Average case analysis of greedy algorithms for optimisation problems on set systems

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Laboratoire d'Analyse et Modélisation de Systèmes pour l'Aide à la Décision CNRS UMR 7024

CAHIER DU LAMSADE

125

Octobre 1994

Average case analysis of greedy algorithms for

optimisation problems on set systems

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Analyse en moyenne de la performance des algorithmes gloutons

pour des probl`

emes d’optimisation sur des syst`

emes d’ensembles

R´esum´e

Nous présentons un cadre général pour l’analyse asymptotique d’algorithmes glou-tons pour plusieurs problèmes d’optimisation sur des systèmes aléatoires d’ensem-bles lorsque le rapport entre la taille de l’ensemble de base et la taille du système d’ensembles reste constant. Les systèmes d’ensembles sont engendrés via des graphes biparties aléatoires et les approximations des chaˆınes de Markov sont réalisées à l’aide d’équations différentielles ordinaires.

mots-clés: Problème NP-complet, algorithme polynomial approché, approximation en moyen-ne, couverture d’ensembles, transversal d’un hypergraphe, stable d’un hypergraphe

Average case analysis of greedy algorithms for optimisation

problems on set systems

Abstract

A general framework is presented for the asymptotic analysis of greedy algorithms for several optimisation problems such as hitting set, set cover, set packing, etc., applied on random set systems. The probability model used is specified by the size n of the ground set, the size m of the set system and the common distribution of each of its components. The asymptotic behaviour of each algorithm is studied when n and m tend to ∞, with m_n a fixed constant. The main tools used are the generation of random families of sets via random bipartite graphs and the approximation of Markov chains with small steps by solutions of ordinary differential equations.

keywords: NP-complete problem, polynomial time approximation algorithm, average case ap-proximation, set covering, hitting set, hypergraph independent set

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1 Introduction

There are many optimisation problems defined on families of sets such as hitting set, set cover, set packing, etc., which are known to be NP-complete, and for the (approximate) solution of which one must thus use either heuristics or greedy algorithms. The purpose of this paper is to present a general framework which permits the asymptotic analysis of greedy algorithms for several of these problems, including in particular those just mentioned.

This framework comprises two main parts:

(i) a representation of random set systems by using degree-constrained random bipartite graphs.

(ii) the introduction of a Markov chain on a certain state space for the analysis of each particular greedy algorithm.

It remains then to analyse the behaviour of this Markov chain. This behaviour can be approximated for problems of large size by the solution of an ordinary differential equation (or a system of such equations) which can be obtained in closed form.

We point out that Markov chains have been used previously, explicitely or implicitely, in the analysis of algorithms by various authors. Let us mention Fernandez de La Vega ([3]) and Frieze Radcliffe and Suen ([5]).

This paper should be considered mainly as a theoretical contribution to the analysis of greedy algorithms on set systems and, accordingly, we have not included numerical results. Such results would be of limited help here since in most cases (a basic obstruction here, comes from the fact that) we do not know the values of the optimum of the objective functions (the approximate values which may be obtained via the use of the first moment method seem to be rather inaccurate).

A preliminary version of this paper has been presented in the LATIN’92 conference ([4]). The plan of this paper is as follows. In § 2 we introduce the model of random set systems that we use and a useful connection with random bipartite graphs. In § 3, we study in details an algorithm for set covering and hitting set and in § 4 we show how this algorithm can be modified to be used for set packing and hypergraph independent set. Finally, in the appendix, we prove the convergence of the differential system describing the set covering algorithm.

2 Random set systems as random bipartite graphs

2.1 Random sets and random multisets

It will be convenient to work with families of random multisets rather than random sets. Moreover, we will view our families as ordered. In other words, we introduce the uniform measure µ on the set of ordered k-tuples from some finite ground set S and work with µm_,

the m-fold product of µ defined on the set of ordered families of k-tuples of size m. Let us denote by ν = νm the more conventional mesure which gives the same weight to each

m-family of pairwise distinct k-sets. It is easy to see that when sampling on genuine k-sets with replacement we get m distinct sets with probability near to 1 if only m = o(nk/2_).

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clearly p = (n − 1)(n − 2) . . . (n − k + 1) nk−1 ≥ exp ( −k 2 2n ) .

Thus, setting λ = m/n, the probability pm _{of getting m genuine k-sets when sampling}

according to µm satisfies pm ≥ exp {−(1/2)λk2}, and this together with the former result,

implies that, for a fixed λ, the µm _{measure of the genuine systems of m k-sets drawn from}

a set of size n exceeds exp{−λk2_{}. Thus, again for a fixed λ, every event “almost sure”}

relatively to the sequence of measures µm_{, m → ∞ is also almost sure relatively to the}

more conventional measure νm and this property suffices for our purposes.

2.2 Random set systems and bipartite graphs

Let X and Y denote sets with the same cardinality. Let P = {X1, X2, . . . , Xm} denote

a partition of the set X into sets with sizes m1, m2, . . . , mm. Let P = (p1, p2, . . . , pm)

where pi denotes the number of classes of P of size i. Similarly, let Q = {Y1, Y2, . . . , Yn}

denote a partition of the set Y into sets with sizes d1, d2, . . . , dnand let Q = (q1, q2, . . . , qn)

be defined similarly as P .

Now, let Π denote a random pairing between the sets X and Y . If we contract each set of vertices Xito a single vertex and similarly each set Yj, we get a bipartite graph, say B, with

partitions M = (m1, m2, . . . , mm) and D = (d1, d2, . . . , dn). If we think of Q as a fixed

ground set, this pairing defines naturally a random set-system P = {X1, X2, . . . , Xm},

where we identify Xi with the set of classes Yj adjacent to it in Π.

Notice that this pairing procedure gives the same measure to all genuine set systems, in which no set contains more than once any element of the ground set and no two sets are equal. Indeed, any such system is given by precisely Qm

i=1mi!Qni=1dj! distinct pairings.

Henceforth, we have at our disposal a procedure for generating random systems of sets with fixed cardinalities, subject moreover to the condition that each element of the ground set belongs to a prescribed number of sets in the system.

2.3 A particular class of set systems distributions

We are in fact mostly interested in the following class of probability distributions on set systems. We postulate that the cardinalities of our random sets X1, X2, . . . , Xm are

independent random variables N1, N2, . . . , Nm with a common distribution F defined by

its individual probabilities {pj = Pr[N1 = j], j = 1, 2, . . .}. Moreover, we postulate that,

for each fixed j, the set Xi is, conditionally on Ni = j, uniformly distributed between the

(multi) subsets of [n] of cardinality j.

Thus, if we choose for F the distribution concentrated on some fixed integer k, we get a random k-uniform set system of size m. If we give a fixed value to the ratio m/n, say m/n = λ, then, by the law of large numbers, the number nh of vertices of degree h in

the corresponding incidence graph satisfies, for each integer h ≥ 0, nh/n → e−λλh/h! in

probability as n → ∞.

Now, if we want to generate a random set system according to some distribution in the class just defined, we are faced with two difficulties:

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i) the degrees are unbounded,

ii) the partition of the incidence graph is not fixed.

In order to circumvent the first difficulty, we restrict our algorithms to work only on vertices with a bounded (but arbitrarily large) degree ∆ and we show that the size of the obtained solutions tends to a limit as ∆ → ∞.

The second difficulty is circumvented by conditioning on the degrees of the graph. Indeed, since the proportion of vertices of each fixed degree in each color class of the incidence graph B = Bn tends to its expectation when n → ∞ and, as it will be seen, the sizes

of the obtained solutions are continuous relatively to the ratios nh/n, we can obtain the

limits of these sizes by replacing these ratios by their expectations.

2.4 Removing Vertices from the Incidence Graph

Let us return to the graph B and let us look at what happens when we remove a ver-tex y randomly chosen from the vertices in Y of degree h in B, i.e, we remove the h corresponding vertices in V (P ) (then, all the sets containing y are removed). Since the pairing P between the sets X and Y is random, the subset of Y \ {y} whose vertices are adjacent to the removed sets in X is also random. It can then easily be checked that the new incidence graph is again random within the graphs which have the same degrees. We note that, since B may eventually be a multigraph, the number of removed sets may (exceptionally) be strictly smaller than h. We will of course take this fact into account in our calculations.

It will be seen that all the algorithms we consider, remove at each step either vertices from Y chosen on the basis of their degrees, or a vertex Xi chosen on the basis of the

size |Xi|, or both. This implies again that the successive incidence graphs which appear

are random within the graphs which have the same partition.

3 Hitting Set and Set Cover

3.1 A Lower Bound for the Random Hitting Set Problem

Let C denote a fixed subset of [n] of cardinality l. The probability that C intersects Ci

for some fixed i is 1 − [1 − (l/n)]k. Since the Ci’s are independent, the probability that

C intersects Ci for each i is p =

h

1 − [1 − (l/n)]kim. Hence, the expectation EYl of the

number Yl of sets of cardinality l which intersects each Ci is given by

n l ! 1 − 1 − l n !k  m . (1)

Setting m = λn and l = βn, we get log EYl ∼ n −λ log 1 1 − (1 − β)k + log 1 ββ_{(1 − β)}1−β ! .

