Laboratoire d'Analyse et Modélisation de Systèmes pour
l'Aide à la Décision
CNRS UMR 7024
CAHIER DU LAMSADE
236
Avril 2006
What about future?
Robustness under vertex-uncertainty in graph-problems
Cécile Murat, Vangelis Th. Paschos
What about future?
Robustness under vertex-uncertainty in graph-problems
Cécile Murat
Vangelis Th. Paschos
LAMSADE, CNRS UMR 7024 and Université Paris-Dauphine
Place du Maréchal De Lattre de Tassigny, 75775 Paris Cedex 16, France
{murat,paschos}@lamsade.dauphine.fr
April 12, 2006
Abstract
We study a robustness model for graph-problems under vertex-uncertainty. We assume
that any vertex vi of the input-graph G(V, E) has only a probability pi to be present in the
final graph to be optimized (i.e., the final instance for the problem tackled will be only a
sub-graph of the initial graph). Under this model, the original “deterministic” problem gives
rise to a new (deterministic) problem on the same input-graph G, having the same set of
feasible solutions as the former one, but its objective function can be very different from
the original one, the set of its optimal solutions too. Moreover, this objective function is a
sum of 2|V | terms; hence, its computation is not immediately polynomial. We give sufficient
conditions for large classes of graph-problems under which objective functions of the robust
counterparts are polynomially computable and optimal solutions are well-characterized. Finally, we apply these general results to natural and well-known representatives of any of the
classes considered.
1
Introduction
Very often people has to make decisions under several degrees of uncertainty, i.e., when only
probabilistic information about the future is available. We deal with the following robustness
model under data uncertainty. Consider a generic instance I of a combinatorial optimization
problem Π. Assume that Π is not to be necessarily solved on the whole I, but rather on a
(unknown a priori) sub-instance I ′ ⊂ I. Suppose that any datum di in the data-set describing I
has a probability pi , indicating how di is likely to be present in the final sub-instance I ′ . Consider
finally that once I ′ is specified, the solver has no opportunity to solve it directly (for example
she/he has to react quasi-immediately, so no sufficient time is given her/him). Concrete examples
of such situations dealing with satellite shots planning or with timetabling are given in [24, 25].
What can the solver do in this case? A natural way to proceed is to compute an anticipatory
solution S for Π, i.e., a solution for the entire instance I, and once I ′ becomes known, to modify S
in order to get a solution S ′ fitting I ′ . The objective is to determine an initial solution S for I such
that, for any sub-instance I ′ ⊆ I presented for optimization, the solution S ′ respects some predefined quality criterion (for example, optimal for I ′ , or achieving, say, constant approximation
ratio, or . . . ).
In what follows we restrict ourselves in graph-problems and consider the following very simple
and quick modification1 of S: take the restriction of S in the present sub-instance of G. Consider
1
As we will see later such a modification strategy does not always produce feasible solutions; in such a case
some more work is needed.
1
a graph-problem Π, a graph G(V, E) of order n, instance of Π, and an n-vector Pr = (p 1 , . . . , pn )
of vertex-probabilities any of them, say pi , measuring how likely is for vertex vi ∈ V , i = 1, . . . , n
to be present in the final subgraph G′ ⊆ G, on which the problem will be really solved. For any
solution S for Π in G and for any V ′ ⊆ V , denote by S ′ the restriction of S in V ′ , i.e., the set
resulting from S after removal of the vertices that do not belong to V ′ . As we have mentioned, S ′ ,
can or cannot (depending of the definition of Π) be a feasible solution of Π in the subgraph G ′
of G induced by V ′ . Whenever S ′ is feasible, denote by m(G′ , S ′ ) the objective value of S ′
in G′ . Then, the value of S for G, denoted by E(G, S) (and frequently called functional ), is the
expectation of m(G′ , S ′ ), over all the possible G′ ⊆ G. Formally, given S, the functional E(G, S)
of S is defined by:
X
E(G, S) =
Pr V ′ m G′ , S ′
(1)
V ′ ⊆V
where Pr[V ′ ] is the probability
that V ′ will be finally the real instance to be optimized and is
Q
Q
′
defined by: Pr[V ] = vi ∈V ′ pi vi ∈V \V ′ (1 − pi ). Obviously, E(G, S) depends also on Pr but,
for simplicity, this dependency will be omitted. Quantity, E(G, S) can be seen as the objective
function of a new combinatorial problem, derived from Π and denoted by robust Π in what
follows, where we are given an instance G of Π and a probability vector Pr on the vertices of G
and the objective is to determine a solution S ∗ in G optimizing E(G, S) (optimal anticipatory
solution). The optimization goal of robust Π is the same as the one of Π.
