Weak CSR expansions and transience bounds in max-plus algebra

This paper aims to unify and extend existing techniques for deriving upper bounds on the transient of max-plus matrix powers. To this aim, we introduce the concept of weak CSR expansions: A^t=CS^tR + B^t. We observe that most of the known bounds (implicitly) take the maximum of (i) a bound for the weak CSR expansion to hold, which does not depend on the values of the entries of the matrix but only on its pattern, and (ii) a bound for the CS^tR term to dominate. To improve and analyze (i), we consider various cycle replacement techniques and show that some of the known bounds for indices and exponents of digraphs apply here. We also show how to make use of various parameters of digraphs. To improve and analyze (ii), we introduce three different kinds of weak CSR expansions (named after Nachtigall, Hartman-Arguelles, and Cycle Threshold). As a result, we obtain a collection of bounds, in general incomparable to one another, but better than the bounds found in the literature.


Introduction
Max-plus algebra is a version of linear algebra developed over the max-plus semiring, which is the set R max = R ∪ {−∞} equipped with the multiplication a ⊗ b = a + b and the addition a ⊕ b = max(a, b). This semiring has zero 0 := −∞ (neutral with respect to ⊕) and unity 1 := 0 (neutral with respect to ⊗), and each element µ except for 0 has an inverse µ − := −µ satisfying µ ⊗ µ − = µ − ⊗ µ = 1. Taking powers of scalars in R max means ordinary multiplication: λ ⊗t := t · λ.
The max-plus arithmetic is extended to matrices in the usual way, so that (AB) ij = k a ik ⊗ b kj = max k (a ik + b kj ) for matrices A = (a ij ) and B = (b ij ) of compatible sizes. In this paper, all matrix multiplications are to be understood in the max-plus sense. For multiplication by a scalar and for taking powers of scalars we will write the sign ⊗ explicitly, while for the matrix multiplication it will be always omitted.
Historically, max-plus algebra first appeared to analyze production systems driven by the dynamics (1.1) x i (k + 1) = max j x j (k) + a ij .
Thus, repeated application of matrix A = (a ij ) in max-plus algebra to an initial vector x(0) computes the vectors x(k). Here x(k) is typically a vector consisting of n real components, expressing the times of certain events happening during the kth production cycle. According to dynamics (1.1), event i has to wait until all the preceding events j happen and the necessary time delays a ij have passed, so that event i can then occur as early as possible. Such situation is usual in train scheduling, working plan analysis, and synchronization of multiprocessor systems [3,5,15]. Recently, Charron-Bost et al. [9] have shown that also the behavior of link reversal algorithms used for routing, scheduling, resource allocation, leader election, and distributed queuing can be described by a recursion of the form (1.1).
In this paper, we investigate the sequence of max-plus matrix powers A t = t times AAA · · · A. Cohen et al. [10] proved that this sequence eventually exhibits a periodic regime whenever A is irreducible, i.e., whenever the digraph D(A) described by A is strongly connected: there exists a positive integer γ and a nonnegative integer T such that where λ = λ(A) is the unique max-plus eigenvalue of A. The smallest T that can be chosen in (1.2) is called the transient of A; we denote it by T (A).
Since it satisfies x(t) = A t v every max-plus linear dynamical system, i.e., every sequence x(t) satisfying (1.1) is periodic in the same sense whenever A is irreducible.

Its transient T (A, v) in general depends on v and is always upper-bounded by T (A).
Bounds on the transients were obtained by Hartmann and Arguelles [14], Bouillard and Gaujal [4], Soto y Koelemeijer [27], Akian et al. [2], and Charron-Bost et al. [8]. Those bounds are incomparable because they depend on different parameters of A or assume different hypotheses. However they all appear, at least in the proofs, as the maximum of a first bound independent of the values of the entries of A and a second bound taking those values into account. The first motivation for this paper was to find a common ground for these bounds in order to understand, unify, combine, and improve them.
Schneider [20] observed that the Cyclicity Theorem can be written in the form of a CSR expansion, which was formulated by Sergeev [23]: there exists a nonnegative integer T such that (1.3) ∀t ≥ T : where the matrices C, S, and R are defined in terms of A and fulfill CS t+γ R = CS t R for all t ≥ 0. In an earlier work, considering infinite-dimensional matrices, Akian, Gaubert and Walsh [2, Section 7] gave a similar formulation originating in the preprints of Cohen et al. [10]. Because of the periodicity of the sequence CS t R, the smallest T satisfying (1.

3) is T (A).
Later, Sergeev and Schneider [24] proved that for t large enough, A t is the sum (in the max-plus sense) of terms of the form λ ⊗t i ⊗ C i S t i R i . This sum, which we call CSR decomposition, has two remarkable properties: it holds for reducible matrices as well as irreducible ones, and the CSR decomposition holds for t ≥ 3n 2 , a bound that does not depend on the values of the entries of A.
As a common ground of transience bounds and CSR decomposition, we propose the new concept of weak CSR expansions. We suggest that all existing techniques for deriving transience bounds implicitly use the idea that eventually we have (1.4) ∀t ≥ T : where C, S, and R are defined as in the CSR expansion, and B is obtained from A by setting several entries (typically, all entries in several rows and columns) to 0.
In this case, we say that B is subordinate to A. Call the smallest T for which (1.4) holds the weak CSR threshold of A with respect to B and denote it by T 1 (A, B). This quantity heavily depends on the choice of B, i.e., on which entries are set to 0. If we choose B = (0), then we recover the ordinary CSR expansion and we have T 1 Analogously, we call T 2 (A, B, v) the least integer satisfying We claim that the bounds in [4,8,14,27] implicitly are of this type, for various choices of B and various ways to bound T 1 and T 2 .
We next summarize the contents of the remaining part of this paper. In Section 2, we recall notions and results of max algebra, focusing on its relation to weighted digraphs. In Section 3, we introduce three schemes of defining B, and thereby weak CSR expansions: the Nachtigall scheme, the Hartmann-Arguelles scheme, and the cycle threshold scheme. The first scheme is implicitly used in [2,4,8,27], the second one is derived from [14], and the third one is completely new.
In Section 4 we state some bounds on T 1 (A, B) and T 2 (A, B), thus on T that we obtain in this paper. Those bounds strictly improve the ones in [4,8,14,27]. Moreover they can be combined in several ways. Notably, for the three schemes defined in Section 3, we bound the weak CSR threshold T 1 (A, B) by the Wielandt number (1.5) Wi(n) = 0 if n = 1 (n − 1) 2 + 1 if n > 1 (named in honor of [28]). The bound Wi(n) is optimal because it is the worst case transient of powers of Boolean matrices, i.e., matrices with entries 0 and 1 (see Remarks 3.1 and 4.2). We also recover another optimal bound for Boolean matrices due to Dulmage and Mendelsohn [11] that do not only depend on n but also on some graph parameter. The section also includes a examples to compare the different bounds.
In Section 5, we compare our results to some bounds found in the literature. In Section 6, we explain the strategy of the proof, which leads us to introduce a graph theoretic quantity, which we name cycle removal threshold of a graph and state bounds on T 1 (A, B) that depend on this quantity for some graphs.
In Sections 7 and 8, we prove the results stated in Section 6 to bound T 1 (A, B) in terms of the cycle removal threshold.
In Section 9 we bound the cycle removal threshold. First we recall the bounds of [7] that depend on several parameters of D(A) and use the ideas of Hartman and Arguelles [14] to give a new bound depending on less parameters. Then, we introduce a new technique leading to other two bounds on T 1 (A, B).
In Section 10 we prove the bounds on T 2 (A, B).
In Section 11, we recall some bounds on the index of Boolean matrices to be used in some bounds on T 1 .
The technique of local reduction, originating from Akian, Gaubert and Walsh [2,Section 7], is recalled in Section 12. We show that this technique can be combined with any of the CSR schemes described in Section 3.

