrices. Theorems 3.6 and 3.10 characterize the matrices for which these bounds are reached.
The paper is organized as follows. After giving preliminary definitions and results in Section 2, we state our characterizations in Section 3.1 and give a quick overview of how to prove them. The characterizations for the Dulmage-Mendelsohn bound on the weak CSR threshold is then proved in Section 4 and that for the Wielandt bound in Section 5. Finally, in Section 6, we prove the characterizations for the transients of critical rows and columns.
2 Preliminaries
2.1 Digraphs and Walks
Most of the techniques for analyzing the max-plus matrix powers and their behavior are based on consideration of walks on the associated weighted digraphs. Hence it is essential to introduce the notion of weighted digraph associated with a given max-plus matrix, as well as the related notions of walks, connectivity, girth and cyclicity.
Definition 2.1** (Associated digraph).**
Let . The digraph associated with , denoted by , is defined as the pair where is the set of nodes and is the set of arcs connecting these nodes. Arc has weight .
Definition 2.2** (Walks).**
A sequence , where , is called a walk on a digraph if for any the arc is in .
For a walk we define the length of the walk as , which is the number of letters in that walk (as a sequence of letters) minus one.
If then is called closed.
A closed walk with no proper closed subwalk is called a cycle.
If and then the weight of is defined as .
Definition 2.3** (Walk sets).**
Let be a set of walks on a weighted digraph , and let be a subdigraph of . Denote (the maximal weight of all walks in ).
The following sets of walks will be particularly useful.
the set of walks from to that have length . 2. 2.
the set of walks from to that have length modulo . 3. 3.
the set of walks of that go through a node in .
Observe that a sequence of nodes is not necessarily a walk. Also observe that an easy way to change a walk into another walk is to remove closed subwalks from a given walk, or to replace a (possibly empty) subwalk by another walk with the same start and same end. This will be the main tool of this article.
The following optimal walk interpretation of matrix powers is well-known:
[TABLE]
See, for example, [6, Example 1.2.3].
We now give some definitions related to connectivity in digraphs.
Definition 2.4** (Connectivity).**
A digraph is called strongly connected if for each there is a walk from to .
A digraph is called a sub(di)graph of if and
Maximal strongly connected component of , further abbreviated to s.c.c., is a maximal strongly connected subgraph of .
A digraph is called completely reducible if for any pair of s.c.c. of there is no walk connecting a node of one s.c.c. to a node of another s.c.c.
Definition 2.5** (Maximal Girth).**
For a strongly connected digraph , the girth of is defined as the minimal length of a cycle in .
For a completely reducible digraph , the maximal girth of , denoted by , is defined as the maximal girth of the s.c.c.âs of .
Although we use the same notation , note that this is not what is usually called the girth of a reducible graph, namely the least common multiple of the girths of its s.c.c.âs, a quantity not used in this paper.
Definition 2.6** (Cyclicity).**
For a strongly connected graph , the cyclicity of , denoted by , is defined as the greatest common divisor of all cycle lengths of .
For a completely reducible digraph of , the cyclicity of is defined as the least common multiple of the cyclicity of the strongly connected components of .
In max-plus algebra we deal not only with but also with special subgraphs of it such as the critical graph of the following definition.
Definition 2.7** (Maximum cycle mean and critical graph).**
The maximum cycle mean of is
[TABLE]
The critical graph of , denoted by , is a subdigraph of consisting of all nodes and arcs of the cycles that attain the maximum in (7). Such nodes and arcs are also called critical.
Definition 2.8** (Visualization).**
We say that is *visualized * if for all and and whenever is an arc of . It is *strictly visualized * if it is visualized and if and only if is an arc of .
A *scaling * of is a matrix of the form where is a diagonal matrix with finite diagonal entries. A *visualization * of is a scaling that is visualized. Likewise, a *strict visualization * of is a scaling that is strictly visualized.
Theorem 2.9** ([18]).**
Every with has a strict visualization.
2.2 Weak CSR Expansion
We now present important definitions, notations and facts related to the main theme of this work.
Definition 2.10** (Kleene star).**
Let with . Then
[TABLE]
is called the Kleene star of . Recall that denotes the max-plus identity matrix (which has on the diagonal and off the diagonal).
Definition 2.11** (CSR).**
Let . If , set , where is the cyclicity of , and define matrices and by
[TABLE]
If , let be the matrix in with only entries for any .
This definition is best understood in combination with Proposition 2.16 part 1, which gives an optimal walk interpretation of : the maximal weight of walks connecting to that have length modulo . Optimal walk interpretation also gives an idea why we have âdivisionâ by in the definition of and hence and : the lengths of corresponding optimal walks are not controlled. Informally speaking, CSR is related to turnpike theorems in a discrete deterministic case [1] at least in some optimal long walks most of the material should be concentrated on the critical graph so that is determined by for large enough . This also shows why there is no âdivisionâ by in .
Below we also deal with some auxiliary matrices, for which the CSR terms are (a priori) different from those derived from . Therefore we will write for a CSR term derived from .
Definition 2.12** ().**
The Nachtigall matrix is defined as the matrix whose entries are
[TABLE]
This is the most obvious choice of a matrix that appears in a weak CSR expansion, since if then expresses the optimal weight of all walks that connect to and do not touch any critical node of .
Definition 2.13** ().**
The *weak CSR threshold * is the least , for which
[TABLE]
holds. In the sequel, is abbreviated to .