This way to tackle robustness in combinatorial optimization is sometimes called a priori
optimization (this term has been introduced by [7]) and can also be seen as a particular case of
stochastic optimization ([9, 26]). In a priori optimization, the combinatorial problem to be solved,
being subject to hazards or to inaccuracies, is not defined on a static and clearly specified instance,
since the instance to be effectively optimized is not known with absolute certainty from the
beginning. Robustness requirements are due to the fact that uncertainty in the presence of data
makes that it is not possible to assign unique values to some parameters of the problems. In such
an approach, the goal is to compute solutions that behave “well” under any assignment of values
to these parameters. Under this model, restrictive versions of routing and network-design robust
minimization problems (in complete graphs) have been studied in [2, 4, 5, 6, 7, 13, 14, 15, 16].
Recently, in [8], the analysis of the robust minimum travelling salesman problem, originally
performed in [4, 13], has been revisited and refined. In [22, 24] the minimum vertex covering
and the minimum coloring are tackled, while in [20, 21] robust maximization problems, namely,
the longest path and the maximum independent set, are studied. An early survey about a priori
optimization can be found in [3] while, a more recent one appears in [23].
Another way to apprehend robustness under uncertainty is as it is done in [1, 10, 17, 18, 19]. In
this framework, one either identifies different feasible scenarii, or associates an interval of values
rather than one unique value with a parameter. Finally, a third very interesting face of robustness
that model inaccuracy by assigning probabilities to scenarii or to data is apprehended by what
has been called two- and multi-stage stochastic optimization (see, for example, [11, 12, 27] for
recent results on this area).
Our goal is to go beyond study of robust versions of particular combinatorial problems and to
propose a structural study of robustness under uncertainty for the a priori optimization paradigm.
Here, the main mathematical issues (assuming that, given an anticipatory solution S, its restriction to G[V ′ ] is feasible) are:
• the complexity of the computation of E(G, S) which, carrying over 2 n additive terms, is
non-trivially polynomial;
• compact combinatorial characterization (based upon the form of E(G, S)) of S ∗ as optimal
solution of robust Π;
2
• the complexity of computing S ∗ , at least for particular classes of subproblems of the initial
problem.
Notice that, for any problem Π, its combinatorial counterpart robust Π contains Π as subproblem (just consider probability vector (1, . . . , 1) for Π). Hence, from a complexity point of view,
robust Π is at least as hard as Π, that is, if Π is NP-hard, then robust Π is also NP-hard,
while if Π is polynomial, then no immediate indication can be provided for the complexity of
robust Π, until this latter problem is explicitly studied.
In what follows, in Sections 2, 3 and 4, we deal with three categories of combinatorial graphproblems exhausting a very large part of the most known ones, namely:
• problems whose solutions are subsets of the input vertex-set verifying some specific property
(Section 2);
• problems whose solutions are collections of subsets of the input vertex-set verifying some
specified non-trivial hereditary property2 (Section 3);
• problems whose solutions are subsets of the input edge-set verifying some specific property
(Section 4).
For any of these categories, we give sufficient conditions under which functionals are analytically
expressible and polynomially computable and anticipatory solutions are well-characterized.