Preliminaries
2.1. Walks in weighted digraphs. Let us recall the optimal walk interpretation of matrix powers in max algebra. This is the fact that the entries of a matrix power A t are equal to maximum weights of walks of length t in the digraph associated to matrix A.
To a square matrix A = (a ij ) ∈ R n×n max we associate an edge-weighted digraph D(A) with set of nodes N = {1, 2, . . . , n} and set of edges E ⊆ N × N containing a pair (i, j) if and only if a ij = 0; the weight of an edge (i, j) ∈ E is defined to be p(i, j) = a ij . A walk W in D(A) is a finite sequence (i 0 , i 1 , . . . i L ) of adjacent nodes of D(A). We define its length l(W ) = L and weight p(W ) = a i0,i1 ⊗a i1,i2 · · ·⊗ a it−1,it . A closed walk is a walk whose start node i 0 coincides with its end node i L . Closed walks are often called circuits in the literature. There exists an empty closed walk at every node of length 0 and weight 0.
The multiplicity of an edge e in W is the number of k's such that (i k , i k+1 ) = e. A subwalk of walk W is a walk V such that the edges of V appear in W with larger multiplicity. A subwalk of W is a proper subwalk if it is not equal to W .
A closed walk is a cycle if it does not contain any nonempty closed walk as a proper subwalk. A walk is a path if it does not contain a nonempty cycle as a subwalk.
An elementary result of graph theory states that a walk can always be split into a path and some cycles. Reciprocally, union of edges of one path and some cycles can always be reordered into a walk provided the graph with all the edges and nodes of those walks is connected. The best way to see this is in terms of multigraph M (W ) defined by a walk W .
For a set W of walks, we write p(W) for the supremum of walk weights in W. Denote by W t (i → j) the set of all walks from i to j of length t and write A t = (a (t) ij ). It is immediate from the definitions that . When we do not want to restrict the lengths of walks, we define the set W(i → j) of all walks connecting i to j. An analog of (I − A) −1 in max-plus algebra is the Kleene star where I is the max-plus identity matrix. It follows from the optimal walk interpretation (2.1) that series (2.2) converges if and only if p(Z) ≤ 0 for all closed walks Z in D(A), in which case it can be truncated as Because every closed walk is composed of cycles, we could replace "cycle" by "closed walk" in definition (2.4). The maximum cycle mean λ(A) is equal to the greatest max-algebraic eigenvalue of A, i.e., a µ ∈ R max such that there exists a nonzero vector x satisfying A ⊗ x = µ ⊗ x. Nonempty closed walks of weight λ(A) are called critical, and so are the nodes and edges on these wlks. The subgraph of D(A) consisting of the set of critical nodes N c and the set of critical edges E c is called the critical graph of A and is denoted by G c (A) = (N c , E c ). A useful fact (used throughout the paper) is that every nonempty closed walk in G c (A) is critical.
As we will see, the behavior of max-algebraic matrix powers is eventually dominated by the walks that visit the critical graph. The set of such walks in W t (i → j) will be denoted by W t (i G c −→ j) More generally, for a node k and a subgraph D of D(A) we write is strongly connected if there exists a walk from i to j for all nodes i, j ∈ N . A strongly connected component (s.c.c.) of G is a maximal strongly connected subgraph of G. Digraph G is called completely reducible if there are no edges between distinct s.c.c.'s of G. The critical graph G c (A) will be the most important example of this.
Matrix A is called irreducible if its associated digraph is strongly connected, and reducible otherwise. Further, it is called completely reducible if so is the associated digraph.
The cyclicity γ(G) of a strongly connected digraph G is the greatest common divisor of the lengths of its closed walks. If G is not strongly connected, its cyclicity γ(G) is the least common multiple of the cyclicities of its s.c.c.'s. It is well-known that any two lengths of walks on G both starting at some node i and both ending at some node j are congruent modulo γ(G). Moreover, if G is strongly connected, there is walk from i to j of all lengths that are large enough and that are congruent to some t ij modulo γ(G).
We call a subgraph G of G c (A) a representing subgraph if G is completely reducible and every s.c.c. of G c (A) contains exactly one s.c.c. of G. The cyclicity γ(G) of a representing subgraph of G c (A) is always a multiple of γ G c (A) . Hence Equation (1.2) also holds with γ = γ(G) instead of γ G c (A) .

2.3.
Visualization and max-balancing. The maximum cycle mean λ(A) also appears as the least µ ∈ R max such that there exists a finite vector x satisfying Ax ≤ µ ⊗ x. When µ = λ(A), we can take When (2.5) holds we say that B is visualized: it exhibits the edges of the critical graph. A diagonal matrix X such that B = D − (λ − (A) ⊗ A)D is visualized and G c (B) = G c (A) is also called a Fiedler-Pták scaling [12] of A. In this case, we call B a visualization of A.
Fiedler-Pták scalings were described in more detail by Sergeev, Schneider and Butkovič [25] using Kleene stars and max algebra. Butkovič and Schneider [6] described applications to various kinds of nonnegative similarity scalings. A Fiedler-Pták scaling, particularly interesting to us, is called the max-balancing. It was described by Schneider and Schneider [21]: Theorem 2.1 (Schneider and Schneider [21]). For all A ∈ R n×n max exists a visualization B of A satisfying the following equivalent properties: ( When the dependency on G needs to be emphasized, we write C G , S G and R G instead of C, S and R.
Essentially C and R can be regarded as sub-matrices of M extracted from the columns, resp. the rows of M with indices in G. If A is visualized, then matrix S is exactly the associated Boolean matrix of G. The matrices C, S, and R are called the CSR terms of A with respect to G.
The following is a CSR version of the Cyclicity Theorem. 20,23]). Let A ∈ R n×n max be irreducible and let C, S, R be the CSR terms of A with respect to G c (A). Then for all t ≥ T (A): As it is shown below, Theorem 2.2 also holds with G c (A) replaced by some representing subgraph G of G c (A).
Note that this theorem implies periodicity of A t after T (A), because the sequence of matrices CS t R is purely periodic, i.e., periodic from the very beginning. In fact, this statement is more generally true for all completely reducible (and hence also for all representing) subgraphs of G c (A): 24]). Let A ∈ R n×n max be irreducible and C, S, R be the CSR terms of A with respect to some completely reducible subgraph G of the critical graph G c (A). Then the sequence of matrices CS t R is purely periodic.
This fact was shown by Sergeev and Schneider [24], where CSR terms with respect to completely reducible subgraphs of G c (A) were studied in detail. It can also be deduced from Theorem 6.1 proved below.
A weak CSR expansion of A is an expansion of the form (1.4) where C, S, R are CSR terms with respect to some representing subgraph of G c (A) and D(B) is a subgraph of D(A) disjoint to G c (A). In particular, the result of Theorem 2.2 is also a weak CSR expansion (take B equal to the max-plus zero matrix).
By iteration of weak CSR expansions, we recover the CSR decomposition of A introduced in [24]. Bounds on T 1 give bounds on the time from which A t admits such a decomposition. (See Corollary 4.3).