We will further work with the following two bounds on , which originate in the works on digraph exponents or indices of imprimitivity and (in the case of unweighted digraphs and matrix powers over Boolean algebra) are due to Wielandt [23] and Dulmage and Mendelsohn [11].
Definition 2.14** ( and ).**
For any (the set of natural numbers) and any , we define
[TABLE]
Theorem 2.15** ([14] Theorem 4.1).**
For and , we have: .
Proposition 2.16** ([14],[20]).**
Let have .
CSR terms have the following optimal walk interpretation:
[TABLE]
for being any multiple of . 2. 2.
* for all (periodicity).* 3. 3.
* for all (group law).* 4. 4.
. (limit property) 5. 5.
**
Parts (3) and (5) also hold with general .
Proof.
(1): This property follows from [20] Theorem 3.3, or [14] Theorem 6.1 where a more general statement is given.
(2), (3): These properties are shown in [20] Proposition 3.2 and Theorem 3.4.
(4): It is obvious that and therefore . The claim then follows from the weak CSR expansion and the periodicity of (ii).
(5) can be deduced from (4) (as could (3)).
Extension to general follows from the homogeneity of (iii) and (v). â
3 Theorem Statements and Proof Strategy
3.1 Statements
For any matrix and any , we define the following matrices.
[TABLE]
[TABLE]
[TABLE]
Figure 1 shows an example of and . The definitions of , and imply that
[TABLE]
Theorem 3.1**.**
Let with . Then satisfies if and only if there exists a renumbering of nodes such that the following conditions hold.
* and are coprime;* 2. 2.
* is strongly connected with a unique critical cycle of length up to choice of its first node;* 3. 3.
* is critical* 4. 4.
; 5. 5.
; 6. 6.
,
where , and are defined as in (9), (10) and (11).
The renumbering satisfying Conditions 1â6 is necessarily unique. More precisely, it is the only one that ensures that
* is an Hamiltonian cycle of with the largest weight, which is unique up to choice of its first node.*
- 2.
* is critical.*
Remark 3.2**.**
Note that in the above theorem we do not assume that is irreducible. The same is true about all the statements in this section. However, it follows from Condition 4 that is irreducible (and aperiodic) and thus, so is .
Remark 3.3**.**
The case turns out to be much more complicated. Although some results do apply (e.g. Proposition 4.2) we were not able to characterize the matrices reaching the bound. Notice that already in the Boolean case the situation is more complicated and not completely understood (see [21]).
On the other hand, if the situation is simpler : with holds if and only if and . In this case Conditions 5 and 6 above hold automatically. For Condition 6, note that is acyclic hence .
Remark 3.4**.**
The index of reaches the bound . It is easy to recover the characterization of such graphs obtained in [21] from the theorem.
Let us see what Theorem 3.1 means on an example.
Example 3.5**.**
We fix and . The theorem says that any matrix of this type that reaches the bound can be decomposed as in Equations (9), (10),(11) with as critical cycle. Let us assume that and all other finite entries of equal . (See. Figure 2.)
Let us compute with Scicoslab (a fork from Scilab which incorporates the MaxPlus Toolbox). We get
[TABLE]
The colored entries are those for which and thus Condition 4 is trivial.
For those entries, Conditions 2 and 3 mean that , while Condition 5 means that .
Finally, get
[TABLE]
where the matrix inequality means that each entry of should be equal to the corresponding entry if it is denoted by = and strictly less than it otherwise.
A computation of shows that Condition 6 can not be removed. For instance, it will be satisfied for not for .
In [15][Lemma 8.2], we have noticed that for any matrix with , the maximal transient of its critical rows or columns is at least the index of its critical graph, so that if the index of reaches the bound, so does the transient of one row and one column. The following shows that the converse is also true.
Theorem 3.6**.**
Let and . Then the transient of the critical rows and columns of is equal to if and only if its critical graph has index .
Theorem 3.7**.**
Let . Then, if and only if there exists a renumbering of nodes such that
* and is critical, or and is critical,*
- 2.
,
where and are defined as in (9) and (11) with in both cases.
The renumbering satisfying these conditions is necessarily unique. More precisely, it is the only one that ensures that
* is an Hamiltonian cycle with the largest weight, which is unique up to choice of its first node.*
- 2.
* is a cycle of length with the largest weight, which is unique up to choice of its first node.*
Remark 3.8**.**
The digraph is exactly the unique (up to renumbering) digraph whose index reaches the bound .
Let us see what Theorem 3.7 means on an example.
Example 3.9**.**
We fix . The theorem says that any matrix of this type that reaches the bound can be decomposed as in Equations (9) and ,(11) with . Let us assume that and . (See. Figure 3.)
Let us compute with Scicoslab. We get
[TABLE]
The colored entries are those for which and thus there is nothing to check. satisfies if and only if each other entry of is strictly less than the corresponding entry of .
For the transient of critical rows or columns, we get:
Theorem 3.10**.**
Let . Then the transient of the critical rows and columns of is equal to if and only if it is of the form such that the index of is , has a critical Hamiltonian cycle, and .
In contrast with the previous case, the critical graph need not have index . It can also be a Hamiltonian cycle, and only has index .
3.2 Overview of Proofs
Each of the next section is devoted to the proof of one of the main theorems (Theorem 3.1 and 3.7). The proofâs strategy is the same for both theorems. It relies on the classical interpretation of entries of powers as weight of walks (cf. Equations (6) and (15)).
The necessity of the conditions is proved in 3 steps :
Firstly, we use (slight refinements of) the results of [14] to prove properties of the critical graph. (Proposition 4.2)
- 2.