In Section 5, we deal with problems having as solutions subsets of the input edge-set verifying
some connectivity properties. For this type of problems, the restriction of S to G[V ′ ] is not
feasible in general, but some additional work (with low algorithmic complexity) is sufficient in
order to render this set feasible. There, as we will see, anticipatory solutions cannot be as wellcharacterized as previously, neither complexity of computing S ∗ . However, we give sufficient
conditions under which functionals for the robust counterparts of these problems are computable
in polynomial time.
These structural results immediately apply to several well-known problems, for instance, min
vertex cover, max independent set, min coloring, max cut, max matching, etc.,
producing so particular results interesting per se. Furthermore, the scope of our results is even
larger than for graph-problems, as problems not originally defined on graphs (e.g., max set
packing or min set cover), are also captured. So, this work can provide a framework for a
systematic classification of a great number of robust derivatives of well-known graph-problems.
In what follows, we deal with problems in NPO. Informally, this class contains optimization
problems whose decision versions belong to NP. Given a combinatorial problem Π ∈ NPO, we
denote by robust Π, its robust counterpart defined as described previously and assume that
the vertex-probabilities are independent.
Let A be a polynomial time approximation algorithm for an NP-hard graph-problem Π,
let m(G, S) be the value of the solution S provided by A on an instance G of Π, and opt(G) be the
value of the optimal solution for G (following our notation for robust Π, opt(G) = E(G, S ∗ )).
The approximation ratio ρA (G) of the algorithm A on G is defined by ρA (G) = m(G, S)/opt(G).
An approximation algorithm achieving ratio, at most, ρ on any instance G of Π will be called
ρ-approximation algorithm.
2
A property π is hereditary if, whenever is satisfied by a graph G, it is satisfied by any subgraph of G; a
hereditary property π is non-trivial if it is true (satisfied) for infinitely many graphs and false for infinitely many
graphs.
3
2
Solutions are subsets of the initial vertex-set
In this section, we deal with graph-problems whose solutions are subsets of the vertex-set of the
input-graph and where, given such a solution S and a set V ′ ⊆ V , the restriction of S in V ′ , i.e.,
the set S ′ = S ∩ V ′ is feasible for G[V ′ ]. The main result of this section is stated in Theorem 1.
Theorem 1. Consider a graph-problem Π verifying the following assumptions: (i) an instance
of Π is a vertex-weighted graph G(V, E, w);
~ (ii) solutions of Π are subsets of V ; (iii) for any
′
′
solution S and any subset V ⊆ V , S = S ∩ V ′ is feasible
for G′ = G[V ′ ]; (iv) the value of any
P
solution S ⊆ V is defined by: m(G, S) = w(S) = vi ∈S wi , where
P wi is the weight of vi ∈ V .
Then, the functional of robust Π is expressed as: E(G, S) = vi ∈S wi pi and can be computed
in polynomial time. Furthermore, the complexity of robust Π is the same as the one of Π.
Proof. Fix a subset V ′ ⊆ V and an anticipatory solution S for robust Π P
on G. According
′
′
′
′
to assumption (iii), S is feasible for G[V ]. Its value is given by: m(G , S ) = vi ∈S wi 1{vi ∈V ′ } .
Then, denoting by 1F the indicator function of a fact F and using (1) we get:
X X
X
m G′ , S ′ Pr V ′ =
wi 1{vi ∈V ′ } Pr V ′
E(G, S) =
V ′ ⊆V vi ∈S
V ′ ⊆V
=
X
wi
vi ∈S
X
V ′ ⊆V
1{vi ∈V ′ } Pr V ′
(2)
For any vertex vi ∈ V , let Vi = V \ {vi } and Vi′ = {V ′ ⊆ V : V ′ = {vi } ∪ V ′′ , V ′′ ⊆ Vi }. Using
also the fact that presence-probabilities of the vertices of V are independent, we get:
X
X
X
Pr {vi } ∪ V ′′
1{vi ∈V ′ } Pr V ′ =
Pr V ′ =
V ′ ⊆V
V ′′ ⊆Vi
V ′ ∈Vi′
=
X
V ′′ ⊆Vi
X
Pr V ′′ = pi
Pr [vi ] Pr V ′′ = Pr [vi ]
(3)
V ′′ ⊆Vi
Combination of (2) and (3) immediately leads to the expression claimed for E(G, S).