Weak CSR Schemes
In this section, we introduce our three schemes for weak CSR expansions and discuss their relation. We define them in terms of the subgraph D of D(A) whose edges denote the indices that are set to 0 in the subordinate matrix B. More explicitly: The three schemes are: (1) Nachtigall scheme. Here, the subgraph D = G c (A). We denote the resulting matrix B by B N . This scheme is consistent with the expansions introduced by Nachtigall [19], which was studied by Molnárová [18] and Sergeev and Schneider [24]. It was used by almost all authors who studied matrix transients [2,4,8,27], excluding Hartmann and Arguelles [14]. The subgraph D = G ha defining B in the Hartmann-Arguelles scheme is the union of the s.c.c of T ha (µ ha ) intersecting G c (A). We denote this matrix B by B HA . Observe that λ(B HA ) = µ ha and the graphs T ha (µ), for all µ, are completely reducible due to max-balancing (more precisely, the cycle cover property).
(3) Cycle threshold scheme. For µ ∈ R max , define the cycle threshold graph T ct (µ) induced by all nodes and edges belonging to the cycles in D(A) with mean weight greater or equal to µ. Again, for µ = λ(A) we have T ct (µ) = G c (A). Let µ ct be the maximum of µ ≤ λ(A) such that T ct (µ) has a s.c.c. that does not contain any s.c.c. of G c (A). If no such µ exists, then µ ct = 0 and T ct (µ ct ) is equal to D(A). The subgraph D = G ct defining B in the cycle threshold scheme is the union of the s.c.c of T ct (µ ct ) intersecting G c (A). This matrix B will be denoted by B CT . We again observe that λ(B CT ) = µ ct . Concerning B HA , Schneider and Schneider [21] proved that a max-balancing of A can be computed in polynomial time (at most O(n 4 )). The same order of complexity is added if we "brutally" examine at most n 2 threshold graphs (for each of them, the strongly connected components found in O(n 2 ) time). A better complexity result can be derived from the work of Hartmann-Arguelles [14].
To show NP-hardness of the computation of T ct (µ), we reduce the Longest Path Problem [13, p. 213, ND29] to it. Consider the Longest Path Problem as a decision problem that takes as input an edge-weighted digraph with integer weights, a pair of nodes (i, j) with i = j in the digraph, and an integer K. The output is YES if there exists a path of weight at least K from i to j. The output is NO if there is none. Observe that if i = j, then by inserting the edge (j, i) with weight −K, the Longest Path Problem can be polynomially reduced to the problem of calculating T ct (0) by checking whether the new edge (j, i) belongs to T ct (0).
The relation between these schemes is as follows. The cycle threshold scheme is most precise, while the Nachtigall scheme is the coarsest. We measure this in terms of the size of B and the value λ(B).
Proof. Evidently both D(B CT ) and D(B HA ) are subgraphs of D(B N ), which is extracted from all non-critical nodes. This implies λ(B CT ) ≤ λ(B N ) and λ(B HA ) ≤ λ(B N ). We show that D(B CT ) is a subgraph of D(B HA ). For this we can assume that the whole digraph is max-balanced, and notice first that T ha (µ) ⊆ T ct (µ) for any value of µ. We also have that T ha (µ 1 ) ⊇ T ha (µ 2 ) and T ct (µ 1 ) ⊇ T ct (µ 2 ) for any µ 1 ≤ µ 2 . Now consider the value µ ct . The components of T ct (µ ct ) which do not contain the components of G c (A), have the property that any other cycle intersecting with them has a strictly smaller cycle mean. It follows that all edges of these components have cycle mean µ ct . Indeed, suppose that there is a component containing an edge with a different weight. In this component, any cycle that contains this edge also has an edge with weight strictly greater than µ ct . The cycle cover property implies that there is a cycle containing this edge, where this edge has the smallest weight. The mean of that cycle is strictly greater than µ ct , a contradiction. But then T ha (µ ct ) contains these components as its s.c.c.'s. In particular they do not contain the components of G c (A), hence µ ct ≤ µ ha .
If µ := µ ct = µ ha then T ha (µ) ⊆ T ct (µ), while we have shown that the components of T ct (µ) not containing the components of G c (A) are also components of T ha (µ). It follows that G ha ⊆ G ct .
If µ ct < µ ha then we obtain that The following example shows that all three schemes can differ and, moreover, that the thresholds In this example we have λ(A) = 0, it is visualized and, moreover, max-balanced. The matrices B N , resp. B HA and B CT are formed by setting the first 2 rows and columns, resp. the first 3 and 4 rows and columns to 0 = −∞, and the corresponding values are λ( Let us provide a class of examples that generalizes the example above to arbitrary dimension. For any matrix A in this class of examples, all three schemes are different but the corresponding thresholds T 1 (A, B) may coincide.
Consider a matrix A such that the node set N of D(A) is partitioned into N = N c ∪ N n ∪ N ha ∪ N ct , see Figure 1. For each x ∈ {c, n, ha, ct}, the nodes in N x form a strongly connected graph where all edges have weight λ x . We set λ c > λ n > λ ha > λ ct . For each set N x with x ∈ {n, ha, ct}, we assume that there is at least one edge from N x to some set N y with λ y > λ x , and one edge from one of such N y to N x . With this assumption, it can be shown that D(A) is strongly connected. Let us also assume that all such edges (from N x and to N x ) have the same weight δ x . Observe that for the matrix of (3.2), we have N c = {1, 2}, N n = {3}, N ha = {4} and N ct = {5}; λ c = 0, λ n = −1, λ ha = −2 and λ ct = −3; δ n = −1, δ ha = −3 and δ ct = −7.
Assume that δ x satisfies Then D(A) is also max-balanced (since it can be shown that each edge (i, j) with i = j is on a cycle where it has the smallest weight).
We also see that λ c = λ(A), λ n = λ(B N ), while λ ha and λ ct are "candidates" for λ(B HA ) and λ(B CT ), respectively. To enforce the correct behaviour of threshold graphs and ensure that λ(B HA ) = λ ha and λ(B CT ) = λ ct , we set: 1) δ n = λ n , 2) δ ha = λ ha − s, where s is chosen in such a way that the inequality holds at least for one cycle Z containing a node in N ha and a node in N c ∪ N n ; 3) δ ct not greater than δ h (for the sake of max-balancing) and such that the mean weight of each cycle containing a node of N ct and a node of N \N ct is strictly less than λ ct . Observe, in particular, that condition 2) ensures that T ct (µ) does not gain any new component as µ decreases from λ c to λ ha so that λ ct < λ(B HA ), and that condition 3) ensures λ ct = λ(B CT ). Note that (3.2) satisfies conditions 1)-3).