Then we introduce a special type of walks with given start and end nodes, which we call âinteresting walksâ. Loosely speaking, these walks are optimal in terms of both weight and length (Definition 4.9). The structure of interesting walks is then studied in Propositions 4.10, 4.12 and 5.1. In particular, interesting walks contain a Hamiltonian cycle that gives the renumbering of nodes up to where to start. The start is determined by the critical nodes.
- 3.
Finally, we use our results on the structure of interesting walks to show that if one condition is not fulfilled, then one can build new shorter walks and thus the interesting walk would not be optimal, either in terms of weight, or length.
The proof of the sufficiency is based on the fact that the index of reaches the bound and the following perturbation lemma, which will also be used in the proof of necessity and we think is interesting for its own sake. We write if for all , or .
Lemma 3.11** (Perturbation lemma).**
If with such that , then, we have , , for all and .
This lemma will be proved, together with all other statements, in the next section.
4 Matrices attaining the Dulmage-Mendelsohn Bound
This section will be devoted to the proof of Theorem 3.1. In the rest of the work we assume (because neither , nor the conditions of the theorem are modified if is multiplied by a scalar) and set .
Observe that ensures that all closed walk have nonpositive weight, so that removing a closed subwalk from a given walk can only increase its weight. A fact that we will use extensively in this paper.
4.1 Perturbation Lemmas
In this section, we prove two lemmas, to be used in both ways of the equivalence. The first one was announced at the end of section 3.1.
If for two matrices and we have for all and that then we write . Observe that if and then and .
Proof of Perturbation Lemma 3.11.
If , then , and there is nothing to prove. Otherwise, since is invariant under , we assume without loss of generality that .
To prove the first equalities, consider a diagonal matrix that provides a visualization scaling for . Then we obtain where . So is visualized and if and only if , which implies that and , which are the first two equalities in the statement of Lemma 3.11.
To prove the remaining two equalities, let us first prove the following statement by induction:
[TABLE]
For , set .
Suppose that (13) holds. Let us prove that it also holds when is replaced by . We have
[TABLE]
We bound from above the last three terms on the right-hand side of (14).
We have:
. by Proposition 2.16 part (5)
- 2.
, by Proposition 2.16 part (5).
- 3.
, by Proposition 2.16 part (3).
Thus where satisfies .
Observe that for . Indeed, for any such ,
[TABLE]
From this equality, the periodicity of and the observation that since , we deduce that and have the same since
[TABLE]
where is the cyclicity of . Thus for all .
To prove the remaining equality , we first observe that , and hence
[TABLE]
Next we use that , that where , and that , so that we have the following equivalences:
[TABLE]
thus . â
Lemma 4.1**.**
Let be defined from the same matrix by Equations (9), (10) and (11). Assume that Conditions 1 to 5 of Theorem 3.1 are satisfied, that is is strongly connected and contains all edges of , and are coprime, and satisfies Condition 5. Then for all .
Proof.
As in the first paragraph of the proof of Lemma 3.11 consider a diagonal matrix providing a visualization scaling for , assuming without loss of generality that . We then have , and this shows that all arcs of are non-critical, hence . In particular, is strongly connected and contains all edges of . The same is true about , and we also have .
Recasting the above statement about CSR in terms of walks with Proposition 4.3 we have to prove for any that
[TABLE]
Since , we have the inequality
[TABLE]
To prove the opposite, we need to take an arbitrary walk and prove that there exists a walk such that .
There are two kinds of arcs in that may not be in .
A. with and . In this case, we can replace by the path . The resulting walk visits even more nodes and hence will go through a node of . It has the same length modulo . Due to Condition 5 of Theorem 3.1, its weight is not smaller than . Thus we can assume that does not contain such arcs.
B. with and .
Since we assumed that contains no arc s.t. , the only arc that go from a node to a larger node are of type , thus if a subwalk of goes from to it goes through all nodes numbered between and .
If  goes to after arc , it has to go through again before reaching , because the only arc to reach from is . In this case, define as the closed subwalk that starts with the arc and follows until it goes back to .
If  does not go to after arc , it has to come from before, so it has to go through which is the only arc leaving . So, it has been in before reaching and arc . Then, define as the closed subwalk that starts with the last occurrence of before arc and follows until it goes back to .
In both cases, lives on so its length is divisible by and it can be removed from .
The resulting walk has the same length modulo , goes through a node of and has larger weight than .
Iterating the process, we build the we are looking for. â
4.2 Proof of Sufficiency
Let us now prove that Conditions 1â6 imply . We assume that the conditions are satisfied.
By Lemmas 3.11 and 4.1, we have .
As (Condition 4 and Lemma 4.1), by Lemma 3.11 we have so we can assume that and we do it from now on.
Entry is the largest weight of a walk from to with length . Let us prove , which ensures that , since in this case for (also recalling that by the periodicity of ). The other inequality follows from Theorem 2.15.
Case 1. If does not go through , then it is a walk on and . Using Condition 6, we conclude that .
Case 2. Assume now that goes through and contains an arc such that , and . Then we can replace this arc by the path thus obtaining a new walk . Using Condition 5, we conclude that . But, we also have , since visits a node of and has length modulo . Thus .
Case 3. Assume that goes through and does not contain an arc such that , and . Then can be decomposed into a path from to , and some cycles. Since we assumed that and contains no arc with , the only path from to is the path that follows the numbers and has length , so that the total length of the cycles is .