It is easy to see that this functional can be computed in time linear in n. Furthermore,
computation of the optimal anticipatory solution for robust Π in G, obviously amounts to
the computation of the optimal weighted solution for Π in G(V, E, w
~ ′ ), where, for any vi ∈ V ,
′
wi = wi pi . Consequently, by this observation and by assumption (iv) in the statement of the
theorem, Π and robust Π have the same complexity.
Although computation of the functional is, as we have mentioned, a priori exponential (since
it carries over 2n subgraphs of G), assumptions (i) through (iv) in Theorem 1 allow polynomial
computation of its value. This is due to the fact that, under these assumptions, given a subgraph G′ induced by a subset V ′ ⊆ V , the value of the solution for G′ is the sum of the weights of
the vertices in S ∩ V ′ . Furthermore, a vertex not in S will never make part of any solution in any
sub-graph of G. Consequently, computation of the functional amounts to determining, for any G ′ ,
which vertices make part of S ∩ V ′ . This is equivalent with the specification, for any vi ∈ S,
of all the subgraphs to which vi belongs and with a summation of the presence-probabilities of
these subgraphs. This sum is equal to pi (the probability of vi ). This simplification is the main
reason that renders functional’s computation polynomial, despite of the exponential number of
terms in its generic expression.
Notice that Theorem 1 can also be used for getting generic approximation results for robust Π. Indeed, since this problem is a particular weighted version of Π, one immediately
concludes that if Π is approximable within approximation ratio ρ, so is robust Π.
4
Corollary 1. Under the hypotheses of Theorem 1, whenever Π and robust Π are NP-hard,
they are equi-approximable.
Theorem 1 has also the following immediate corollary dealing with the case of robust versions of
unweighted combinatorial optimization problems.
Corollary 2. Consider a robust combinatorial optimization problem robust Π verifying assumptions (i) toP
(iv) of Theorem 1 with w
~ = ~1. Then, the functional of robust Π, is expressed
as: E(G, S) = vi ∈S pi and can be computed in polynomial time. Furthermore, robust Π is
equivalent to a weighted version of Π where vertex-weights are the vertex-probabilities.
Corollary 2 is somewhat weaker than Theorem 1 since it does not establish the equivalence
between Π and robust Π but rather a kind of reduction from Π to robust Π stating that the
latter is a priori harder than the former one. As a consequence, whenever Π is NP-hard, so is
robust Π whereas if Π is polynomial, the status of robust Π remains unclear by Corollary 2.
2.1
Applications of Theorem 1
Theorem 1 can be applied to a broad class of problems that fit its four conditions, as robust
max independent set ([21]), robust min vertex covering ([22]), etc. We describe in what
follows two further applications, namely, robust max induced subgraph with property π
and robust min feedback vertex-set.
2.1.1
robust max induced subgraph with property π
Consider a graph G(V, E) and a non-trivial hereditary property. A feasible solution for max
induced subgraph with property π is a subset V ′ ⊆ V such that, the subgraph G[V ′ ] of G
induced by V ′ satisfies π. The objective for max induced subgraph with property π is to
determine such a set V ′ of maximum-size. Note that, “independent set”, “clique”, “planar graph”
are hereditary properties. In the weighted version of the problem (i.e., the one where positive
weights are associated with the vertices of G), called max weighted induced subgraph with
property π, we search for maximizing the total weight of V ′ .
Given a solution S for max weighted induced subgraph with property π and an
induced subgraph G[V ′ ] of the input graph G(V, E), the set S ∩V ′ is a feasible solution for G[V ′ ],
since, by the definition of π, if a subset S ⊆ V induces a subgraph verifying it, then any subset
of S also induces a subgraph verifying π. Henceforth, max weighted induced subgraph
with property π fits the conditions of Theorem 1, or of Corollary 2 (for the unweighted case).