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This implies EYl = o(1) whenever λ > λβ = log 1 ββ_(1−β)1−β log_1−(1−β)1 k . (2)

The following assertions can be easily deduced from this formula. Assertion 1. For any fixed k, β tends to 1 when λ tends to ∞.

Assertion 2. For any fixed λ, the ratio βk/ log k tends to 1 when k tends to ∞. Proof of assertion 1.

Putting γ = 1 − β, and using the inequalityn_l≤ (ne/l)l, we get from (1), E(Yl)1/n ≤ 1 − γkλ e γ !γ .

Suppose for a contradiction that γ ≥ γo > 0. Then, when λ → ∞, the first term on

the righthandside tends to 0 whereas one can easily check by taking derivatives that the second term is bounded above by e. Therefore, EYl → 0 as λ → ∞.

Proof of Assertion 2.

Let us prove first that we have β ≥ k−1_{log k(1 − o(1)). Inserting in (2) the value β =}

(log k − 2 log log k)/k and using the inequality β log(β−1_{) ≤ 1, we get}

λβ ≤ log _1−β1 log 1 1−e−βk ≤ β + 2β 2 e−βk _{− e}−β2_k2 ≤ 2k−1_{log k}

k−1_log2_{k(1 − o(1))} = o(1).

The proof of the reverse inequality is similar and is omitted.

3.2 The GRECO algorithm

For any family C of sets, a set H is a hitting set for C if H has a non-empty intersection with each element of this family and the minimum hitting set problem is that of finding a hitting set of minimum cardinality (see also [6]). We note that this problem is the dual (via the interchange of the two vertex sets of the incidence graph) of the set cover problem.

We note that recently, Lund and Yannakakis in [10] have proved that unless NP ⊆ DTIME[npoly log n_{] (conjecture weaker than P = NP but highly improbable), there is}

no polynomial time algorithm approximating the optimal solution of these two problems with a ratio smaller than c log n for a constant c < 1/4 (this result is of course a “worst case” result). Consequently, average case studies of approximation algorithms for these problems are of high theoretical and practical importance. Let us mention that another algorithm due to Karp has been analysed with the same probability model ([7]).

Let the family of sets P = {X1, X2, . . . , Xm}, together with the corresponding partitions P

and Q, be defined as above via the random pairing Π. We consider the greedy algorithm 1 for finding a hitting set for P. Informally, our algorithm GRECO selects at each step

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begin

split every set Yi with degree di > η into di new vertices of degree 1

for h ← η to 1 do while|X| 6= 0 do

choose randomly one of the Yi’s with maximum di;

delete Yi;

delete X = Xi1, . . . , Xih, the set of the classes in P, incident to Yi in Π;

delete any other vertex in Y adjacent to at least one of the classes of X; let X, Y, P, Q denote now the new configuration;

let Π denote the survived incidence graph od

od end;

Algorithm 1: GRECO algorithm. We suppose that after the splitting of the Yi

per-formed in the first line of the algorithm, the incidence graph Π is left untouched).

begin

γ ← ⌈n log k/k⌉; SC ← {x1, . . . , xγ}

for j ← 1 to m do

ifCj∩ SC = ∅ then add to SC an arbitrarily chosen element of Cj

od; end.

Algorithm 2. QUICK algorithm

one element chosen at random between the elements hitting the biggest number of sets which have not been hitted before. It depends on an integer parameter η.

Before proceeding to the analysis of this algorithm let us remark that when the product λk is high, the distribution of the degrees tends to uniformity. It turns out that in this case the much simpler algorithm QUICK (algorithm 2) provides asymptotically optimal solutions.

Let us proceed to the analysis of QUICK.

Plainly the size of the solution is bounded above by γ + µ, where µ denotes the number of elements of C which do not intersect the set {x1, . . . , xγ}. We have

Eµ = λn _n−γ k _n k ∼ λn 1 − log k k !k .

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of its expectation. This implies, using Tchebicheff inequality and the familiar inequality 1 + x < ex_{, that Pr [µ ≤ λn/k] →}

n→∞ 1.

Thus, again with probability tending to 1, the total size of the solution found by QUICK is bounded above by (n log k/k) + (λn/k).

Assertion 2 then implies that the approximation ratio of QUICK tends to 1 when k tends to ∞. Moreover, it can be shown that algorithm GRECO works better than QUICK; thus, assertion 2 provides also an approximation guarantee for GRECO when k tends to ∞.

3.3 Analysis of GRECO on random instances

Let us consider again a random configuration X, Y, P, Q, Π and the corresponding par-titions P and Q. We are interested here in a system of m random sets of size k, drawn from a ground set of size n and we assume that the ratio m/n is fixed, say m/n = λ, and we let m (and n) go to infinity. We must analyse separately the phases of the algorithm corresponding to the successive removals of the Q-vertices of degree η, η − 1, . . . , 1. Let h be a positive integer satisfying 1 ≤ h ≤ η and assume that, after having used vertices with degrees greater than h, there remain, for 1 ≤ j ≤ h, precisely mj Q-vertices of degree j.

Then, as it was pointed out before, the conditional distribution of the remaining incidence graph (conditioned by the previous steps of the algorithm) coincides with the “random pairing” distribution corresponding to the nj’s and mj’s (the nj’s being here all equal

to k).

Let us remove a (randomly chosen) Q-vertex of degree h. Then: • Pr[h new sets in P are captured] = 1 − o(1);

• the total decrease of the degrees of the remaining vertices has expectation h(k −1)−o(1) and the probability that the degree of any given Q-vertex is decreased by 1 is equal to [(h(k − 1) − o(1))j]/P

1≤i≤himi, where j denotes the degree of X;

• the decrease of the total size of the remaining Q-vertices has expectation h(k − 1) + o(1) and this implies that, setting S = h(k − 1)/P

1≤i≤himi, the expectations E∆mj of the

increments ∆mj satisfies

E∆mj = (j + 1)mj+1S − jmjS, 1 ≤ j ≤ h − 1,

E∆mh = −1 − hmhS.

Let Rη,h= (1/n)P1≤i≤himi denote the average degree of the Q vertices. Notice that the

total decrease of the degrees of these vertices in one step is h + h(k − 1) = hk. Standard results concerning the approximation of “small steps Markov chains” (see for instance Proposition 4.1 in [9]) imply, setting yη,h,j = mj/n and θ = t/n, that the yη,h,j’s are well

approximated as n → ∞ by the solution ~y of the following system of differential equations (where we use Newton’s notation for dy/dθ)

(Sη,h)    ˙yη,h,j(θ) = (j + 1)_Rh(k−1) η,h−hkθyη,h,j+1(θ) − j h(k−1) Rη,h−hkθyη,h,j(θ), 1 ≤ j ≤ h − 1 ˙yη,h,h(θ) = −1 − h_Rh(k−1)_η,h_−hkθyη,h,h(θ)

in the sense that, sup

θ

|yη,h,j(θ) − (1/n)mj(θ)| → 0 as n → ∞, in probability for each

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This concludes the description of the systems (Sη,h), 1 ≤ h ≤ η.

Let us mention that the use of three indices η, h and j which may seem luxurious to the reader, is in fact needed for the mathematical study of the above systems (see the appendix).

Let us emphasize now how we use these systems to analyse GRECO with parameter η. We integrate first (Sη,η) under the following initial conditions:

yI η,η,1(0) = e−λ+ ∞ X j=η+1 e−λ_λj (j − 1)! yI_η,η,j(0) = e −λ_λj j! , 1 ≤ j ≤ η (3)

where λ = m/n is the average degree of the Q-vertices of Π.

Observe that for each j, the corresponding RHS of the above equalities is equal to the almost sure limit of mj(0)/n as n → ∞. Also, Rη,η =Pηj=1jyη,η,jI . We denote by θη,η the

first time at which the last coordinate becomes 0:

θη,η = min{θ > 0 : yη,η,η(θ) = 0}.

We set yI

η,η−1,j = yη,η,j(θη,η), 1 ≤ j ≤ η − 1, and Rη,η−1 = Pη−1j=1jyη,η−1,jI . Then, we turn

to the system (Sη,η−1) of dimension (η − 1) × (η − 1) which we integrate with the initial

conditions: θ = 0 and yη,η−1,j(0) = yIη,η−1,j, 1 ≤ j ≤ η − 1. We define θη,η−1 by

θη,η−1 = min{θ > 0 : yη,η−1,η−1(θ) = 0}

and we repeat inductively this procedure for each h ≥ 1, taking as initial conditions for the system (Sη,h) the final conditions for (Sη,h+1).

We define τη = η X h=1 θη,h.

Our aim is to study the asymptotic behaviour of the sequence (τη)_η∈_{IN . We observe first}

that, using inductively the approximation theorem of [9], we get immediately the following theorem.