Since goes through a node of , and is the only cycle that connects and , the cycle decomposition of contains at least one copy of the cycle . But, again since , is the only cycle whose length is not a multiple of . As this length is, moreover, coprime with , the walk should contain at least copies of the cycle . But then their total length is , a contradiction. Hence this case is impossible, and the attainment of Dulmage-Mendelsohn bound has been proved for all possible cases.
4.3 Critical Graph
The end of the section will be devoted to the proof of the necessity of Conditions 1â6, along the lines presented in Section 3.2.
In this subsection, we prove the following proposition.
Proposition 4.2**.**
If , then is strongly connected and contains only one cycle of length up to choice of its first node.
To prove Proposition 4.2, we will use techniques from [14] related to CSR expansions and walks. We will first recall the main statements that will be required, and then the proof of Proposition 4.2 will be given after the statement of Proposition 4.8.
Proposition 4.3 below is an extended version of Proposition 2.16 part (i). In particular, it builds on the idea that the CSR terms in Definition 2.11 can be defined using any completely reducible subgraph of instead of the full .
Proposition 4.3** (cf. [14], Theorem 6.1).**
Let be a matrix with and and be the CSR terms of with respect to some completely reducible subgraph of the critical graph .
Let be a multiple of and a set of some nodes of that contains one node of every s.c.c. of .
Then we have, for any and :
[TABLE]
where \mathcal{W}^{t,\gamma}(i\xrightarrow{\mathcal{N}}j):=\left\{W\in\mathcal{W}(i\xrightarrow{\mathcal{N}}j)\,\big{|}\,l(W)\equiv t\pmod{\gamma}\right\}
In particular, CSR terms can be defined using a s.c.c. of rather than the whole , since any s.c.c. of is a completely reducible subgraph of . Following [14] let be the s.c.c.âs of with node sets , and let , , be the CSR terms defined with respect to . Let , and for define a matrix by setting the entries of with rows and columns in to , and let , , be the CSR terms defined with respect to in . By the dimension of we will mean the number of elements in .
Proposition 4.4** ([14], Corollary 6.3).**
If are the s.c.c.âs of , then we have:
[TABLE]
Let us now prove the following bound on :
Proposition 4.5**.**
Let for be the dimension of , and for be the girth of . Then
[TABLE]
Proof.
For each , the weak CSR expansion applied to reads
[TABLE]
The smallest for which the weak CSR expansion (18) holds is bounded by by Theorem 2.15. If then (18) holds for all and by successive replacement, we have:
[TABLE]
Observing that and that the CSR terms sum up to by Proposition 4.4 we see that (19) is exactly the weak CSR expansion . â
Bounds for in [14] are based on the concept of the cycle removal threshold defined as follows.
Definition 4.6**.**
Let be a subgraph of and .
The cycle removal threshold of is the smallest nonnegative integer for which the following holds: for all walks with length , there is a walk obtained from by removing cycles and possible inserting cycles of such that and .
The idea behind this definition is to be able to shorten an optimal walk while keeping it optimal and keeping its length modulo , thus proving inequalities between and .
The following proposition is stated in [14] and proved there by âarithmetical methodâ.
Proposition 4.7** ([14], Proposition 9.5).**
For and a subgraph of with nodes, we have:
[TABLE]
The next proposition can be proved using a slight generalization of [14], Proposition 6.5â(i), which differs in the fact that is considered as a whole and not each s.c.c. at a time. The proof of [14] actually shows this stronger statement.
Proposition 4.8**.**
Let be a square matrix such that all s.c.c.âs of , have the same girth . Let be the subgraph of consisting of all cycles of length . Then
[TABLE]
We are finally able to prove Proposition 4.2.
Proof of Proposition 4.2.
Let us assume that . We want to prove that is strongly connected and has only one cycle with length , up to choice of its first node.
To apply Proposition 4.8 we need to show that all s.c.câs of have girth .
Otherwise, w.r.t. the notation of Proposition 4.5 let be a component with girth . We have (recall that ). For all other components of we have since . Therefore using the bound of Proposition 4.5 we would have .
Now, we will combine Propositions 4.8 and 4.7 to show both the connectivity of the and the uniqueness of the shortest critical cycle. Let us denote by the number of nodes of the graph of Proposition 4.8 and set in (20) (see Proposition 4.7). Then we have
[TABLE]
If is not strongly connected then is not and . If is strongly connected but contains more than one cycle, then . Indeed, one can not have two critical cycles of length with the same set of nodes, because it would build a shorter cycle so that would not be the girth anymore. Thus, in both cases, we would have and .
Finally, is strongly connected and contains exactly one cycle of length , up to choice of its first node. â
4.4 The Interesting Walk and Its Structure
In this section we assume that . and deduce from this the structure of special walks which we call interesting.
By Proposition 4.2, there is a unique critical cycle of length up to choice of its first node. The subgraph consisting of all nodes and edges of this cycle will be denoted by .
Definition 4.9**.**
A walk is called twice optimal if it has minimal length among all the walks with maximal weight in the set . It is called interesting if it is twice optimal and has length .
Interesting walks are twice optimal with maximal possible length among all entries and matrices. Their particular structure, described in Proposition 4.12 will define matrices .
Proposition 4.10**.**
If , then there exists such that . For any such there is an interesting walk from to .
Proof.
We set .
If there is no such that , then for all we have .
By definition of , we have for all . We hence have . We always have the opposite inequality by distinguishing whether a walk visits the critical graph or not. Hence . But this means , which contradicts .
Let us prove the second part of the proposition.
Proposition 4.7, applied with , and , implies that twice optimal walks have length at most (alternatively, see the proof of [14] Theorem 4.1).