2.1.2
robust min feedback vertex-set
Given an oriented graph G(V, A), a feedback vertex-set is a subset V ′ ⊆ V such that V ′ contains
at least a vertex of any directed cycle of G. In min feedback vertex-set, the objective is to
determine a feedback vertex-set of minimum size.
Remark that, absence of a vertex v from a feedback vertex-set V ′ , breaks any cycle containing
this vertex. If v makes part of an anticipatory solution S then, since no such cycle that contained v
exists in G′ , feasibility of the solution S ∩V ′ does not suffer from the absence of v. So, Corollary 2
applies for this problem.
Note that the weighted version of this problem can be tackled in a similar way.
2.2
Extensions of Theorem 1 beyond graphs
Theorem 1 can also be used to capture problems that are not originally defined on graphs as
robust max set packing and robust min set cover. The robust versions dealt for both
5
of them consist of considering presence probabilities p1 , . . . , pn for the corresponding elements
of the ground set C. Extensions to these problems are shown in the following Sections 2.2.1
and 2.2.2.
2.2.1
robust max set packing
Given a family S = {S1 , . . . , Sm } of subsets of a ground set C = {c1 , . . . , cn }, max set packing
consists of determining a maximum size sub-collection S ′ ⊆ S such that for any (Si , Sj ) ∈ S ′ ×S ′ ,
Si ∩ Sj = ∅.
The robust version assumed for max set packing consists of considering presence probabilities p1 , . . . , pn for the corresponding elements of C. A set Si ∈ S is present if at least one of its elements is present. So, denoting by Pr[Si ] the presence probability of the set Si = {ci1 , ci2 , . . . , cik },
we get:
k
Y
1 − pij
Pr [Si ] = 1 −
(4)
j=1
Consider the following very well-known transformation of an instance (S, C) of max set packing into an instance G(V, E) of max independent set:
• for any Si ∈ S, add a new vertex vi ∈ V ;
• for any pair (Si , Sj ) ∈ S × S, if Si ∩ Sj 6= ∅, then add an edge (vi , vj ) ∈ E.
It is easy to see that, under the above transformation, any set packing in (S, C) becomes an
independent set of G of the same size and vice-versa.
So, given a robust model for max set packing, one can transform, as described just previously, its instance (S, C) into a graph G(V, E) and one can consider vertex probabilities
p(vi ) = Pr[Si ], i = 1, . . . , m, where vi is the vertex added for Si . A feasible max independent set-solution S ′ in G corresponds to a sub-family S ′ consisting of the sets represented by
the vertices of S ′ and is indeed a set packing. On the other hand, an induced subgraph G[V ′ ] of G
corresponds to a sub-instance I ′ of (S, C) induced by the sets represented by the vertices of V ′
(according to the rule that a set is present if at least one of its elements is in I ′ ). Then, taking
the restriction of S ′ in V ′ amounts to consider the restriction of S ′ in I ′ . It is easy to see that this
restriction is a feasible set packing for I ′ . So, max set packing, being an independent set problem, meets conditions of Corollary 2 (or of Theorem 1 for max weighted set packing). Hence,
given an instance (S, C) of max set packing, with element-probabilities
p i , for any ci ∈ C, and
P
a feasible solution S ′ of (S, C), then E((S, C), S ′ ) = Si ∈S ′ Pr[Si ], where Pr[Si ] is as defined
in (4). Then, the robust version of max set packing amounts to a particular weighted version
of max set packing where each set Si = {ci1 , . . . , cik } in S is weighted by Pr[Si ].
2.2.2
robust min set cover: when sets become vertices
Given a collection S = {S1 , . . . Sm } of subsets of a ground set C = {c1 , . . . , cn } (it is assumed
that ∪Si ∈S Si = C), min set covering consists of determining a minimum-size set cover of C,
i.e., a minimum-size sub-collection S ′ of S such that ∪Si ∈S ′ Si = C.