Main Results
In this section, we present the main results of this paper. These bounds of this section use the following graph parameters of a digraph D: The computation of the circumference cr(D) and the cab driver's diameter cd(D) are both NP-hard in the number of nodes of D. However, they can be upper bounded by |D| and |D| − 1, respectively.
Denote by A the difference between the largest and the smallest finite (i.e., = 0) entry of A, and by n B the size of the smallest submatrix of B containing all its finite entries.
We explained in the introduction that T (A) ≤ max T 1 (A, B), T 2 (A, B) . Our main results are bounds on T 1 and T 2 . All of them are mutually incomparable.
where n c = |G c (A)| (i.e., the number of critical nodes) and ep(G c (A)) is the exploration penalty of G c (A) (see Definition 6.4).
The exploration penalty ep(G c (A)) is a quantity that depends only on the critical graph and can be bounded by its index, see Section 11 for further details. Remark 4.2. As we noted in Remark 3.1, those bounds apply to the transient of Boolean matrices. We thus recover the bound of Wielandt [28] in (4.1) and the bound of Dulmage and Mendelsohn [11] in (4.2). Notice that (4.2) implies (4.1) ifĝ ≤ n − 1. The remaining case is trivial for Boolean matrices because there is only one such matrix, but in the non-Boolean case we need a different strategy. (Proposition 9.4 below).
Bound (4.1) is optimal in the sense that the bound is reached for any n, as was already noted in [28], while bound (4.2) is reached if and only ifĝ and n are coprime (see [26]).
Iterating the process of weak CSR expansion, we get the following improvement of [24][Theorem 4.2]: Corollary 4.3 (CSR decomposition). For any matrix A ∈ R n×n max , there are some matrices C i , S i , R i defined by induction with S i diagonally similar to Boolean periodic matrices and some scalars λ i ∈ R, where i varies between 1 and K ≤ n such that we have: Theorem 4.4. For any matrix A ∈ R n×n max , we have the following bounds where cr = cr(D(A)) and cd = cd(D(A)).
The proof of those theorems is explained in Section 6 and performed in Sections 7 and 8. Now we also state bounds on T 2 (A, B) and on T 2 (A, B, v). If λ(B) = 0, then T 2 (A, B) ≤ cd B +1 ≤ n B . Otherwise, we have the following bounds If A has only finite entries, then we have: The following theorem generalizes Proposition 5 of [8] and Theorem 3.5.12 of [27].
Theorem 4.6. Let A ∈ R n×n max be irreducible, B be subordinate to A and v be a vector with only finite entries, i.e., v ∈ R n . If Otherwise, we have the following bound: If A has only finite entries, then we have: The proofs of Theorems 4.5 and 4.6 are deferred to Section 10. For the same reason, both bounds are quadratic if all entries are finite.

Comparison to Previous Transience Bounds
Hartmann and Arguelles [14] proved one transience bound for irreducible maxplus matrices and one for irreducible max-plus systems with finite initial vector.
These two bounds are, respectively, .
Combination of our bounds in (4.6) and (4.9), respectively (4.14), yields bounds that are strictly lower than that of Hartmann and Arguelles. Note that our results, being more detailed, allow a considerably more fine-grained analysis of the transient phase. For instance, there exist matrices for which λ(B CT ) = 0 but λ(B HA ) = 0 (cf. Example 3.4). Our bounds show in particular that the transients of these matrices and systems are at most Wi(n), which cannot be deduced from previous bounds, including that of Hartmann and Arguelles. Bouillard and Gaujal [4] and Akian et al. [2] gave transience bounds for irreducible matrices in the case that the cyclicity of the critical graph is equal to 1. They explained how to extend their bounds to arbitrary cyclicities, but that reduction involves multiplying the bound by the cyclicity of the critical graph or its subgraph. Akian et al. [2] derive bounds for the periodicity transient of {a (t) ij } t≥1 for fixed i, j instead of the whole matrix powers, and show that their bounding techniques extend to the case of matrices of infinite dimensions. We discuss the relation of this approach to weak CSR expansions in more detail in Section 12.
Soto y Koelemeijer [27] (Theorem 3.5.12) established a transience bound for matrices whose entries are all finite. In our notation, it reads Combination of our bounds in (4.6) and (4.13) yields a bound of max Wi(n) , 2 A /(λ(A)−λ(B CT ))+cd B , which is strictly lower. In many cases, it is even better to use (4.12). Charron-Bost et al. [8] gave two transience bounds for systems. They also explained how to transform transience bounds for systems into transience bounds for matrices. Combination of our bounds (4.3), (4.5), and (4.14) yields bounds that are strictly lower than those of [8].

Proof Strategy
In this section, we outline the proof of the bounds on T 1 stated in Theorems 4.1 and 4.4. Moreover, we provide some general statements that can be used to get a better bound if more information on the matrix is available.
In all proofs, we assume λ(A) = 1 (replacing A by λ(A) − ⊗ A if necessary). The first stage of the proof is the following representation theorem for CS t R expansions.
Theorem 6.1 (CSR and walks). Let A ∈ R n×n max be a matrix with λ(A) = 1 and C, S, R be the CSR terms of A with respect to some completely reducible subgraph G of the critical graph G c (A).
Let γ be a multiple of γ(G) and N a set of critical nodes that contains one node of every s.c.c. of G.
Then we have, for any i, j and t ∈ N: The proof of this theorem is deferred to Section 7.
Observe that it implies Proposition 2.3 as well as the following corollary.
Let G 1 , . . . , G l be the s.c.c. of G c (A) with node sets N 1 , . . . N l , and let C G1 , S G1 , R G1 be the CSR terms defined with respect to G 1 . For ν = 2, . . . , l, we define a subordinate matrix A (ν) by setting the entries of A with rows and columns in N 1 ∪ . . . ∪ N ν−1 to 0, and let C Gν , S Gν , R Gν be the CSR terms defined with respect to G ν in A (ν) . Corollary 6.3. If G 1 , · · · , G l are the s.c.c.'s of G c (A), then we have: Proof. Using Theorem 6.1, observe that the set of walks W t,γ (i where γ is the cyclicity of G c (A), can be decomposed into the sets W ν consisting of walks Corollary 6.3, which will be useful in the final section of the paper, and Corollary 4.3 are different examples of the CSR decomposition schemes considered by Sergeev and Schneider [24].
If D is the graph defining B in (3.1), it contains G c (A) and by the optimal walk interpretation (2.1), we have: The proof that T 1 (A, B) ≤ T has two parts: (1) Scheme-dependent part: show that for t ≥ T we have (2) Scheme-independent part: show that for t ≥ T we have Let us go deeper into the strategy for each part. − −−− → j) and a closed walk V from a node of W to G c (A) and back whose edges have weight greater than or equal to the greatest weight of the edges in W . Then, we insert V γ(G c (A)) (i.e., γ(G c (A)) copies of V ) in W , and remove as many cycles of the new walk as possible, preserving the length modulo γ(G c (A)) until we get a walkW with length at most t. As a result, we thus replaced some edges of W by edges with greater weight and removed other edges, so p(W ) ≥ p(W ). (c) For B = B CT , we also take a walk W with maximal weight in W t (i − −−− → j) but now we replace some cycles of W by some copies of a cycle with greater mean weight, to get a new walk with length t. We therefore introduce the concept of a "staircase" of cycles, and Lemma 8.4 will ensure us that we can iterate this process and eventually reach a critical node. Note that we need to remove cycles before we replace them and to have some steps with non-critical cycles, which explains why the bound for T 1 (A, B CT ) are larger than the one for T 1 (A, B HA ) and T 1 (A, B N ). However, the worst case remains Wi(n).
(2) By Theorem 6.1, to have (6.4), it is enough to prove that for each s.c.c. H of G c (A) there is a γ ∈ N and a set of nodes N ⊂ H such that To ensure that Equation (6.5) is satisfied, we use the following steps: and take a walk W such that p(W ) = p(W t,γ (i N − → j)).
(b) Remove as many cycles as possible from W , keeping it in W t,γ (i N − → j). (c) Insert critical cycles so that the new walk has length t. Since λ(A) ≤ 1, steps 2b and 2c cannot strictly increase the weight of the walk, so (6.5) is satisfied.
It is clear from the strategy that the main point is to remove closed walks from a given walk, while preserving the length modulo some given integer. This will be the subject of Section 9. We will use three different tactics, one of them is completely new. Different bounds depending on different parameters arise from different choices of N and γ in step 2a and different tactics in step 2b. To reach the (optimal) Wielandt number Wi(n), we have to combine two of them.
To state general results, we introduce two graph-theoretic quantities. (1) The cycle removal threshold T γ cr (G), (resp. the strict cycle removal thresh-oldT γ cr (G)) of G is the smallest nonnegative integer T for which the following holds: for all walks W ∈ W(i G − → j) with length ≥ T , there is a walk V ∈ W(i G − → j) obtained from W by removing cycles (resp. at least one cycle) and possible inserting cycles of G such that l(V ) ≡ l(W ) (mod γ) and l(V ) ≤ T .
(2) The exploration penalty ep γ (i) of a node i ∈ G c (A) is the least T ∈ N such that for any multiple t of γ greater or equal to T , there is a closed walk on G c (A) with length t starting at i.
The exploration penalty ep γ (G) of G ⊆ G c (A) is the maximum of the ep γ (i) for i ∈ G. We further set ep((G c (A)) = max l ep γ(G c l ) (G c l ), which is the quantity used in Theorem 4.1.
Obviously, T γ cr (G) ≤T γ cr (G) ≤ T γ cr (G) + 1 but it will be useful to have both definitions.
Bounds on ep γ are given in Section 11 while T γ cr is investigated in Section 9 We can already notice the following. This gives two extremal choices for G and γ: either G is a s.c.c. of G c (A) and γ is its cyclicity, or G is a critical cycle and γ is its length.
The first choice is used in [4], the second one in [14] and both choices in [8]. Here we systematically test those two choices. The first one is used to prove the bounds in Theorem 4.1 that depend on ep(G c (A)). The second one is used for the other bounds on T 1 .
If other choices prove to be useful under additional assumptions on D(A), one can apply Proposition 6.5 with other parameters.
The strategy explained in this section leads to the following proposition, which implies Theorems 4.1 and 4.4, except for (4.6).