Now, if there is no interesting walk from to it means that the set contains a walk with optimal weight and length strictly less than . However, this length is congruent to modulo , hence it is less than or equal to and, furthermore, can be made equal to by inserting copies of . The weight of this walk is by Proposition 4.3, so . â
Proposition 4.11**.**
If and , then if and only if .
Proof.
Observe that implies that consists of the nodes and arcs of the unique cycle of length up to choice of its first node, and that both nodes of are critical. Thus , and .
Let us notice that and , and compute .
Consider first the off-diagonal entries. In this case we have
[TABLE]
for any such .
Consider now
[TABLE]
for and . This is not equal to if and only if . Finally, and the proof is complete. â
The following proposition, to be proved in Subsection 4.5, shows uniqueness and characterizes the interesting walk in the remaining cases of and .
Proposition 4.12**.**
Let be such that , , and not . For any interesting walk , there is a renumbering of the nodes such that are the nodes of the (only) critical cycle of length and
[TABLE]
*where .
Corollary 4.13**.**
Under the conditions of Proposition 4.12, has an Hamiltonian cycle with maximal weight, unique up to choice of its first node, which is labeled by the renumbering stated in Proposition 4.12.
Proof.
By contradiction, suppose that there is a different Hamiltonian cycle with the largest weight. It can replace one of the copies of in , and the walk should still be interesting. However as , the resulting walk contains edges of at least two different Hamiltonian cycles, so it cannot be represented as (21), which is in contradiction with this walk being interesting. â
Remark 4.14**.**
By Proposition 4.12 and Corollary 4.13, the Hamiltonian cycle with maximal weight induces a renumbering of the nodes, which is unique up to choice of the first node. Then, the first node is defined as the end of the only edge of that does not belong to the Hamiltonian cycle. Thus, the renumbering is unique and so is the interesting walk.
4.5 Proof of Proposition 4.12
Let us first note that the case is impossible unless . Indeed, if then , which is the Wielandt bound for the periodicity transient, so in this case cannot be attained. The case has been considered in Proposition 4.11.
The following elementary number-theoretic lemma will be especially useful in what follows.
Lemma 4.15**.**
Let . Then there is a nonempty subset with .
This lemma will allow us to remove some cycles from a walk an keep its length modulo as soon as we have cycles that do not intersect in the walk.
The first step of the proof of Proposition 4.12 is to establish properties of the structure of interesting walks.
Lemma 4.16**.**
In any interesting walk , there are exactly occurrences of each node of and exactly occurrences of each node not in .
Proof.
This is an improvement on the proof of Theorem 2.15 in [14] with extra care on the extremal cases. Let us first argue that the number of occurrences does not exceed and , respectively.
By contradiction, let occur times, then we have where occurs in no .
We have and by Lemma 4.15 some of the cycles (for ) can be removed in such a way that the resulting walk has the same length modulo . Moreover, the resulting walk still goes through a node of (namely ) and has the same length modulo meaning that is not twice optimal.
Let and decompose so that contains only nodes not in and . Then we have two cases:
a) One of the walks or does not contain . Then the remaining walk can have no more than occurrences of , otherwise these occurrences lead to at least cycles some of which can be removed in such a way that the resulting walk has the same length modulo and goes through a node of , contradicting the optimality of (the weight of the resulting walk is also not smaller since by the weight of each cycle is not bigger than ).
b) Both and contain at least once. If there are at least two occurrences of in then the cycle between these occurrences can be moved to . Hence we can assume that contains no more than once. As in a), can occur in no more than times, and the total number of âs occurrences is thus bounded by .
The total number of occurrences of all nodes in is thus bounded from above by . Observe now that the total number of these occurrences is exactly , since the length of is . Hence each node in occurs exactly times and each node not in exactly times. â
Lemma 4.17** (Interlacing).**
Let and .
- (i)
In any interesting walk, there is exactly one occurrence of node between two consecutive occurrences of .
- (ii)
In any interesting walk, there is exactly one occurrence of before the first and exactly one after the last occurrence of .
Proof.
We are going to prove that, for each , there are exactly occurrences of before the -th occurrence of and occurrences of after that occurrence of . This implies both parts of the lemma.
We first show after occurrences of we have no more than occurrences of , for otherwise we have consecutive closed walks going through and at least consecutive closed walks going through . Figure 4 depicts the situation. Thus the overall number of closed walks is at least and by Lemma 4.15 some of these closed walks can be removed from the walk. The resulting walk still goes through a node of (namely ) and has the same length modulo , so is not twice optimal, a contradiction.
Let us also show that we have no more than nodes of before the -th occurrence of . Indeed, after the -th occurrence of we still have occurrences of by Lemma 4.16. If the hypothesis is not true then we have closed walks going through and at least closed walks going through . Figure 5 depicts the situation. Thus we have at least closed walks in total and by Lemma 4.15 some of these closed walks can be removed from the walk. We conclude that is not twice optimal, a contradiction.
However, the total number of occurrences of before and after the -th occurrence of is , and therefore there are exactly occurrences of before the -th occurrence of , and occurrences of after that occurrence. â
Corollary 4.18**.**
In any interesting walk, there is exactly one occurrence of node between every two consecutive occurrences of , and no occurrences of neither before the first nor after the last occurrence of .
Lemma 4.19**.**
Any interesting walk can be represented as
[TABLE]
*where and contain all nodes not in exactly once and only them, contains all nodes of exactly once and only them, and is a walk starting with a node in .
Proof.