Theorem 1. There exists a function f (., ., .) such that, for any fixed values of the pa-rameters k, η and λ, the size S = S(k, η, λ, n) of the solution found by GRECO when applied with the parameter η to a random instance of m k-sets of size k drawn randomly from a set of size m = nλ−1_{, satisfies}

S

n.f (k, η, λ) → 1 in probability as n → ∞.

We will prove in the appendix that the sequence (τη)_η∈_IN∗ tends to a limit as η → ∞.

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Theorem 2. There exists a function g(., .) such that, for any fixed values of the pa-rameters k and λ and any positive ǫ, the size S = S(k, η, λ, n) of the solution found by GRECO when applied with the parameter η to a random instance of m k-sets of size k drawn randomly from a set of size m = nλ−1_{, satisfies}

1 − ǫ ≤ S

n.g(k, λ) ≤ 1 + ǫ in probability as n → ∞, if only η is sufficiently large.

3.4 Proof of the convergence of of the sequence (τη)_η∈_IN∗

3.4.1 Explicit integration of the differential systems

Let us firs reformulate the system (Sη,h). By introducing β = (k − 1)/k and αη,h =

(1/hk)Rη,h, 1 ≤ h ≤ η, the vectors of IRh : yη,h(θ) = (yη,h,j(θ))T1≤j≤h, eh = (0, 0, . . . , −1)T

and the matrix Mh of dimension h × h with

Mh =      Mh,j,j = −j Mh,j,j+1 = j + 1 Mh,i,j = 0 elsewhere

we can reformulate (Sη,h) in a vectorial form as follows:

(Sη,h) ˙yη,h(θ) = β αη,h− θ Mhyη,h(θ) + eh. We define also τη = η X h=1 θη,h.

Our aim is to study the asymptotic behaviour of the infinite sequence (τη)_η∈_{IN .}

We consider (Sη,h) when θ ∈ [0, αη,h[. It is a vectorial first-order linear

non-homogene-ous non-autonomnon-homogene-ous o.d.e. (ordinary differential equation). The system’s coefficients are continuous functions of the variable θ on [0, αη,h[. So, Cauchy’s theorem about the

existence and uniqueness of the solutions of linear Cauchy’s problems is applicable here, and moreover the non-extendable solution of (Sη,h) under the initial conditions yη,h(0) =

yI

η,h, is defined everywhere on [0, αη,h[, ([1], p. IV, 17).

To solve (Sη,h), we consider the associated homogeneous o.d.e.:

(Hη,h) ˙y(θ) =

β αη,h− θ

Mhy(θ).

The resolvant of this homogeneous system is Φ(θ) = exp {β ln (αη,h/αη,h− θ) Mh}. The

calculation of the characteristic polynomial of Mh shows that Mh possesses h distinct

eigenvalues, namely −1, −2, . . . , −h, hence Mh is diagonalizable. If bj is an eigenvector

associated to eigenvalue −j, for each j in {1, 2, . . . , h}, then B = (b1, . . . , bh) is a basis

of IRh and, in terms of B, Mh is expressed by means of the diagonal matrix Dh =

−iδj_i

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where δji is the Kronecker symbol. Denoting by P the matrix of the change of coordinates

from B to the canonical basis, we have Mh = P DhP−1. For each j, we take bj =

(bj,1, . . . , bj,h)T where bj,i =        (−1)i+jj i if i < j 1 if i = j 0 if i > j The matrix P is then defined as

Pi,j =        (−1)i+jj i if i < j 1 if i = j 0 if i > j and its inverse matrix P−1 _{= Q is defined as}

Qi,j =        _j i if i < j 1 if i = j 0 if i > j

Then Φ(θ) = exp {β ln (αη,h/αη,h− θ) P DhP−1} = P exp {β ln (αη,h/αη,h− θ) Dh} P−1,

and we can explicitely compute its components, obtaining so:

     Φi,j(θ) = 0 if i > j Φi,j(θ) = _j i _α η,h αη,h−θ −βi 1 − αη,h αη,h−θ −βj−i if i ≤ j (4)

So, we have expicitely integrated (Hη,h). The solution of (Sη,h) under the initial conditions

yη,h(0) = yIη,h is , for θ ∈ [0, αη,h[, ([1], p. IV, 21): yη,h(θ) = Φ(θ)yη,hI + Φ(θ)   θ Z 0 Φ−1(σ)dσ  e_h (5) Let uη,h(θ) = Φ(θ)   θ Z 0 Φ−1(σ)dσ  e_h. (6)

Since uη,h is the solution of (Sη,h) under the initial conditions uη,h(0) = 0, one can verify

by a straightforward calculation that, for every j = 1, . . . , h and for every θ ∈ [0, αη,h[,

uη,h,j(θ) = − h j ! θ Z 0 αη,h− θ αη,h− t !−βj 1 − αη,h− θ αη,h− t !β  h−j dt. (7)

So, with expressions (4,5,6,7), we have explicitely resolved the system (Sη,h) under the

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3.4.2 The computation of θη,h

Using the results of § 3.4.1, we have, ∀θ ∈ [0, αη,h[,

yη,h,h(θ) = αη,h αη,h− θ !−βh y_η,h,hI + 1 βh − 1   α−βh+1_η,h (αη,h− θ)−βh − (αη,h− θ)  . Assuming that yI

η,h,h > 0, we search for these θ ∈ [0, αη,h[ verifying yη,h,h(θ) = 0. By a

straightforward computation, we obtain: if yI

η,h,h > 0, k ≥ 2 and h ≥ 2, then there exists

a unique θη,h∈ [0, αη,h[ such that yη,h,h(θη,h) = 0 given by

θη,h= αη,h  1 − 1 + βh − 1 αη,h y_η,h,hI ! 1 −βh+1  . (8) 3.4.3 The signs of yI η,h,j

Let us start by establishing the following assertions. (A1) Let 1 ≤ h ≤ η and suppose that ∀j, 1 ≤ j ≤ h, yI

η,h,j > 0. Then:

(i) ∀j ∈ {1, 2, . . . , h − 1}, ∀θ ∈ [0, θη,h[, yη,h,j(θ) > 0;

(ii) ∀j ∈ {1, 2, . . . , h − 1}, yI

η,h−1,j = yη,h,j(θη,h) ≥ 0.

To show (i), we reason by contradiction in assuming that there exist θ ∈ [0, θη,h[ and j,

1 ≤ j ≤ h − 1, such that yη,h,j(θ) ≤ 0. Since yη,h,j(0) = yη,h,jI > 0, we have θ > 0. By an

argument of connectedness, there exists θ1 ∈]0, θ] such that yη,h,j(θ1) = 0. Let us denote

by J = {j : 1 ≤ j ≤ h − 1, yη,h,j([0, θη,h[) ∋ 0}, J 6= ∅. For each j ∈ J, let us introduce

θ(j) = inf{θ ∈ [0, θη,h[ : yη,h,j(θ) = 0} = inf{[0, θη,h] : yη,h,j(θ) = 0}

= inf([0, θη,h] ∩ y−1η,h,j({0})).

Since this last set is closed and bounded below, it contains its greatest lower bound, thus yη,h,j(θ(j)) = 0. As, on the other hand, yη,h,j(0) > 0 and yη,h,j is continuous, then yη,h,j is

positive on a neighbourhood of 0, hence θ(j) = 0. By definition of θ(j), for all θ ∈ [0, θ(j)[, we have yη,h,j(θ) > 0. Let us introduce ˆθ = min{θ(j) : j ∈ J}; we have 0 < ˆθ < θη,h

and let ˆj = max{j ∈ J : θ(j) = ˆθ}. Note that j cannot be equal to h − 1. In fact, if the contrary holds, we have ∀θ ∈ [0, ˆθ[, yη,h,h−1(θ) > 0 and yη,h,h−1(ˆθ) = 0, which implies

˙yη,h,h−1(ˆθ) ≤ 0. By using (Sη,h), we get

0 ≥ ˙yη,h,h−1(ˆθ) = h β αη,h− ˆθ yη,h,h(ˆθ) − (h − 1) β αη,h− ˆθ yη,h,h−1(ˆθ) = h β αη,h− ˆθ yη,h,h(ˆθ) > 0 which is impossible.

Consequently, we have proved that 1 ≤ ˆj ≤ h − 2. Let us note that if ˆj ≤ j ≤ h and if θ ∈ [0, ˆθ], then yη,h,j(θ) > 0 because j > ˆj =⇒ (j /∈ J ∨ θ(j) > ˆθ).

Since y_η,h,ˆ_j(θ) > 0 for θ ∈ [0, ˆθ], and y_η,h,ˆ_j(ˆθ) > 0, we have ˙y_η,h,ˆ_j(ˆθ) ≤ 0. This, by using (Sη,h), implies that

0 ≥ ˙y_η,h,ˆ_j(ˆθ) = (ˆj + 1) β αη,h− ˆθ y_η,h,ˆ_j+1(ˆθ) − ˆj β αη,h− ˆθ y_η,h,ˆ_j(ˆθ) = (ˆj + 1) β αη,h− ˆθ y_η,h,ˆ_j+1(ˆθ) > 0

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a contradiction.