Proof of Theorem 6.1
Let A, G, N , γ, t, i, j be as in the statement of Theorem 6.1. We first prove: By definition of C, S and R, there are walks W 1 , W 2 and W 3 such that (CS t R) ij = p(W 1 W 2 W 3 ) and Let k be the start node of W 2 . By hypotheses, k is critical and there is a node l of N in the same s.c.c. H of G c (A) as k. Thus there are walks W 4 and W 5 with only critical edges, going from k to l and from l to k respectively. Thus, W 4 W 5 is a circuit of G c (A) and p(W 4 ) + p(W 5 ) = 0.
Let G be the s.c.c. of k in G. As G ⊆ H, γ(H) divides γ(G), thus also γ(G) and γ. Hence γ(H) divides l(W 1 ) and l(W 3 ). It also divides l(W 4 W 5 ) and we have Therefore, for m ∈ N large enough, there is a closed walk W 6 on H starting at k with length mγ − L. Set It remains to show: By definition of W t,γ (i N − → j) there are a node l ∈ N and two walks V 1 and V 2 going from i to l and from l to j respectively such that l(V 1 V 2 ) ≡ t (mod γ) and Let k be a node of G in the same s.c.c H of G c (A) as l. As above, there are critical walks W 4 and W 5 , going from k to l and from l to k respectively and γ(H) divides γ.
Let V 3 be a closed walk in G with start node k, whose length is ≥ t + γ. Let V 4 be its shortest prefix such that l(V 1 ) + l(W 5 ) + l(V 4 ) ≡ 0 (mod γ) and V 5 be the complementary (i.e., V 3 = V 4 V 5 ). Let W 2 be the prefix of length t of V 5 and V 6 be its complementary ( By construction W 1 , W 2 satisfy (7.2). Moreover, we have ≡ 0 (mod γ) and W 3 also satisfies (7.2).

Proof of Proposition 6.5
In this section, we prove Proposition 6.5, following the strategy described in Section 6. 8.1. Scheme independent part. In this section, we prove the following lemma.  s.c.c. G 1 , . . . , G m and γ l be multiples of γ(G l ), for l = 1, . . . , m.
For any t ≥ max l (T γ l cr (G l ) − γ l + 1 + ep γ l (G l )) and any i, j, inequality (6.4), with G instead of G c (A), holds for γ = lcm l γ l .
Proof. Indeed, any walk W ∈ W t,γ (i If t ≥ T γ l cr (G l ) − γ l + 1 + ep γ l (G l ), then t − l(V ) ≥ ep γ l (G l ) − γ l + 1. Since t − l(V ) and ep γ l (G l ) are both multiples of γ l , it implies t − l(V ) ≥ ep γ l (G l ), so there is a closed walk on G c (A) with length t − l(V ) at each node of G l . Inserting such a walk in V where it reaches G l , we get a new walkW ∈ W t (i

8.2.
Hartmann and Arguelles scheme. In this section, we perform step 1 of the strategy in the case B = B HA . We prove the following lemma. A be a square matrix with λ(A) = 0 and G be a representing subgraph of G c (A) with s.c.c. G 1 , · · · , G m and γ l be multiples of γ(G l ) for l = 1, . . . , m.