Define such that , where all nodes of are not in and the first node of is in . By Lemma 4.17 part (ii), contains all nodes not in exactly once (and only them).
Define as the subpath of such that , where all nodes of are in and the first node of is not in . That node also occurs once in . By Lemma 4.17 part (i), all nodes of should occur between the two occurrences of that node exactly once, and therefore contains all such nodes exactly once (and only them).
Define such that , where all nodes of are not in and the first node of is in . That node also occurs once in . By Lemma 4.17 part (ii), all nodes that are not in should occur between the two occurrences of that node exactly once, and therefore contains all such nodes exactly once (and only them). â
Proof of Proposition 4.12.
Let be an interesting walk and  and defined as in (22). We want to show
[TABLE]
We start with the decomposition (22). Define as the last node of and by .
As a first step, using (22) and observing occurrences of we can immediately obtain
[TABLE]
see Figure 6. Decomposition (22) also implies that the subpath in the beginning should be followed by a sequence containing all non-critical nodes once (and only them), in any interesting walk. Therefore we have for some .
The cycle contains each node (critical and non-critical) no more than once. Indeed, if a node occurred in twice then we would decompose , replace the node in by and delete from , see Figure 7. This would result in a new walk with consecutive closed walks some of which can be removed, resulting in a walk of a smaller length and showing that the initial walk was not interesting, a contradiction.
Since any node is contained in no more than once, consists of nodes of only. Comparing (23) with (24) we need to prove that and that . For that, take any node . It also occurs in since that path contains all nodes but . Consider the following decompositions and . If we assume that or then for some the sets of nodes of and differ or the sets of nodes of and differ. Assume the latter (the case of different and is treated similarly). By replacing with and by as in Figure 8 (in other words, by exchanging and ) we obtain a new interesting walk. We now prove that it is not of the form (22), in contradiction with Lemma 4.19.
Indeed, we have where consists only of nodes not in , and consists only of nodes in . Similarly, where consists only of nodes not in , and consists only of nodes in . However, the set of nodes of is a complement of the set of nodes of (recall that and all nodes occur in exactly once) and that of is not (since and have different node sets), and this implies that or miss some nodes in contradiction with Lemma 4.19. Hence and .
Generalizing the cycle , define for all . Since we can exchange any two without changing neither the length, nor the weight of the walk, the decomposition of is also true for any , that is for all .
Now, each critical node occurs in the walk exactly times, hence, by Lemma 4.17 part (ii), contains all non-critical nodes exactly once, and only them. So we have obtained that where contains all non-critical nodes exactly once, and only them.
It remains to show that . In (the graph of the transpose of ) there is also an interesting walk. Since for all , Walk , that starts with the end of and goes to the beginning of via exactly the same nodes listed in the opposite order is an interesting walk on . On one hand, by construction , where , and contain the same nodes as , and listed in the opposite order. On the other hand, applying the argument above we get a decomposition of the form , and, since , we conclude that . This implies , so the decomposition (23) is established.
In order to obtain (21) we renumber the nodes of in such a way that . â
Corollary 4.20**.**
If then and are coprime (Condition 1).
Proof.
If and are not coprime, then . We have and for some and . Let be given by (21) and be . Since , we have and are congruent modulo , and since , we obtain that is not twice optimal, so by Proposition 4.12, a contradiction. â
4.6 Proof of Necessity
In this subsection, we finish the proof of necessity of Conditions 1.â6. and the last statement of the theorem. We assume .
Condition 1 was proved as Corollary 4.20. By Proposition 4.2, is strongly connected and contains only one cycle with length denoted by . (Condition 2).
We now turn to the proof of Conditions 3 4 and 5, which will be proved together. The core of the proof is split into Lemmas 4.21, 4.22 and 4.23 below.
By Proposition 4.12 there is a unique twice optimal walk of length . After renumbering the nodes we can assume that for and are the arcs of a Hamiltonian cycle of , and that nodes , are the nodes of .
Notice that we have not yet proved (condition 3) since we do not know the arcs of .
Any occurrence of a node in can be encoded by its position in that walk. We now define what we mean by position. We assume that the first occurrence of node has position [math], and the position of any node in the th copy of (called period) is , and the position of the th node in the part of the walk before the first occurrence of (for ) is . Note that these positions (and only these) are non-positive.
We will be interested in the set of subwalks of from to , with length modulo . Denote this set by . This set is nonempty if and only if there is an occurrence of node at some position denoted by and there is an occurrence of node at some position denoted by such that and .
Consider the following properties of subwalks of :
Property A: We say that a subwalk has this property if it goes through one of the first nodes (i.e. a node of ).
Property B: We say that a subwalk has this property if after replacing in by the arc where and are the beginning node and the end node of respectively, the resulting walk goes through one of the first nodes.
A subwalk does not have Property B if and only if it begins in the tail, ends in the last period of and has .
Define , and by (9), (10) and (11).
Lemma 4.21**.**
If has Property A then .
Proof.
Property A means that belongs to on , since the first nodes are the node set of . As is the largest weight of walks in on (recall and Proposition 2.16 part (1)), the claim follows. â
Lemma 4.22**.**
If has Property B and and then .
Proof.
If then replacing by in we get a walk with the same length modulo as , whose weight is not less than and whose length is strictly less than . This contradicts the fact that is twice optimal. â
Lemma 4.23**.**
If then and ;
- 2.
If and or belong to but and , then and ;
- 3.
The arcs for and are critical.
Proof.
We will examine the existence of walks with Properties A and B in the cases of our interest, and apply Lemmas 4.21 and 4.22 .