This justifies (i). On the other hand, (ii) is a consequence of (i) because of the continuity of yη,h,j.

(A2) Under the hypotheses of (A1) let 1 ≤ j ≤ h−1. If yη,h,j(θη,h) = 0, then ∀i, j ≤ i ≤ h

we have yη,h,i(θη,h) = 0.

In fact, with (A1) we know that for all θ ∈ [0, θη,h[, yη,h,j(θ) > 0 and yη,h,j(θη,h) = 0,

which implies 0 ≥ ˙yη,h,j(θη,h) = (j + 1) β αη,h− θη,h yη,h,j+1(θη,h) − j β αη,h− ˆθη,h yη,h,j(θη,h) = (j + 1) β αη,h− θη,h yη,h,j+1(θη,h) ≥ 0. Thus, yη,h,j+1(θη,h) = 0.

We can repeat the same reasoning for yη,h,j+2 and so on.

(A3) Under the hypotheses of (A1) we have:

(∀j = 1, . . . , h − 1, yη,h,j(θη,h) > 0) ⇐⇒ (yη,h,h−1(θη,h) > 0).

In fact, the first implication is obvious and the converse results from (A2).

We also remark that yη,h,h−1(θη,h) = 0 ⇐⇒ ˙yη,h,h−1(θη,h) = 0. In fact, it suffices to remark

that by the definition of θη,h, ˙yη,h,h−1(θη,h) = −(h − 1)(β/αη,h− θη,h)yη,h,h−1(θη,h).

3.4.4 Relations between θη,h and αη,h

We multiply the j-th equation of (Sη,h) by j and next, we add the h resulting equations;

we thus obtain Ph

j=1j ˙yη,h,j(θ) = −h − [β/(αη,h− θ)]Phj=1jyη,h,j(θ).

Denoting by zη,h(θ) =Phj=1jyη,h,j(θ), we see that zη,h is a solution of the o.d.e.:

˙zη,h(θ) = −h −

β αη,h− θ

zη,h(θ) (9)

under the initial condition zη,h(0) = Rη,h. We note that equation (9) is an

one-dimen-sional linear non-homogeneous o.d.e. with variable coefficient (continuous function of θ) and consequently, the theorem about the existence and uniqueness of the non-extendable solutions of Cauchy’s problems is usable in (9). Integrating (9), for instance using the method of constants variations, we obtain, ∀θ ∈ [0, αη,h[,

zη,h(θ) = Rη,h 1 − θ αη,h !β + αη, h −β + 1h   1 − θ αη,h ! − 1 − θ αη,h !β = R_η,h− hkθ.

On the other hand, zη,h(θη,h) = Phj=1jyη,h,j(θη,h) = Ph−1j=1 jyIη,h−1,j = Rη,h−1, that implies

Rη,h−1 = Rη,h− khθη,h, 2 ≤ h ≤ n − 1, or equivalently αη,h−1= h h − 1 ! (αη,h− θη,h). (10)

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By introducing the 2-variable function Th(α, y) = α

h

1 − [1 + (βh − 1)/αy]1/(−βh+1)i for α > 0, y > 0, we see, using expression (8), that θη,h= Th(αη,h, yIη,h,h). One can also verify

that if k ≥ 3, h ≥ 2, α > 0, y > 0 and kα ≥ y then 1 −βh + 1ln 1 − −βh + 1 α y ! ≥ 1 1 − (k − 1)(h − 1)ln((k − 1)(h − 1)).

By taking for a short while X = (k − 1)(h − 1) ≥ 2, the previous inequality allows us to obtain [1/(−βh + 1)] ln [1 − (−βh + 1)/αy] ≥ ln(X)/(1 − X) ≥ ln (1/2); therefore, taking the exponential mapping, we get [1 + (βh − 1)/αy][1/(−βh+1)] ≥ 1/2; consequently, if k ≥ 3, h ≥ 2, α > 0, y > 0 and kα ≥ y, then

Th(α, y) ≤

α

2. (11)

From the above expression, since kαη,h = (1/h)Rη,h ≥ yη,h,hI , we deduce, ∀k ≥ 3, ∀h ≥ 2

θη,h≤

1 2αη,h. 3.4.5 An upper bound of yη,h,j

By adding the h equations of the system (Sη,h), we obtain after some calculation the

following expression: Ph

j=1 ˙yη,h,j(θ) = −1 − (β/αη,h)yη,h,1(θ). Since yη,h,1 is non-negative

on [0, θη,h], the derivative of the functionPhj=1yη,h,j is non-positive on the same interval,

hence this function is monotonically decreasing on [0, θη,h], and consequently, ∀θ ∈ [0, θη,h] h X j=1 yη,h,j(θ) ≤ h X j=1 y_η,h,jI . (12) By taking θ = θη,h, we getPh−1j=1yη,h−1,jI ≤ Ph

j=1yη,h,jI which implies that ∀h ≤ η h X j=1 y_η,h,jI ≤ η X j=1 y_η,η,jI ≤ 1. (13) Also, from the above expression and expression (12), we deduce, ∀n ∈ IN∗, ∀h ∈ IN∗, h ≤ η, ∀j ∈ IN∗, j ≤ h, ∀θ ∈ [0, θη,h]

0 ≤ yη,h,j(θ) ≤ 1. (14)

On the other hand, using expressions (5) and (6), we get: yη,h,j(θ) =

h

X

p=1

Φj,p(θ)yη,h,pI + uη,h,p(θ).

We easily verify, using expression (7), that uη,h,p(θ) ≤ 0; thus using expression (4), we

obtain yη,h,j(θ) ≤ h X p=j p j ! αη,h− θ αη,h !jβ 1 − αη,h− θ αη,h !β  p−j yI η,h,p. (15)

(16)

In what follows, we will need inequality (16), whose demonstration is elementary:

xie(1−x)λ≤ 1, x ∈]0, 1], λ > 0, i ∈ IN , i ≥ λ. (16) We are going to establish, by downward induction, inequalities (17) and (18), for all integers η, h, j ∈ IN , λ ≤ j ≤ h ≤ η and ∀θ ∈ [0, θη,h].

yI_η,h,j ≤ e

−λ_λj

j! . (17)

In fact, we initialize the induction at h = η with expression (3). We suppose true the inequalities for a fixed h. Then, in the step h − 1, by putting x = [(αη,h− θη,h)/αη,h]β, we

use expression (15) to obtain, for θ = θη,h,

y_η,h−1,jI = yη,h,j(θη,h) ≤ h X p=j p j ! xj(1 − x)p−jyI_η,h,p; next, using the induction hypothesis, we get for yI

η,h−1,j: y_η,h−1,jI ≤ h X p=j p j ! xj(1 − x)p−je −λ_λp p! ≤ e−λ_xjλj j! h X p=j 1 (p − j)!(1 − x) p−j_λp−j ₌ e−λλj j! x j_e(1−x)λ

and, finally, with expression (16) we obtain (17). yη,h,j(θ) ≤

e−λ_λj

j! . (18)

In fact, by taking now x = [(αη,h− θ)/αη,h]β and by using (15), we get:

yη,h,j(θ) ≤ h X p=j p j ! xj(1 − x)p−jyI_η,h,p. Next, using expressions (16) and (17), we obtain

yη,h,j(θ) ≤ h X p=j p j ! xj(1 − x)p−je −λ_λp p! ≤ e−λ_λj j! x j_e(1−x)λ _≤ e−λλj j! q.e.d.

Let us now introduce the constant

Ξ = 1 min j∈IN,1≤j≤λ n e−λ_λj j! o. (19)

We note that Ξ does not depend on λ and also that Ξ > 1, since e−λ_λj_{/j! < e}−λ_eλ _{= 1.}

Using Ξ, we can prove the following expression. 0 ≤ yη,h,j(θ) ≤ Ξ

e−λ_λj

(17)

In fact, if j ≥ λ, then by using (18), (Ξ > 1) we get yη,h,j(θ) ≤ e−λλj/j! ≤ Ξe−λλj/j!. On

the other hand, if j < λ, then by expression (14) we get yη,h,j(θ) ≤ 1 ≤ Ξ

e−λ_λj

j! q.e.d.