We use the same robust model for min set cover, i.e., we consider presence probabilities
p1 , . . . , pn for the corresponding elements of the ground set C. As previously in Section 2.2.1, a
set Si ∈ S is present if at least one of its elements is present. So, Pr[S i ] is given by (4). We give
a formulation of min set cover as a graph-problem and according this formulation, we show
that robust min set cover fits conditions of Theorem 1.
Consider an instance (S, C) of min set cover and construct GS (VS , ES , ~ℓ), an edge-labelled
multi-graph, as follows:
6
• for any set Si ∈ S, add a vertex vi ∈ VS ;
• for any pair Si , Sj of sets in S, add a new edge (vi , vj ) labelled by ck only if ck ∈ Si ∩ Sj .
Under this formulation, min set cover amounts to determine the smallest subset of vertices
covering all the labels in GS . Then, an analysis very similar to the one for robust max set
packing in Section 2.2.1, concludes that, under the considered formulation, min set cover
meets conditions of Theorem 1 (for the weighted case) or of Corollary 2 (for the unweighted
one).
3
Solutions are collections of subsets of the initial vertex-set
We now deal with problems the feasible solutions of which are collections of subsets of the initial
vertex-set. Consider a graph G(V, E) and a combinatorial optimization graph-problem Π whose
solutions are collections of subsets of V verifying some specified non-trivial hereditary property
(e.g., independent set, clique, etc.). The following theorem characterizes functionals and optimal
anticipatory solutions for such problems.
Theorem 2. Consider a graph- problem Π verifying the following assumptions: (i) an instance
of Π is a graph G(V, E); (ii) a solution of Π on an instance G is a collection S = (V1 , . . . , Vk ) of
subsets of V any of them satisfying some specified non-trivial hereditary property; (iii) for any
solution S and any subset V ′ ⊆ V , the restriction S ′ of S in V ′ , i.e., S ′ = (V1 ∩ V ′ , . . . , Vk ∩ V ′ ),
is feasible for G′ = G[V ′ ]; (iv)
Pk the value
Q of any solution S ⊆ V of Π is defined by: m(G, S) =
|S| = k. Then, E(G, S) = j=1 (1 − vi ∈Vj (1 − pi )) and can be computed in polynomial time.
robust Π amounts to a particular weighted version of Π, where the weight
Q of any vertex vi ∈ V
is 1 − pi , the weight w(Vj ) of a subset Vj ⊆ V is defined by w(Vj ) = 1 − vi ∈Vj (1 − pi ) and the
P
objective function to be optimized is equal to Vj ∈C w(Vj ).
Proof. Consider an anticipatory solution S = (V1 , V2 , . . . , Vk ) and a subgraph G′ = G[V ′ ]
of G. Denote by k ′ = m(G′ , S ′ ), the valuePof the solution obtained on G′ as described in
assumption (iii). Then, from (1), E(G, S) = V ′ ⊆V Pr[V ′ ]k ′ .
Consider the facts Fj : Vj ∩ V ′ 6= ∅ and F̄j : Vj ∩ V ′ = ∅. Then, k ′ can be written as
P
P
k ′ = kj=1 1Fj = kj=1 (1 − 1F̄j ) and E(G, S) becomes:
E(G, S) =
X
V ′ ⊆V
=
X
V ′ ⊆V
=
k
X
Pr V ′
1 − 1F̄j
k
k
X
X
X
Pr V ′
1Vj ∩V ′ =∅
Pr V ′
1−
k X
X
j=1 V ′ ⊆V
= k−
j=1
V ′ ⊆V
j=1
j=1
k X
X
Pr V ′ 1Vj ∩V ′ =∅
Pr V ′ −
k Y
X
j=1 vi ∈Vj
j=1 V ′ ⊆V
(1 − pi ) =
k
X
j=1
1 −
Y
vi ∈Vj
(1 − pi )
(5)
It is easy to see that computation of E(G, S) can be performed in at most O(n) steps; consequently, robust Π is in NPO. Furthermore, by (5), the characterization of the feasible solutions
for robust Π claimed in the statement of the theorem is immediate.