1: Examine the case . We take the occurrence of in the first period, that is, . Then, since , the (unique) occurrence of for which and exists in some other period. They have both Property A and Property B, and Lemmas 4.21 and 4.22 imply that . See Figure 9 for this case as well as cases 2.1 and 2.2 described below.
2: We need to examine the following two cases: 2.1 , and and 2.2 , , , .
Case 2.1. We can take the occurrences of and in any period at positions and .
Case 2.2. Take the occurrence of with and the occurrence of with (in the last period).
In both cases, the walks defined by these occurrences have both Property A and Property B. By Lemmas 4.21 and 4.22, we have if . Note that here and in 1. above we still have to argue that can be replaced with and .
3: If an arc is critical then . Hence if then is non-critical. As are nodes of a critical cycle of length and parts 1. and 2. above imply that for and are the only arcs between the first nodes that can have , so these arcs are critical.
Thus is a critical cycle of and , and therefore can be replaced with and first in statement and proof of Lemma 4.21 and therefore also in the proofs of 1. and 2. above. â
Condition 3 of the theorem follows now from Lemma 4.23 , Condition 4 follows from Lemma 4.23 , and Condition 5 is implied by the following Lemma.
Lemma 4.24**.**
If and then .
Proof.
We can take the occurrences of and at positions and . The resulting walk has Property B. Hence by Lemma 4.22 we have for . As is also a unique (and hence optimal) walk on from to and having length we obtain that . â
It remains to obtain Condition 6. Since , we have ,
and hence .
Let us argue that for all . Indeed, the equality follows from Lemma 3.11 since by Lemma 4.23, and the equality , follows from Lemma 4.1. We also have since and .
The remaining uniqueness statements have been proved in Remark 4.14.
5 Matrices Attaining the Wielandt Bound
This section is devoted to the proof of Theorem 3.7.
If , then the Wielandt bound cannot be attained. Hence we are only interested in the case when . Observe that , and therefore
[TABLE]
for any .
Thus we have two cases: and .
In case , we have , and observe that Conditions 1, 5 and 6 of Theorem 3.1 trivially hold, in view of Remark 3.3. Next, both sufficiency and necessity as well as the last part of the statement, for , follow as a special case of the corresponding claims in Theorem 3.1.
In case , let us prove the sufficiency and necessity of Condition 2.
Sufficiency: By Lemma 3.11, Condition 2 implies that , so it suffices to show that . For this, note that for with entries defined by
[TABLE]
(Wielandtâs example, see e.g. [5]) we have , meaning that there exist such that . This implies that also and hence (recalling that ). However we also have and hence .
Necessity: To prove necessity we will need the following result, which is analogous to Proposition 4.12. Since the conditions are invariant under scalar multiplication, we will assume in the rest of this section.
Proposition 5.1**.**
Let for and . If has , then there exists a unique twice optimal walk of length . It is of the form
[TABLE]
where, after appropriate renumbering, for and are the arcs of the unique critical cycle.
Proof.
This is an improvement on the proof of Theorem 2.15 in [14] with extra care on the extremal cases.
Let be a critical cycle, which is of length . This is the only critical cycle (up to choice of its first node) as any extra arc would lead to a shorter cycle.
Let be a twice optimal walk. Denote its first node by and its last node by . Let , for be the noncritical arcs of . For every such arc of , insert a copy of as a walk beginning and ending at . Denote the resulting walk by . Observe that for each noncritical arc we can detect the node occurring in the subsequent copy of . This gives rise to a cycle consisting of a shortcut and a number of critical arcs. Thus we obtain the following decomposition in term of the multiset of its arcs:
[TABLE]
where denotes the multiset of arcs of a walk , is the number of noncritical arcs in the original walk, the are critical cycles, and is a critical path from to . Since is the only critical cycle, we have for all . We can remove copies of and get an optimal walk in of smaller length. Denote the resulting walk by .
Further, since has the largest weight in , one cannot remove cycles from it maintaining the length modulo , and hence by Lemma 4.15. We distinguish two cases:
-
is connected, in which case also the th copy of can be removed. The length of the resulting walk is bounded by . Thus, , a contradiction, which shows that this case is impossible.
-
is disconnected. Then we cannot remove from . However, in this case there is a cycle such that . Therefore the length of is bounded by .
Further, in case 2, the length of all cycles is bounded by , unless one of the connected components of has size . In the former case the length is bounded by , which is again impossible. It thus remains to treat two subcases:
2a) There is a loop (or possibly, several copies of the same loop) disconnected from , and the rest of the cycles of connected to ;
2b) and there are cycles of length at most disconnected from .
In subcase 2a, the length of all cycles is bounded by , since any cycle of length could be combined with the loop and removed. In this case, the length of the walk is again bounded by , which is impossible.
In subcase 2b, we have . Length can reach the length only if all cycles , not containing , are of length . However, by construction, every such cycle should contain just one noncritical arc, and this has to be a -shortcut bypassing . However, there is only one -shortcut bypassing which implies that all these cycles are identical and has to be of the form (26), after renumbering the nodes in such a way that and . â
As in the proof of Theorem 3.1, any occurrence of a node in of Proposition 5.1 can be encoded by its position in that walk. Node occurs there only twice. Its first occurrence has position [math], and its second occurrence has position . The th occurrence of node , for and has position .
A subwalk of length modulo from an occurrence of node at position to an occurrence of node at position exists if and only if and .
We will need the following observation:
Lemma 5.2**.**
If there are occurrences of node and of node such that the subwalk from to has length modulo and , then .