3.4.6 An upper bound of the sequence (τη)_η∈_IN∗

Using expression (10) and for h ≥ 2, we get: θη,h= αη,h− h − 1 h αη,h−1 = αη,h− αη,h−1+ 1 hαη,h−1. Consequently, τη = η X h=1 θη,h= θη,1+ η X h=2 (αη,h− αη,h−1) + η X h=2 1 hαη,h−1 = θη,1+ αη,η− αη,1+ η−1 X h=1 1 h + 1αη,h. Thus, τη = (θη,1− αη,1) + αη,η+ η−1 X h=1 1 h + 1αη,h. We know that θη,1 ≤ αη,1 and, by using expression (3) we obtain

τη ≤ 1 ηk η X j=1 je −λ_λj j! + η−1 X h=1 1 h + 1 1 hk h X j=1 jy_η,h,jI ≤ e −λ η λ k η−1 X j=0 λj j! + η−1 X h=1 1 (h + 1)hk h X j=1 jΞe −λ_λj j! the last inequality obtained using (20); hence

τη ≤ e−λ η λ k η−1 X j=0 λj j! + Ξe−λ_λ k η−1 X h=1 1 h(h + 1) h−1 X j=0 λj j!. Note that η−1 X h=1 1 h(h + 1) h−1 X j=0 λj j! ≤ e λ η−1 X h=1 1 h(h + 1) = e λ η−1 X h=1 1 h − 1 h + 1 = eλ 1 − 1 η ! which leads to τη ≤ e−λ_λ k   1 η η−1 X j=0 λj j!  + Ξλ k 1 − 1 η ! . (21)

Note also that (1/η)Pη−1

j=0(λj/j!) ≤ max0≤j≤n−1{λj/j!}. By denoting by ⌊λ⌋ the floor of λ,

we easily verify that: max0≤j≤λ−1{λj/j!} = λ⌊λ⌋−1/(⌊λ⌋ − 1)! and maxλ−1≤j{λj/j!} =

λ⌊λ⌋_{/⌊λ⌋! and, since λ ≥ ⌊λ⌋, we have λ}⌊λ⌋_{/⌊λ⌋! ≥ λ}⌊λ⌋−1_{/(⌊λ⌋ − 1)!; consequently, we}

get max_j∈_{IN {λ}j_{/j!} = λ}⌊λ⌋_{/⌊λ⌋!. Whenever 0 < λ < 1, we have ⌊λ⌋ = 0, ⌊λ⌋! = 1 and}

λj_{/j! ≤ 1 = λ}0_{/0! = λ}⌊λ⌋_{/⌊λ⌋!, therefore, ∀λ > 0,} max j∈IN ( λj j! ) = λ ⌊λ⌋ ⌊λ⌋!. (22)

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On the other hand, 1 − (1/η) ≤ 1 and by (21) and (22) we deduce that τη ≤ e−λ_λ k λ⌊λ⌋ ⌊λ⌋! + Ξλ k = λ k " e−λ_λ⌊λ⌋ ⌊λ⌋! + Ξ # and consequently, sup η≥2τη ≤ λ k " e−λ_λ⌊λ⌋ ⌊λ⌋! + Ξ # < +∞. (23)

Since Ξ depends only on λ (see also (19)), the proposed upper bound only depends on λ and k.

3.4.7 Variations with respect to α

We consider for each h ∈ IN∗ and for each α > 0, the following differential system in IRh: (Ch,α) ˙y(θ) =

β

α − θMhy(θ) + eh. when θ ∈ [0, α[.

One can see that (Sη,h) = (Ch,αη,h).

We denote by ϕh(θ, α, zI) the value at the time θ of the solution of (Ch,α) that takes the

value zI _{at the time 0. Then, function T}

h(α, zhI) introduced above denotes the first time

the h-th coordinate (ϕh,h) of ϕh becomes 0. We can see that yη,h(θ) = ϕ(θ, αη,h, yη,hI ),

∀θ ∈ [0, αη,h[. We have already noted that θη,h= Th(αη,h, yη,h,hI ).

The homogeneous system associated with (Ch,α) is denoted (Lh,α) and the resolvant

of (Lh,α) is denoted by Φh(θ, α).

With computations similar to the ones performed for the systems (Sη,h), we establish the

following: Φh(θ, α) = exp {β ln [α/(θ − α)] Mh}, with

Φh,i,j(θ, α) =      0 if i > j j i α α−θ −βi 1 −_α−θα −β j−i if i ≤ j (24)

We have also ∀j = 1, . . . , h, ϕh,j(θ) ≥ 0 in [0, Th(α, zhI)] when zIh,i > 0.

The solution of (Ch,α) that is equal to zIat the time 0 is ϕ(θ, α, zI) = Φh(θ, α)zI+ψh(θ, α),

where ψh,j(θ, α) = − h j ! Z θ 0 α − θ α − t !βj 1 − α − θ α − t !β  h−j dt. (25)

Starting from expression (25), we have, ∀j ≤ h and ∀θ ∈ [0, Th(α, zhI)]

ψh,j(θ) ≤ 0. (26)

Moreover, with a similar reasoning as in the case of expression (18), if ∀j = 1, . . . , h, zI

j ≤ Ξe−λλj/j!, then ∀j = 1, . . . , h, ∀θ ∈ [0, Th(α, zhI)] we have

ϕh,j(θ, α, zI) ≤ Ξ

e−λ_λj

(19)

Also by arguments similar to the ones of § 3.4.4 we establish that function θ 7→ h X j=1 jϕh,j(θ, α, zI)

is solution of the scalar o.d.e. ˙x(θ) = −h − [β/(α − θ)]x(θ). We then deduce that

h X j=1 jϕh,j(θ, α, zI) =   h X j=1 jz_jI   1 − θ α !β + hα −β + 1   1 − θ α ! − 1 − θ α !β .

Supposing also that Ph

j=1jzjI ≤ hkα, after some easy algebra we obtain: ∀h ∈ IN∗,

∀α > 0, ∀zI _{∈ IR}+h _{such that} Ph j=1jzIj ≤ hkα h X j=1 jϕh,j(θ, α, zI) ≤ hk(α − θ). (28)

We now compute a lower bound for Rη,η and αη,η. In fact, we prove that ∀λ > 0,

∃η(λ) ∈ IN∗ such that ∀η ∈ IN∗, η ≥ η(λ) Rη,η ≥ λ 2 αη,η ≥ λ 2kη. (29)

Since limη→∞Pη−1i=0(λi/i!) = eλ > eλ/2 and

Pη−1

i=0(λi/i!)

η is a monotonic increasing

sequence, then ∃n(λ) ∈ IN∗ such that ∀η ≥ η(λ), Pη−1

i=0(λi/i!) ≥ eλ/2. Consequently,

Rη,η = e−λλ η−1 X i=0 λi i! ≥ e −λ_λeλ 2 = λ 2 and αη,η = Rη,η ηk ≥ λ 2kη q.e.d.

Let us now prove the following: ∀η ∈ IN∗, ∀h ∈ IN∗, h ≤ η

y_η,h,hI ≥ θη,h. (30)

Recall that

0 ≤ ∂Th

∂y (α, y) ≤ 1 (31)

and also that ∀α > 0, Th(α, 0) = 0. Using the mean value theorem, we have

(20)

This concludes the proof of expression (30).

Moreover, by some easy calculations, we can prove the following ∀p ∈ IN∗, p ≥ 2λ, ∀h, η ∈ IN∗ _{such that η − 1 ≥ h ≥ 2λ} λp p! ≤ λp−1 (p − 1)! − λp p! η−1 X p=h λp p! ≤ λh (h − 1)! − λη−1 (η − 1)!. (32)

We now prove that ∀η ∈ IN∗, ∀h ∈ IN∗ such that h < η we have

hαη,h≥ (h + 1)αη,h+1− (h + 1)yη,h+1,h+1I . (33)

In fact, using expression (10), we have Rη,h= Rη,h+1− (h + 1)kθη,h+1 or

Rη,h

k =

Rη,h+1

k − (h + 1)θη,h+1. Recall also that αη,h = Rη,h/(hk). So, we get

hαη,h= (h + 1)αη,h+1− (h + 1)θη,h+1.

Moreover, using expression (30): −θη,h+1 ≥ −yη,h+1,h+1I . Combining the last two

expres-sions, we obtain expression (33).

Let us now prove that ∀η ∈ IN∗, η ≥ 2λ and ∀h ∈ IN∗, 2λ ≤ h ≤ η we have αη,h≥

η hαη,η−

e−λ_λh

h! . (34)

Using expression (33), ∀l ∈ IN∗ such that η > l ≥ h ≥ 2λ, we have lαη,l− (l + 1)αη,l+1 ≥ −(l + 1)yIη,l+1,l+1 ≥ −(l + 1) e−λ_λl+1 (l + 1)! (17) ≥ e −λ_λl+1 l! = e −λ_λλl l!. Summing these inequalities, we obtain (using expression (32))

η−1 X l=h lαη,l− η−1 X l=h (l + 1)αη,l+1≥ −e−λλ η−1 X l=h λl l!

Using also expression (32), we get hαη,h− ηαη,η ≥ −e−λλh/(h − 1)!, and expression (34)

follows.

Also, ∀η ∈ IN∗, η > 2λ, η > η(λ) (expression (29)) and ∀h ∈ IN∗ such that 2λ ≤ h ≤ η, we have αη,h≥ λ 2hk − e−λ_λh h! . (35)

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In fact, using expressions (34) and (29), αη,h ≥ (η/h)αη,η− e−λλh/h! ≥ (η/h)(λ/2ηk) −

e−λ_λh_{/h! and expression (35) follows.}

We finally prove that ∃h∗ ∈ IN∗ such that ∀η ∈ IN∗, η > 2λ, η ≥ η(λ) (expression (29)),

∀η ∈ IN∗, η > h ≥ h∗ , h ≥ 2λ, we have

αη,h≥

λ

4hk. (36)

Since λh−1_{/(h − 1)!}

h is the general term of a convergent series, λ

h−1_{/(h − 1)! →}

h→∞ 0.