Lemma 8.2. Let
For any t ≥ max l (T γ l cr (G l ) − γ l + 1) and any i, j, inequality (6.3) holds with γ = lcm l γ l and D = G ha (the graph defining B HA in Section 3).
Proof. We assume without loss of generality that A is max-balanced.
Let W be a walk with maximal weight in W t (i Denote the maximum weight of edges in W by µ(W ). Define the graph By the definition of Hartmann-Arguelles threshold graphs, G c (A) ⊆D ⊆ G ha . In both cases of (8.1), walk W contains a node k of digraphD, which is completely reducible (due to the max-balancing).
Let W = W 1 · W 2 with W 1 ending at node k. By definition ofD, there exists a critical node ℓ in the same s.c.c. H ofD as k. Moreover, H contains a whole s.c.c. G c l (A) of G c (A), and hence also a component G l of the representing subgraph G. Hence we can choose ℓ in G l .
Let V 1 be a walk inD from k to ℓ and V 2 be a walk inD from ℓ to k. Set V = V 1 V 2 and W 3 = W 1 · V γ l · W 2 . By the definition of the cycle replacement threshold, there exists a walkW ∈ W t,γ l (i G l − → j) obtained from W 3 by removing cycles and possibly inserting cycles in G l such that l(W ) ≤ T γ l cr (G l ) ≤ t + γ l − 1.
Recall that since A is max-balanced and λ(A) = 0, all edges have nonpositive weights, and the weight of each edge ofD is not smaller than that of any edge of W . Each edge of W is either removed, kept or replaced by an edge ofD inW , thus we conclude that p(W ) ≥ p(W ). This shows However, Theorem 6.1 implies that for each l and hence and this concludes the proof. Proposition 6.5 part (i) now follows from Lemmas 8.1 and 8.2.
8.3. Cycle threshold scheme. In this section, we perform step 1 of the strategy in the case B = B CT . We prove the following lemma.
For any t ≥ max T l(Z) cr (Z)|Z cycle of G ct and any i, j, inequality (6.3) holds with γ = γ(G c (A)) and D = G ct the graph defining B CT in Section 3.
A finite sequence of cycles Z 1 , . . . , Z m in G is called a staircase in G if, for all 1 ≤ s ≤ m − 1, Z s and Z s+1 share a node, p(Z s )/l(Z s ) ≤ p(Z s+1 )/l(Z s+1 ) and, moreover, the cycle mean of Z s+1 is the greatest among all the cycles sharing a node with Z s . Lemma 8.4. Let µ > µ ct and Z be a cycle in T ct (µ) or µ = µ ct and Z be a cycle in G ct (µ) with p(Z)/l(Z) = µ. Then there exists a staircase Z 1 , . . . , Z m in T ct (µ) such that Z 1 = Z and Z m is critical.
Proof. Suppose by contradiction that no such staircase exists. Let Z 1 , . . . , Z m be a staircase in T ct (µ) such that Z 1 = Z and p(Z m )/l(Z m ) is maximal.
Denote µ ′ = p(Z m )/l(Z m ), so µ ′ < λ(A). If the s.c.c. of T ct (µ ′ ), in which Z m lies, contains a cycle of mean weight strictly greater than µ ′ , then we can build a staircase with a greater cycle mean of the final cycle, a contradiction. So that component of T ct (µ ′ ) does not contain a cycle of mean weight strictly greater than µ ′ , which is a contradiction to the definition of µ ct and the fact that µ ′ ≥ µ ct . Thus we must have µ ′ = λ(A).
Proof of Lemma 8.3 and Proposition 6.5 part (ii). Let t ≥ max ZTcr (Z) and let W ∈ W t (i → j) visiting a node of G ct but no critical node.
Denote by ν(W ) the largest cycle mean of subcycles of W . We assume in the following that ν(W ) is maximal among all W ∈ W t (i → j) with p(W ) = a By the definition of cycle threshold graphs, G c (A) ⊆D ⊆ G ct .
By Lemma 8.4, there exists a staircase Z 1 , . . . , Z m inD such that Z 1 has p(Z 1 ) = ν(W ) and shares a node with W , and Z m is critical. We inductively define walks W 0 , . . . , W m as follows: obtained from W ℓ−1 by removing at least one cycle and inserting at least one cycle in G (i.e., one copy of Z ℓ ) such that l(V ) ≤T l(Z ℓ ) cr (Z ℓ ) ≤ t. Now define W ℓ as walk V after inserting enough copies of Z ℓ , to have l(W ℓ ) = t. Thus Z ℓ is a subwalk of W ℓ for all ℓ, and walk W m contains a critical node.
We now show that p(W ℓ ) ≥ p(W ℓ−1 ) on each step. For this we will prove by induction that, on each step, the mean weight of Z ℓ+1 is not less than that of any cycle (and hence closed walk) in W ℓ . The base of induction (ℓ = 0) is due to the definition ofD. In general, observe that the cycles in W ℓ are 1)Z ℓ and cycles using the edges of Z ℓ , 2) cycles that were already in W ℓ−1 . For the latter cycles we use the inductive assumption, while the cycles using edges of Z ℓ share a common node with it and hence their mean weight does not exceed that of Z ℓ+1 by the definition of staircase. SettingW = W m we obtainW ∈ W t (i Let us first recall an elementary application of the pigeonhole principle. The origins of this lemma were briefly discussed by Aigner and Ziegler [1], p. 133. In the context of max-algebraic matrix powers, it was considered for the first time by Hartmann and Arguelles [14]. It is in the heart of almost all of our cycle reductions. One of the bounds that we use is in fact proved in [7] (see also [8], Theorem 2). The proof is recalled for the reader's convenience.
Proof. Let W be a walk going through i. Write this walk as W = W 0 ·Z 1 ·· · ··Z m ·W m where (i) all Z s are nonempty cycles, (ii) node i is a node of the walk W r , and (iii) m is maximal. Write also W r = V 0 V 1 so that i is the end of V 0 and the start of V 1 . The whole configuration is shown on Figure 2.
If l(W ) > (γ − 1) cr +(γ + 1) cd, then such a subset of cycles can be chosen, and a strictly shorter subwalk of the same length modulo γ is obtained by cycle deletion, hence the claim. (G c ) ≤ 2γ(G c )(n−1)+γ(G c )−1 for any s.c.c. G c ⊂ G c (A) but both bounds can be improved using various methods.
The first bound is improved in Section 9.2, following a method used in [14], which leads to: Proposition 9.3. For A ∈ R n×n max , Z a cycle of D(A) and γ a divisor of l(Z), we have: T γ cr (Z) ≤ (n − 1 − l(Z) + γ) cr + cd +l(Z), where cd = cd(D(A)) and cr = cr (D(A)).
This method also leads to: Proposition 9.4. For A ∈ R n×n max and Z a cycle with length n of D(A), we havẽ T n cr (Z) ≤ n 2 − n + 1.
The second bound, is improved in Section 9.3 thanks to a new method, which leads to: (n − 1) cr + cd +1 n cr + cd +1 9.4 Wi(n) n 2 − n + 1 9.5 l(Z)(n − 2) + n l(Z)(n − 1) + n Table 1. Expressions of Proposition 6.5 (with l(Z)) (γ − 1)(cr −1) + (γ + 1) cd 9.5 γ(n − 1) + n − |G c | Table 2 Tables 1 and 2 with Proposition 6.5. For each s.c.c. G c of G c (A), Table 3 explains which choices of N , γ and proposition to bound T γ cr (N ) should be made. To obtain bounds (4.1)-(4.3) we take, for the representing subgraph G in Proposition 6.5, any collection of critical cycles such that each s.c.c. of G c (A) contains exactly one cycle of the collection and each cycle has the minimal length in the corresponding s.c.c. In the case of (4.4) and (4.5), we set G = G c (A). Bounds (4.7) and (4.8) can be obtained from the last column of Table 3. Note that (4.7) is obtained as the minimum of two bounds.
The only difficult case is bound (4.6). Indeed, in the worst case, cycle Z with length n, we only getT n cr (Z) ≤ n 2 − n + 1 by Proposition 9.4. instead of Wi(n). Thus, Proposition 6.5 would give T 1 ≤ n 2 − n + 1 instead of T 1 ≤ Wi(n) and we have to go into more details. The proof of (4.6) is thus postponed to the end of the next subsection.
Z in staircase or Z critical l(Z) 9. 3, 9.5 (4.8) any i in any Z l(Z) 9.2 Table 3. How to deduce the bounds on T 1 9.2. Cycle removal by cycle decomposition. In this section, we present and improve the method of [14] to prove Propositions 9.3 and 9.4. It will also be used to prove that T 1 (A, B CT ) ≤ Wi(n) (Equation (4.6)) at the end of the next subsection. For any set of walks W α with α ∈ S for S a subset of natural numbers, let us denote by G(∪ α∈S W α ) the subgraph of D(A) consisting of all nodes and edges that belong to some walk W α , α ∈ S.
Proof of Propositions 9.3 and 9.4. To any walk W ∈ W(i Z − → j), we apply the following procedure, adapted from [14].
(1) We choose a decomposition of the walk W ∈ W(i → j) into a path P and a set of cycles Z α for α ∈ S (with S a subset of natural numbers). Note that P may be empty. If it is, walk W is closed. Then, it has the same start and end node. We denote by n W the number of nodes that appear at least once in W and by cd W the maximum length of an acyclic walk whose edges belong to W .
(2) We take a subset R 1 of S with |R 1 | ≤ n−l(Z) such that G(P ∪Z ∪ α∈R1 Z α ) is connected and contains all nodes appearing in W . This is possible because the connection of G(P ∪ Z) with all the nodes of W can be ensured by adding at most n − l(Z) edges of W to P ∪ Z, and hence by adding to it at most n − l(Z) cycles Z α , for α ∈ S. (3) Let R 2 be a result of recursively removing from S \ R 1 sets of indices whose corresponding cycles have a combined length that is a multiple of γ.
Set cr W = max α∈R l(Z α ) (circumference of the walk W ). (4) If G 0 = G(P ∪ α∈R Z α ) is connected, then we build a walk V ∈ W(i Z − → j) by starting from P and successively inserting (in some order) all cycles Z α with α ∈ R. (5) Otherwise, we build V ∈ W(i Z − → j) by starting from P and successively inserting (in some order) all cycles Z α with α ∈ R, and Z. By construction, l(V ) ≡ l(W ) (mod γ) in both cases. Let us bound the length of W .