Proof.
Denote by the subwalk from to . This subwalk uses only the arcs of the digraph , where it is optimal among the walks with the same length modulo , and goes through critical nodes (since all nodes are critical). Hence . If is replaced with then the resulting walk also goes through the critical nodes, and therefore we must have , for otherwise is not twice optimal. Hence the claim. â
Using Lemma 5.2, for the necessity of Condition 2, it suffices to prove that the subwalks with length modulo but larger than exist for any except possibly for , , or and , which are the arcs of .
Case 1. . We have two occurrences of with and . As , the walk with required properties exists.
Case 2. and . In this case and we can choose with , since and and are coprime.
Case 3. and . This case is symmetrical to case 2.
Case 4. , , , and . Observe that is equivalent to because .
Take and where . Then . As and are coprime, there is a such that , and since we have and hence . The case is only possible if and , the case which was excluded.
The uniqueness of the renumbering follows from the uniqueness of .
6 Matrices with Critical Columns Attaining the Bounds
This section is devoted to the proof of Theorems 3.6 and 3.10. As before, we assume that .
Let us first notice that the transient of critical rows and columns is at most because if or is critical then . Thus, if the transient of a critical row or column reaches the bound, so does and  belongs to the class defined by Theorem 3.1 (except if ) or 3.7.
Proof of Theorem 3.6.
Let be a matrix with and a critical row whose transient is and show that the index of is . Then, the same is true for a critical column by transposition of the matrix and the converse of the theorem follows from [15][Lemma 8.2]. We also have , by the above argument.
We assume without loss of generality that  is strictly visualized.
By Proposition 4.2, is strongly connected and contains a unique (up to choice of the starting node) cycle of length denoted by .
Since is critical, the transient of row reaches the bound means that there is a such that and by Proposition 4.10, there is an interesting walk from to .
If , then satisfies the conditions of Theorem 3.1. We assume without loss of generality that has been reordered as in the theorem.
By Proposition 4.12, we have , so that is critical.
It remains to understand . By Lemma 3.11, it is contained in . Since is visualized, Condition 5 ensures that it only contains entries of and entries with , so that the only possible critical cycle that would share a node with is . Finally, since is strongly connected and contains and , is critical and we have . Since and are coprime, .
If , since and are critical, and is strongly connected, there is a walk from to any node of with only critical arcs. Since is strictly visualized, this walk has weight [math] and by optimality of , also uses only critical arcs from to . But, by Lemma 4.17, goes through every node not in before reaching , so that all nodes are critical.
In both cases, is strongly connected, has cyclicity and contains all nodes. Thus, the strict visualization ensures that for all , and that if and only if is a critical arc. Therefore, we can redefine and by and if is critical and and otherwise. We have , and , so that Lemma 3.11 ensures that the index of is .
â
Proof of Theorem 3.10.
Let us first check that the conditions are sufficient : by Lemma 3.11 and is at least the index of and thus . But, since all nodes are critical (because of the Hamiltonian critical cycle), this means that there is a (critical) row (or column) whose transient is .
Conversely, let us assume there is a critical row (or column) whose transient is . Since the existence of and with follows from Theorem 3.7 and we already noticed that has index . It remains to show the existence of the Hamiltonian cycle. By Lemma 3.11, we know that , so that
If , then and we are in the situation of Theorem 3.6. Thus, has index , which is only possible if , that is if has a critical Hamiltonian cycle.
If , then there is an Hamiltonian cycle in , which is necessarily a critical Hamiltonian cycle of because . â
Acknowledgement
We are grateful to Stéphane Gaubert and anonymous referees of the paper for many constructive and helpful suggestions and comments.
The reference list from the paper itself. Each links out to its DOI / PubMed record.
- 1[1] Marianne Akian, StĂ©phane Gaubert, and Cormac Walsh. Discrete max-plus spectral theory. Idempotent Mathematics and Mathematical Physics (G.L. Litvinov and V.P. Maslov, eds.), Contemporary Mathematics, vol. 377, AMS, Providence, 2005, pp. 53â77.
- 2[2] François Baccelli, Guy Cohen, Geert Jan Olsder, and Jean-Pierre Quadrat. Synchronization and Linearity: An Algebra for Discrete Event Systems . Wiley, Chichester, 1992.
- 3[3] François Baccelli and Dohy Hong. TCP is max-plus linear and what it tells us on its throughput. Proceedings of the Conference on Applications, Technologies, Architectures, and Protocols for Computer Communication (SIGCOMM 2000) . ACM, New York, 2000, pp. 219â230.
- 4[4] Anne Bouillard and Bruno Gaujal, Coupling time of a (max,plus) matrix. Proceedings of the Workshop on Max-Plus Algebra at the 1st IFAC Symposium on System Structure and Control . Elsevier, Amsterdam, 2001, pp. 335â400.
- 5[5] Richard A. Brualdi and Herbert J. Ryser. Combinatorial Matrix Theory . Cambridge University Press 1991.
- 6[6] Peter ButkoviÄ. Max-linear Systems: Theory and Algorithms . Springer, London, 2010.
- 7[7] Bernadette Charron-Bost, Matthias FĂŒgger, and Thomas Nowak. New transience bounds for max-plus linear systems. Discrete Applied Mathematics 219:83â99, 2017.
- 8[8] Bernadette Charron-Bost, Matthias FĂŒgger, Jennifer L. Welch, and Josef Widder. Time complexity of link reversal routing. ACM Transactions on Algorithms 11(3):18, 2015.