Thus, λh_{/(h − 1)! = λλ}h−1_{/(h − 1)! →}

h→∞ λ.0 = 0 or hλh/h! = λh/(h − 1)! →h→∞ 0;

consequently, ∃h∗ ∈ IN∗ such that ∀h ≥ h∗, hλh/h! ≤ λ/(4ke−λ) or e−λλh/h! ≤ λ/(4hk).

Thus, using expression (35), we deduce expression (36).

Let us now introduce, ∀h ∈ IN∗, the norm k · kh in IRh, defined as kxkh =Phi=1(i/h)|xi|.

Then, ∃c1 > 0 such that ∀h ∈ IN∗, h > 2λ, ∀η ∈ IN∗, η ≥ h, ∀α ≥ min{αη,h, αη+1,h},

∀zI _{∈ IR}+h _{such that ∀j = 1, . . . , h, z}I

j ≤ Ξe−λλj/j! and zhI ≤ kα we have

∂ ∂θϕh(Th(α, z I h), α, zI) _h ≤ c1. (37)

Let us fix η and h such that η ≥ h > 2λ. Let also θ = Th(α, zhI).

Then, ∀j = 1, . . . , h − 1, we have j h ∂ ∂θϕh,j(θ, α, z I₎ ≤ j(j + 1) h β α − θ(ϕh,j+1(θ, α, z I_{) + ϕ} h,j(θ, α, zI)). Since zI

h ≤ kα (expression (11), we have θ ≤ (1/2)α, hence β/(α − θ) ≤ 2β/α.

Let us suppose that α ≥ αη,h, the reasoning being similar in the case α ≥ αη+1,h. So,

β/(α − θ) ≤ 2β/αη,h and since h > 2λ, using expression (36), we get β/(α − θ) ≤

8βhk/λ = (8(k − 1)/λ)h. Thus, using the hypotheses of expression (37) and expres-sion (27), we have j h ∂ ∂θϕh,j(θ, α, z I₎ ≤ j(j + 1) h 8(k − 1) λ h(ϕh,j+1(θ, α, z I_{) + ϕ} h,j(θ, α, zI)) ≤ 8(k − 1) λ j(j + 1) Ξ e−λ_λj+1 (j + 1)! + Ξ e−λ_λj j! ! . Hence, h X j=1 j h ∂ ∂θϕh,j(θ, α, z I₎ ≤ 8(k − 1) λ Ξe −λh−1X j=1 j(j + 1) λ j j! + λj+1 (j + 1)! ! ≤ 8(k − 1) λ Ξe −λ_2eλ_(λ2_{+ λ) ≤ 16(k − 1)Ξ(λ + 1).}

On the other hand, h h| ∂ ∂θϕh,h(θ, α, z I_{)| ≤ 1 +} β α − θϕh,h(θ, α, z I_{) ≤ 1 +}8(k − 1) λ hkΞ e−λ_λh h! ≤ 1 + 8(k − 1)Ξ

(22)

and, finally, h X j=1 j h ∂ ∂θϕh,j(θ, α, z I₎ ≤ 8(k − 1)Ξ[2(λ + 1) + 1] + 1 = c1 q.e.d.

We now prove that ∃c2 > 0 such that ∀h ∈ IN∗, h ≤ 2λ, ∀η ∈ IN∗, η ≥ h, ∀α > 0,

∀zI _{∈ IR}+h _{such that} Ph j=1jzjI ≤ hkα, we have ∂ ∂θϕh(Th(α, z I h), α, zI) h ≤ c2. (38)

Let us fix η, h, α and put θ = Th(α, zhI). If j ≤ h − 1, then

∂ ∂θϕh,j(θ, α, z I₎ = β α − θ h (j + 1)ϕh,j+1(θ, α, zI) − jϕh,j(θ, α, zI) i ≤ β α − θ[(j + 1)ϕh,j+1(θ, α, z I_{) + jϕ} h,j(θ, α, zI)] ≤ β α − θ h X i=1 iϕh,i(θ, α, zI) and using (28), β α − θ h X i=1 iϕh,i(θ, α, zI) ≤ β α − θhk(α − θ) = βhk = h(k − 1) ≤ 2λ(k − 1) ∂ ∂θϕh,h(θ, α, z I₎ ≤ 1 + β α − θϕh,h(θ, α, z I₎ ≤ 1 + β α − θhk(α − θ) = 1 + h(k − 1) ≤ 2λ(k − 1) + 1. So, ∂ ∂θϕh(θ, α, z I₎ h = h X i=1 i h ∂ ∂θϕh,i(θ, α, z I₎ ≤ h X i=1 h h ∂ ∂θϕh,i(θ, α, z I₎ = h−1 X i=1 ∂ ∂θϕh,i(θ, α, z I₎ + ∂ ∂θϕh,h(θ, α, z I₎ ≤ h−1 X i=1 2λ(k − 1) + 2λ(k − 1) + 1 ≤ (2λ)2(k − 1) + 2λ(k − 1) + 1 = c2 q.e.d.

Using (37) and (38), putting c3 = max{c1, c2}, noting that hzhI ≤

Ph

i=1jzjI ≤ hkα, or

zI

h ≤ kα, we deduce that ∃c3, ∀h ∈ IN∗, ∀η ∈ IN∗, η ≥ h, ∀α ≥ min{αη,h, αη+1,h},

∀zI _{∈ IR}+h _{such that ∀j = 1, . . . , h, z}I jΞe−λλj/j! and Ph i=1jzjI ≤ hkα we have ∂ ∂θϕh(Th(α, z I h), zI) _h ≤ c3. (39)

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We now prove that ∀v ∈ IRh, ∀α > 0, ∀θ ∈ [0, Th(α, v)] kΦh(θ, α)vkh ≤ kvkh. (40) We have kΦh(θ, α)vkh = h X i=1 i h h X j=1 Φh,i,j(θ, α)vj ≤ h X i=1 i h h X j=1 |Φh,i,j(θ, α)| |vj| .

By expression (24), putting x = [(α − θ)/α]β (0 ≤ x ≤ 1 or 1 − x ≥ 0), we have Φh,i,j(θ, α) ≥ 0. So, kΦh(θ, α)vkh ≤ h X i=1 i h h X j=1 Φh,i,j(θ, α) |vj| = 1 h h X j=1 h X i=1 i j! i!(j − i)!x i_{(1 − x)}j−i_|v j| = i h h X j=1 jx   j−1 X l=0 j − 1 l ! xl_{(1 − x)}j−1−l  |v_j| = 1 h h X j=1 jx(x + 1 − x)j−1|vj| = xkvkh q.e.d.

Next, we prove that ∀h ∈ IN∗, ∀α > 0, ∀θ < α

1 αψh(θ, α) _h ≤ 1. (41)

Let us put x(t) = [(α − θ)/(α − t)]β (x(t) ≤ 1). Then

1 αψh(θ, α) h = 1 α h X i=1 i h|ψh,i(θ, α)| = 1 α h X i=1 i h θ Z 0 x(t)i(1 − x(t))h−idt = 1 α θ Z 0 " _h X i=1 h − 1 i − 1 ! x(t)i−1(1 − x(t))h−1−(i−1)x(t) # dt = 1 α θ Z 0 (x(t) + 1 − x(t))h−1_{x(t)dt =} 1 α θ Z 0 x(t)dt = 1 α θ Z 0 α − θ α − t !β dt ≤ θ α ≤ 1 q.e.d. Also, ∀h ∈ IN∗, ∀α > 0, ∀zI _{∈ IR}h_{, ∀θ ∈ [0, T} h(α, zhI)], we have ∂ ∂αϕh(θ, α, z I_{) =} −θ α ∂ ∂θϕh(θ, α, z I_{) +} 1 αψh(θ, α). (42)

We fix h, α, zI_{. We have already seen that θ ∈ [0, T}

h(α, zhI)] and also that

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Let us put χh(θ, α, zI) = Φh(θ, α)zI, so,

ϕh(θ, α, zI) = χh(θ, α, zI) + ψh(θ, α).

Let us introduce the function w : [0, 1[7→ IRh, w(s) = exp {β ln [1/(1 − s)] Mh} zI; then,

χh(θ, α, zI) = w (θ/α).

We also introduce the function v : [0, 1[7→ IRh, v(s) = (v1(s), . . . , vh(s))T, where

vj(s) = − h j ! s Z 0 1 − s 1 − r βj" 1 − 1 − s 1 − r β#h−j dr.

Performing a variable changing t = αr, we obtain ψh(θ, α) = αv (θ/α).

From the above, we deduce that ϕh(θ, α, zI) = w (θ/α) + αv (θ/α).

Therefore, ∂ ∂θϕh(θ, α, z I_{) = ˙}_w θ α ! 1 α + α ˙v θ α ! 1 α = ˙w θ α ! 1 α + ˙v θ α ! . On the other hand,

∂ ∂αϕh(θ, α, z I_{) = ˙}_w θ α ! −θ α2 ! + v θ α ! + ˙v θ α ! −θ α2 !

and after some easy algebra, expression (42) is proved.