If G 0 is connected, If G 0 is not connected, we have l(V ) ≤ cr W (n−l(Z)+γ −2)+(cd W + cr W )+l(Z). But there is someα ∈ R such that l(P ) + l(Zα) ≤ n W − 1, because otherwise every Z α with α ∈ R would share a node with P . Because |R\{α}| ≤ n−l(Z)+γ−2, we have This gives the following Lemma 9.6. For any cycle Z, any divisor γ of l(Z) and any walk W ∈ W(i there is a walk V ∈ W(i Z − → j) with length at most l(Z) + (n − l(Z) + γ − 2) cr W + max(min(n W −1, cr W + cd W ), cr W + cd W −l(Z)) obtained by removing cycles from W and possibly inserting Z such that l(V ) ≡ l(W ) (mod γ).
Moreover, if no copy of Z is inserted then l(V ) ≤ cr W (n − l(Z) + γ − 1) + cd W Using that cr W ≤ cr(D(A)) and cd W ≤ cd(D(A)), we get Proposition 9.3.
9.3. Cycle removal by arithmetic method. In this section, we present a new method to bound T cr leading to Proposition 9.5.
We begin with: Lemma 9.7. Let γ ∈ N and let W ∈ W(i → j). Then there exists a walk W ′ ∈ W(i → j) obtained from W by removing cycles such that l(W ′ ) ≡ l(W ) (mod γ) and each node appears at most γ times in W ′ .
Proof. Consider W as a sequence of adjacent nodes (i 0 , · · · , i L ), where L is the length of the walk. If a given node appears twice, first as i a and then as i b and if a ≡ b (mod γ), then the subwalk (i 0 , · · · , i a , i b+1 , · · · , i L ) is strictly shorter than W and has the same length modulo γ.
Iterating this process, we get a sequence of subwalks of W . Since the sequence of length is strictly decreasing, the sequence is finite and we denote the last walk by W ′ .
Obviously, l(W ′ ) ≡ l(W ) (mod γ) and a node does appear twice as i a and i b only if a ≡ b (mod γ), so the pigeonhole principle implies that it appears at most γ times (otherwise there would exist i a and i b with a ≡ b (mod γ)).
Proof of Proposition 9.5. We take W ∈ W(i G − → j) and construct a subwalk V with length at most γn + n − n 1 − 1 by the following steps.
1. Find the first occurrence of a node of G in W , and denote this node by k. Let W 1 be the subwalk of W connecting i to k, and let W 2 be the remaining subwalk. So we have (9.5) W 1 ∈ W(i → k), W 2 ∈ W(k → j), l(W 1 ) + l(W 2 ) = l(W ) 2. As long as there is a node ℓ that appears twice in W 1 and at least once in W 2 , we can write W 1 = U 1 · U 2 · U 3 and W 2 = V 1 · V 2 , where U 1 , U 2 , V 1 end with ℓ and U 2 , U 3 , V 2 start with ℓ. Thus, we can replace W 1 by U 1 · U 3 and W 2 by V 1 · U 2 · V 2 . Equation (9.5) still holds, but now i appears only once in W 1 .
Step 2 is over when all nodes that appear more than once in W 1 do not appear in W 2 . Let us denote the resulting walks by W 3 and W 4 respectively.
. Now we take a node of V and bound the number of its appearances.
(1) If it is a node of G \ {k}, then it appears only in W ′ 2 , thus at most γ times. k appears once in W ′ 1 , as ending node, and at most γ times in W ′ 2 . In the concatenation of the walks, one occurrence disappears, so all nodes of G appear at most γ times.
(2) If it is a node of W ′ 2 , then it is also a node of W 4 , it appears at most once in W 3 , thus also in W ′ 1 . Therefore it appears at most γ + 1 times in V . (3) If it is not a node of W ′ 2 , then it appears only in W ′ 1 , thus at most γ times. The total number of appearances of all nodes in V is at most (γ + 1)(n − n 1 ) + γn 1 = γn + (n − n 1 ), so l(V ) is bounded by γn + (n − n 1 − 1), as claimed.
Proof of Bound (4.6). To prove bound (4.6), we apply Lemma 8.1 as before and the difficulty only comes from Lemma 8.3 that is not good enough.
To prove that inequality (6.3) holds with t ≥ Wi(n) and D = G ct , we do as in the proof of Proposition 6.5: we remove cycles from a walk W to replace them by cycles with greater weight, following the staircase given by Lemma 8.4. In this process, the walks to reduce have no critical node.
Since W has no critical node, n W ≤ n − 1, and this bound is less than Wi(n) except when n W = n − 1. But in this last case one has a critical loop on the only critical node and the rest of the nodes are in W . Let Z be the penultimate cycle of the staircase, it shares nodes with W and contains the unique critical node. The weight of this cycle is greater than or equal to that of all cycles in W . Applying Proposition 9.5 with G = Z and γ = 1, it is possible to reduce the walk to a length at most 2n − l(Z) − 1, insert Z and then as many critical loops as necessary to get back to a walk with length t. This is possible if t ≥ 2n − 1. Thus, Equation (4.6) holds true for any n.  If moreover A has only finite entries, then equations (4.12) and (4.13) hold.
We begin with the following lemmas.
Lemma 10.2. Let A ∈ R n×n max be an irreducible matrix, and C, S, R be defined relative to any completely reducible G ⊆ G c (A). For any B subordinate to A and ij , By the optimal walk interpretation (2.1), there is a walk W connecting i to j, of length t, such that p(W ) = a (t) ij . As A is irreducible, there is a closed walk V containing i and a node k of G. If γ is the cyclicity of G then V γ W ∈ W t,γ (i G − → j), and (CS t R) ij = 0 by (6.1). Lemma 10.3. For any B ∈ R n×n max and any t ∈ N, letB be B − λ(B), we have: If B has only finite entries, thenb * ij ≤ (λ(B) − min kl b kl ). Proof. The first part of the claim immediately follows from the optimal walk interpretation (2.1) and (2.3).
For the second part, observe thatb * ij is equal to p(W ) − λ(B)l(W ) for some walk W connecting i to j in B. For any t ∈ N, the finite entries of CS t R satisfy If A has only finite entries, then for all i, j we have: Before proving this lemma, let us state another one to use for the matrices with finite entries. Proof. Let Z be a critical cycle of A. Since l(Z m ) = l(Z)m, there are walks W 1 , · · · , W l(Z) of length m such that W 1 · · · W l(Z) = Z m . Since l p(W l ) = p(Z)t = 0, there is a W k with nonnegative p(W k ).
Proof of Lemma 10.4. We first show inequality (10.1). By the optimal walk interpretation (6.1) we have (CS t R) ij = max{p(W ) : W ∈ W t,γ (i G − → j)} for any walk W . If (CS t R) ij is finite then the walk set W t,γ (i G − → j) is non-empty and contains a walk with the length bounded by T γ cr (G), hence (10.1).
To prove inequality (10.2), let us assume that A has only finite entries, and that t ≥ 2 + n (using that the sequence {CS t R} t≥1 is periodic).
Apply Lemma 10.5 with m = t − 2 and set W = (i, r) · W 0 · (s, j), where r, resp. s, are the beginning node, resp. the end node of W 0 . By the optimal walk interpretation (6.1), we get (CS t R) ij ≥ p(W ) ≥ a ir + a sj ≥ 2 min kl a kl .
The inequalities (10.3) and (10.4) are proved similarly. For (10.3), select a walk V with minimal length among those with weightb * ij on D(B) and a walk W 0 with nonnegative p(W 0 ) and length t − l(V ) − 2. Set W = (i, r) · W 0 · (s, i) · V and get For (10.4), select a walk W 0 with nonnegative p(W 0 ) and length t − 1 and set W = (i, r) · W k (where r is the beginning node of W 0 ).
Proof of Proposition 10.1 and Theorem 4.5. Assume that λ(A) = 0 and t is greater than one of the bounds. We want to prove that equation in the general case (the first inequality) and in the case of finite entries (the second inequality). If t is greater than one of the required bounds, then one of the inequalities (10.6) holds, and (10.5) follows.
To obtain Theorem 4.5 it remains to deduce the shorter parts of (4.9)-(4.11) from the longer ones. Observe that all the longer parts of the bounds are of the form for some i, j, k, l, where n 1 is greater than cd B . Using n 1 > cd B , expression (10.7) can be bounded by This completes the proof of all the bounds of Theorem 4.5.
It remains to prove Theorem 4.6. We do it by generalizing the proof of [8,Proposition 5].
Proof of Theorem 4.6. The case λ(A) = 0 is trivial. In the rest of the prove, we assume λ(A) = 1 by replacing A with λ(A) − ⊗ A. Figure 3. WalksṼ = W 1 W ′ 1 and W = W 1 W 2 Z r W 3 in proof of Theorem 4.6.
We denote by ∆ and δ the greatest and smallest edge weight in D(A), respectively. We have A = ∆ − δ. If ∆ = δ, then G c (A) = D(A) and hence B t ≤ A t ≤ CS t R by the optimal walks interpretations (2.1) and (6.1).
Denote by v max and v min the greatest and smallest entry of v, respectively. It is Let i be a node of D(A). Let V be a walk in D(B) of length t starting at i, and letṼ be the remaining walk after repeated cycle deletion. Let W 2 be a shortest path connecting some node k ′ ofṼ to a critical node k and let W 1 be the prefix ofṼ ending at k ′ and letṼ = W 1 · W ′ 1 . See Figure 3 for an illustration of these walks. We obtain using that λ(B)t ≤ −(||v|| + ||A||(n − 1)). Let Z be a critical cycle starting at k and set r = t − l(W 1 · W 2 ) /l(Z) . Then let W 3 be the prefix of Z of length t−l(W 1 ·W 2 ·Z r ), which is between 0 and l(Z)−1.
The claim for the case that all entries of A are finite follows from Lemmas 10.3 and 10.4.