We now prove the following: ∃c4 > 0 such that ∀h ∈ IN∗, ∀η ∈ IN∗, η ≥ h, we have

kϕh+1(θη+1,h+1, αη+1,h+1, yIη+1,h+1) − ϕh+1(θη,h+1, αη,h+1, yIη+1,h+1)kh+1

≤ c4|αη+1,h+1− αη,h+1|. (43)

To shorten the writing, we put zI _{= y}I

η+1,h+1. So, θη+1,h+1 = Th+1(αη+1,h+1, zh+1I ) and

θη,h+1 = Th+1(αη,h+1, zh+1I ). Let us also denote by [x1; x2] a convex segment (notation

which does not signify that x1 ≤ x2).

Let us introduce, for α ∈ [αη,h+1; αη+1,h+1], the function g(α) = ϕh+1(Th+1(α, zh+1I ), α, zI).

Thus

= kg(αη+1,h+1) − g(αη,h+1)kh+1.

Using the mean value theorem, we get

kg(αη+1,h+1) − g(αη,h+1)kh+1≤ sup α∈[αη,h+1;αη+1,h+1]

k ˙g(α)kh+1

!

|αη+1,h+1− αη,h+1|.

On the other hand ˙g(α) = ∂ϕh+1 ∂θ (Th+1(α, z I h+1), α, zI) ∂Th+1 ∂α (α, z I h+1) + ∂ϕh+1 ∂α (Th+1(α, z I h+1), α, zI) (42) = ∂Th+1 ∂α (α, z I h+1) − θ α ! ∂ϕh+1 ∂θ (Th+1(α, z I h+1), α, zI) + 1 αψh+1(Th+1(α, z I h+1), α).

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Consequently k ˙g(α)kh+1 ≤ ∂Th+1 ∂α (α, z I h+1) + θ α ! ∂ϕh+1 ∂θ (Th+1(α, z I h+1), α, zI) h+1 + 1 αψh+1(Th+1(α, z I h+1), α) h+1 .

Let us remark here that one can prove by elementary techniques that

∂Th(α, y) ∂α ≤ 4. (44)

So, using this inequality as well as θ < α and (41), we get k ˙g(α)kh+1≤ 5 ∂ϕh+1 ∂θ (Th+1(α, z I h+1), α, zI) _h+1 + 1.

We can also remark that the hypotheses made on α and zI _{for the proof of (39) remain}

valid here also; therefore, ∀α ∈ [αη,h+1; αη+1,h+1], we have k ˙g(α)kh+1≤ 5c3+ 1, q.e.d.

the last two inequalities holding because yI

η+1,h, yη,hI ∈ IRh and because k · kh is a norm

of IRh respectively, q.e.d.

We now prove that ∀η, h ∈ IN∗, h < η, we have

≤ kyI_η+1,h+1− y_η,h+1I kh+1. (46)

Plainly

≤ kΦh+1(θη,h+1, αη,h+1)(yIη+1,h+1− yη,h+1I )kh+1.

Using this inequality and expression (40), we obtain (46).

Let us continue by showing the following: ∃c5 > 0 such that ∀η, h ∈ IN∗, h < η, we have

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Let us start the proof of expression (47) by noting that, if x = (x1, . . . , xh, 0)T is a vector

of IRh+1 and if we denote by x′ _{the sub-vector (x}

1, . . . , xh)T of x, then kx′kh = 1 + 1 h kxkh+1≤ 2kxkh+1. kyI η+1,h− yη,hI kh = kyη+1,h+1(θη+1,h+1) − yη,h+1(θη,h+1)kh ≤ 2kϕh+1(θη+1,h+1, αη+1,h+1, yη+1,h+1I ) − ϕh+1(θη,h+1, αη,h+1, yη,h+1I )kh+1≤ 2kϕh+1(θη+1,h+1, αη+1,h+1, yη+1,h+1I ) − ϕh+1(θη,h+1, αη,h+1, yη+1,h+1I )kh+1 +2kϕh+1(θη,h+1, αη,h+1, yη+1,h+1I ) − ϕh+1(θη,h+1, αη,h+1, yη,h+1I )kh+1 (26,45,46) ≤ 2c 4 k + 1 kyI η+1,h+1− yη,h+1I kh+1 = c5kyη+1,h+1I − yη,h+1I kh+1 q.e.d.

Also, ∃c6 > 0 such that ∀η ∈ IN∗, η > 2λ, we have

sup θ∈[0,θη+1,η+1] k ˙yη+1,η+1(θ)kη+1≤ c6. (48) Let j = 1, . . . , η; then ˙yη+1,η+1,j(θ) = (j + 1) β αη+1,η+1− θ yη+1,η+1,j+1(θ) −j β αη+1,η+1− θ yη+1,η+1,j(θ) or j η + 1| ˙yη+1,η+1,j(θ)| ≤ j(j + 1) η + 1 β αη+1,η+1− θ (yη+1,η+1,j(θ) + (yη+1,η+1,j+1(θ)). Also, β αη+1,η+1− θ ≤ β αη+1,η+1− θη+1,η+1 ≤ 2β αη+1,η+1 ≤ 8β(n + 1)k λ = 8(n + 1)(k − 1) λ .

Therefore, using (20), we get j η + 1| ˙yη+1,η+1,j(θ)| ≤ 8(k − 1) λ Ξe −λ_{j(j + 1)} " λj j! + λj+1 (j + 1)! # or, η X j=1 j η + 1| ˙yη+1,η+1,j(θ)| ≤ 8(k − 1) λ Ξe −λ_2eλ_(λ2_{+ λ) = 16(k − 1)Ξ(λ + 1).}

On the other hand, ˙yη+1,η+1,η+1(θ) ≤ 1 + β αη+1,η+1− θ yη+1,η+1,η+1(θ) ≤ 1 + 8(k − 1)Ξ e−λ_λη η! ≤ 1 + 8(k − 1)Ξ.

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Thus finally, k ˙yη+1,η+1(θ)kη+1 ≤ 8(k − 1)Ξ(4(λ + 1) + 1) + 1 = c6, q.e.d.

Let us now prove that ∃c7 > 0 such that ∀η ∈ IN∗, η ≤ 2λ − 1, we have

sup θ∈[0,θη+1,η+1] k ˙yη+1,η+1(θ)kη+1≤ c7. (49) Let j = 1, . . . , η; then ˙yη+1,η+1,i(θ) ≤ β αη+1,η+1− θ

((i + 1)yη+1,η+1,i+1(θ) + i

β αη+1,η+1− θ yη+1,η+1,i(θ)) ≤ β αη+1,η+1− θ h X j=1 jyη+1,η+1,j (13) ≤ β αη+1,η+1− θ (h + 1)k(αη+1,η+1− θ) ≤ β(n + 1)k < 2λ(k − 1). On the other hand

˙yη+1,η+1,η+1(θ) ≤ 1 + β αη+1,η+1− θ yη+1,η+1,η+1(θ) ≤ 1 + β αη+1,η+1− θ η+1 X j=1 jyη+1,η+1,j(θ) (13) = 1 + β αη+1,η+1− θ (n + 1)k(αη+1,η+1− θ) = 1 + β(n + 1)k ≤ 1 + (k − 1)2λ.

Hence, k ˙yη+1,η+1(θ)kη+1 ≤ [Pηi=1(i/η + 1)][(k − 1)2λ]+ [1 + (k − 1)2λ] ≤ 4λ(k − 1) + 1 = c7,

q.e.d.

Using (48) and (49) and putting c8 = max{c6, c7}, we obtain the following, ∀η ∈ IN∗

sup

θ∈[0,θη+1,η+1]

k ˙yη+1,η+1(θ)kη+1≤ c8. (50)

We now prove that ∃c9 > 0 such that ∀η ∈ IN∗, we get

kyI

η+1,η− yη,ηI kη ≤ c9

e−λ_λη+1

(η + 1)!. (51)

Using the remark in the beginning of the proof of (47), we have kyI η+1,η− yIη,ηkη = kyη+1,η+1(θη+1,η+1) − (yη,ηI , 0)kη ≤ 2kyη+1,η+1(θη+1,η+1) − (yIη,η, 0)kη+1 = 2 η X i=1 i η + 1 yη+1,η+1,i(θη+1,η+1) − e−λ_λi i! ≤ 2 η+1 X i=1 i η + 1 yη+1,η+1,i(θη+1,η+1) − e−λ_λi i!

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= 2 η+1 X i=1 i η + 1|yη+1,η+1,i(θη+1,η+1) − yη+1,η+1,i(0)| = 2 η+1 X i=1 kyη+1,η+1(θη+1,η+1) − yη+1,η+1(0)kη+1 = 2 θη+1,η+1 Z 0 ˙yη+1,η+1(θ)dθ η+1 ≤ 2 θη+1,η+1 Z 0 k ˙yη+1,η+1(θ)kη+1dθ (50) ≤ 2c8 θη+1,η+1 Z 0 dθ = 2c8θη+1,η+1 ≤ 2c8 e−λ_λη+1 (η + 1)! (30) = c9 e−λ_λη+1 (η + 1)! q.e.d.

So, using (45), we obtain expression (52).

3.4.8 The convergence of the sequence (τη)_η∈_IN∗

We prove first that the series of general term (Pη

h=1|θη+1,h− θη,h|)_η≥1is convergent in IR+.

In fact, this series is a positive terms one, hence in order to prove its convergence we shall use the comparison principle.

By combining expressions (52), (51) and (47), we can obtain

η X h=1 |θη+1,h− θη,h| ≤ η X h=1 cη−h₅ c9 e−λ_λη+1 (η + 1)! = 4 1 + 1 k c9cη5 e−λ_λη+1 (η + 1)! η X h=1 1 c5 h . Since c5 ≥ 2 > 1, we get η X h=1 |θη+1,h− θη,h| ≤ 4 1 + 1 k c 9 c5 e−λ_(λc 5)η+1 (η + 1)! 1 c5− 1 .

Putting c10 = 4 [1 + (1/k)] (c9/c5)[1/(c5− 1)] > 0, the quantity Pηh=1|θη+1,h − θη,h| is

bounded above by c10e−λ(λc5)η+1/(η + 1)!; hence, ∞ X η=1 η X h=1 |θη+1,h− θη,h| ! ≤ c10 e−λ_(λc 5)η+1 (η + 1)! ≤ c10e −λ_eλc5 _{< +∞}

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q.e.d.

Let us now prove that the series of general term (θη+1,η+1)η≥1 converges in IR+.

Plainly, using (30), we get θη+1,η+1≤ yIη+1,η+1,η+1 = e−λ(λ)η+1/(η + 1)!.

Since it is a positive term series, it suffices to apply the comparison principle, by noting that the series of general term (e−λ_λη+1_{/(η + 1)!)}

η≥1 converges in IR+, q.e.d.

The series of general term (|τη+1− τη|)η≥1 converges in IR+.

In fact, |τη+1− τη| = η+1 X h=1 θη+1,h− η X h=1 θη,h = θη+1,η+1+ η X h=1 (θη+1,h− θη,h) ≤ θη+1,η+1+ η X h=1 |θη+1,h− θη,h|.

The convergence follows from the comparison principle and the propositions proved above. We conclude the section by proving that the sequence (τη)_η∈_IN∗ converges in IR+.

Using the proposition proved above, the series of general term (τη+1− τη)η≥1 is absolutely

convergent in IR, thus convergent in IR.

Let us remark that τη =Pη−1q=1(τq+1− τq) + τ1. Moreover,

lim η→∞τη = ∞ X q=1 (τq+1− τq) + τ1.

These two remarks conclude the convergence of the sequence (τη)_η∈_IN∗ in IR+.

4 Set packing and hypergraph independent set

4.1 The PACK algorithm

We consider the greedy algorithm 3 for finding a set packing for P. Informally, PACK algorithm selects at each step one set chosen at random between the sets of smallest cardinality which do not intersect any previously selected set. By exchanging the two color classes of the incidence graph, this algorithm gives an independent set in the hypergraph P, Q, i.e. a set of vertices no two of which belong to the same hyperedge.

4.2 Analysis of PACK on random instances 4.2.1 Analysis of PACK on standard distribution

The analysis of PACK on random instances can be carried over along the same lines than for GRECO. However, this leads prima facie to a more complicated system of differential equations because, in the case of PACK, we have three succesive removals at each step (see the description of PACK).

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[Step1] choose at random one of the Xi’s with minimum di;

add Xi to the solution, delete it and delete also all the classes in Q, say Yi1, . . . , Yih,

incident to it in the graph Π;

next delete any other vertex in X adjacent to one of the classes Yi1, . . . , Yih;

let X′_{, Y}′_{, P}′_{, Q}′ _{denote the analogues of X, Y, P, Q for the new configuration;}

let Π′ denote the subgraph of Π induced by X′∪ Y′_;

[Step2] if X′ _{is empty, then stop, else, execute Step1 with X}′_{, Y}′_{, P}′_{, Q}′_{, Π}′ _{in place of}

X, Y, P, Q, Π.

Algorithm 3. PACK algorithm.

of PACK, the conditional distribution of the remaining sets X′

1, . . . , Xh′ (relatively to the

previous unraveling of the algorithm) coincides with the uniform distribution; we can then compute the expectation of the number of these sets which will be tried before finding one which does not intersect the partial packing as a function t(j) of the size j of the partial packing which satisfies t(j) ∼ [1 − (jk/n)]−k_.

Thus, the expectation of the number T (s) of trials before a packing of size s is obtained satisfies ET (s) = s X j=1 t(j) ∼ n k(k − 1). " 1 (1 −sk_n)k−1 − 1 # . (53)

Using the fact that the trials are independent, Tchebichev’s inequality implies T (s) ∼ ET (s) and solving (53) for s, one gets the following estimate for the size of the obtained packing: s = n k  1 − 1 [1 + αk(k − 1)]k−11  .

4.2.2 The differential system simulating PACK for solving the hypergraph independent set problem

When analysing PACK on the hypergraph independent set problem, we interchange the color classes of the incidence graph and, since it is not true that for any given value h the joint distribution of the sets Yi with |Yi| = h is uniform, we must abandon the

preceding stopping times method and come back to Markov chains. The expectations of the increments during the stage when vertices of degree h enter the independent set are

E∆mj = ((j + 1)mj+1− jmj)S, j 6= h,

E∆mh = −1 + ((h + 1)mh+1− hmh))Sd,

where S = h(k − 1)/(P

imi).

With the same notation as in § 3.3, we get the differential system ˙yn,h,j = (j + 1)yn,h,j+1 h(k − 1) Rn,h− hkθ − jyn,h,j h(k − 1) Rn,h− hkθ , 0 ≤ j ≤ h − 1,

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˙yn,h,h = −1 + (h + 1)yn,h,h+1 h(k − 1) Rn,h− hkθ − hyn,h,h h(k − 1) Rn,h− hkθ .

A similar method, as for system (Snh) in the case of hitting set, could be developed to

solve the above system.

5 Conclusions

We have proposed a general method to analyse the behaviour of approximation algorithms for hard optimization problems defined on random set systems. We note that this method is quite general and its application could be extended even to graph problems. In fact, a graph G = (V, E) can be seen as a hypergraph whose vertex set is E, each vertex of V being a hyperedge containing its incident edges. Hence, problems like minimum dominating set, minimum vertex covering, maximum independent set, etc., could be solved by algorithms, the behaviour of which could be analyzed using the framework described.

We remark here that, when working on average case approximation, a well-known difficult mathematical problem is the precise estimation of the expectation of the size of the optimal solution. Such an estimation would enrich our method, by permitting to obtain close estimations for the expected values of the approximation ratios for the algorithms under study.

References

[1] N. Bourbaki, Fonctions d’une Variable R´eelle (Hermann, Paris, 1976).

[2] W. Fernandez de la Vega, Sur la cardinalit´e maximum des couplages d’hypergraphes al´eatoires uniformes, Discrete Mathematics 40, (1982) 315-318.

[3] W. Fernandez de la Vega, Induced trees in sparse random graphs, Graphs and Com-binatorics 2, (1986) 227-231.

[4] W. Fernandez de la Vega, V. Th. Paschos, R. Saad, Average case analysis of a greedy algorithm for the minimum hitting set problem, in: Proc. LATIN’92, Lecture Notes in Computer Science, Vol. 583 (Springer, Berlin, 1992) 130-138.

[5] A. Frieze, A. J. Radcliffe, S. Suen, Analysis of a simple greedy matching algorithm on random cubic graphs, in: Proc. 4th Ann. Symposium on Discrete Algorithms (ACM Press, 1993) 341-351.

[6] M. R. Garey and D. S. Johnson, Computers and Intractability - A Guide to the Theory of NP-Completeness (Freeman, New York, 1979).

[7] R. M. Karp, The probabilistic analysis of some combinatorial search algorithms, in: J. F. Traub, ed., Algorithms and Complexity: New Directions and Recent Results (Academic Press, New York, 1976) 1-19.

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[9] T. G. Kurtz, Solutions of ordinary differential equations as limits of pure jump Markov processes, J. Appl. Prob. 7 (1970) 49-58.

[10] C. Lund and M. Yannakakis, On the hardness of approximating minimization prob-lems, in: Proc. STOC’93 (ACM Press, 1993) 286-293.

Average case analysis of greedy algorithms for optimisation problems on set systems

CAHIER DU LAMSADE

125

Average case analysis of greedy algorithms for

optimisation problems on set systems

Contents

Analyse en moyenne de la performance des algorithmes gloutons

pour des probl`

emes d’optimisation sur des syst`

emes d’ensembles

Average case analysis of greedy algorithms for optimisation

problems on set systems

1

Introduction

2

Random set systems as random bipartite graphs

3

Hitting Set and Set Cover

4

Set packing and hypergraph independent set

5

Conclusions

References