Cycle Insertion
In this section, we state some bounds on ep γ . The exploration penalty has been introduced in [8], where the following is proven.
In connection with these schemes, define the following subsets: Remark 12.1. Unless i = s = j, a * is a * sj is the biggest weight of a walk connecting i to j via s. It follows from Theorem 6.1 and this optimal walk interpretation that i, j / ∈ J(i, j), J(i, j, l) and i / ∈ J(i, v), J(i, l, v). Moreover, if some critical s belongs to one of the sets defined here, then all its s.c.c. in G c (A) does, since for each pair of nodes in the same s.c.c. of G c (A) we can find a closed walk in G c (A) containing both of them.
Note that i, j / ∈ J(i, j), J(i, j, l) and i / ∈ J(i, v), J(i, l, v), by the optimal walk interpretation (2.3) of A * and CSR terms (6.1). Now letG c (A) be the remainder of the critical graph, without the s.c.c. with indices in J, for J = J(i, j), J(i, j, l), J(i, v) or J(i, l, v).
RedefineC,S andR usingG c (A) instead of G c (A), and the subordinate matrix' A of A where all rows and columns with indices in J are canceled, instead of A. RedefineB as a subordinate ofÃ whose indices are in D(B) (but not in J). This procedure will be referred to as local reduction of a weak CSR expansion. When J = J(i, j), or resp. J = J(i, j, l), J = J(i, v) or J = J(i, v, l), this will be called i, j-reduction, or resp. i, j, l-reduction, i, v-reduction or i, l, v-reduction. ij , for t ≡ l(mod γ) and t ≥τ (i, j, l), for t ≡ l(mod γ) and t ≥τ (i, l, v), (12.3) withC,S,R andB defined in the local reduction procedure.
Proof. We prove the theorem only in the case of i, j-reduction, i.e., in the first case of (12.3) corresponding to the first case of (12.1) and (12.2). The rest is analogous. Let N B , resp. N c be the set of nodes of D(B), resp. G c (A).
Define the subordinate matrix A ′ of A formed by setting to 0 all rows and columns with indices in J(i, j) ∩ N B . We first show that the first equation of (12.1) for a (t) ij holds also with CSR terms and B defined from A ′ instead of A. First, recall that the weights of walks going through s ∈ J(i, j)∩N B are less than min l (CS l R) ij , and (CS l R) ij is the greatest weight of any walk with certain length constraint, connecting i to j via a critical node. Defining (CS l R) ij from A ′ amounts to canceling all walks going through s ∈ J(i, j) ∩ N B and contributing to (CS l R) ij . Since such walks have low weight, (CS l R) ij does not decrease, for any l, when defined from the subordinate matrix A ′ , so it is exactly the same. Next, observe that (since the weights of walks going through s ∈ J(i, j) ∩ N B are less than min l (CS l R) ij ) we can replace b ij in the first equation of (12.1). We next show that the CSR term defined from A ′ can be reduced. Use expansion (6.2) of the CSR terms defined from A ′ , where the first terms in (6.2) are defined from the components of G c (A) with indices in J(i, j) (these components can be taken in any order). The sum of these terms expresses p(W t (i J(i,j)∩Nc −−−−−−→ j)) (with walk sets defined in D(A ′ )) for all large enough t. Since these walk weights are strictly less than the entries of CSR, all those terms in expansion (6.2) with indices in J(i, j) can be canceled. The remaining part of expansion (for the entry i, j) sums up to the reduced CSR term defined from the subordinate matrixÃ (as defined in the reduction procedure).
Notice that the proofs of the Theorems 4.5 and 4.6 in Section 10 work with walks in W t (i → j). Hence the corresponding bounds can be combined with all reductions of Theorem 12.2. In particular, local reductions may lead to smaller B and λ(B) when i and j or i and v are fixed. Moreover, they can also result in decrease of the initial bounds onτ based on the cycle removal threshold, since some of the critical components get removed.