7
What does play a central role for yielding result of Theorem 2, is the fact that property
satisfied by the sets of the collection is hereditary. This allows to preserve sets V 1 , . . . , Vk in
the solution returned by S ∩ Vi , i = 1, . . . , k, unless they are empty and, consequently, to
express E(G, S) as in (5), in terms of facts Fj and F̄j .
Assume that pi = 1, for any vi ∈ V . Then, by (5), E(G, S) = k and robust Π coincides in
this case with Π.
Corollary 3. If Π is NP-hard, then robust Π is also NP-hard.
As for Corollary 2, Corollary 3 settles complexity only for the case where Π is NP-hard, leaving
unclear the status of robust Π when Π ∈ P.
3.1
Applications of Theorem 2
Theorem 2 has also application for numerous combinatorial optimization problems, as robust
min coloring ([24]), robust min partition into cliques, etc. In what follows we describe
two further applications, namely, to robust min complete bipartite subgraph cover and
robust min cut cover.
3.1.1
robust min complete bipartite subgraph cover
Given a graph G(V, E), a solution of min complete bipartite subgraph cover is a collection
C = (V1 , V2 , . . . , Vk ) of subsets of V such that the subgraph induced by any of the V i ’s, i =
1, . . . , k, is a complete bipartite graph and for any edge (u, v) ∈ E there exists a V i containing
both u and v. The objective here is to minimize the size |C| of C.
Consider a solution S = (V1 , . . . , Vk ) min complete bipartite subgraph cover and a
subset V ′ ⊆ V . The set S ′ = (V1 ∩ V ′ , . . . , Vk ∩ V ′ ), is feasible for G′ = G[V ′ ]. Indeed, if a
vertex v disappears from some subset Vi of an anticipatory solution S, the surviving set Vi always
induces a complete bipartite graph. Furthermore:
• except the edges that have been disappeared (the ones incident to v) any other edge remain
covered by the surviving sets of S;
• property “is a complete bipartite graph” is hereditary.
So, robust min complete bipartite subgraph cover meets the conditions of Theorem 2.
3.1.2
robust min cut cover
Given a graph G(V, E), a feasible solution for min cut cover is a collection (V 1 , . . . , Vk ) of V
such that any Vi , i = 1, . . . , k is a cut, i.e., for any (u, v) ∈ E, there exists a Vi such that either
u ∈ Vi and v ∈
/ Vi , or u ∈
/ Vi and v ∈ Vi . The objective is to minimize the size of the collection.
Consider a solution S = (V1 , . . . , Vk ) for min cut cover. If a vertex v ∈ V is absent, then
any edge incident to v is also absent. But, absence of a vertex together with any edge incident to
it, does not affect the edges present to the final graph G′ (V ′ , E ′ ), that remain feasibly covered by
endpoints any of them belonging to distinct cuts. Hence, S ′ = (V1 ∩V ′ , . . . , Vk ∩V ′ ) is feasible for
min cut cover, that meets the conditions of Theorem 2, since property “is a cut” is hereditary.
3.2
robust min set cover: when elements become vertices
As in Section 2, application of Theorem 2 can go beyond graphs. We give in this section another
formulation of min set cover as a graph-problem that can be seen as a kind of dual with
respect to the formulation given in Section 2.2.2. We show that, according this new formulation,
8
robust min set cover fits conditions of Theorem 2. As we will see, both ways to tackle
robust min set cover lead to the same result.
Consider an instance (S, C) of min set cover and this time construct an edge-colored
multi-graph GC (VC , EC , ~ℓS ) as follows:
• for any ci ∈ C, add a vertex vi ∈ VC ;
• for any pair ci , cj of elements in C, add a new edge (vi , vj ) colored with Sk only if Sk ⊇
{ci , cj }.
In the so-constructed graph GC a set Si = {ci1 , . . . cik } ∈ S becomes a clique on vertices
vi1 , . . . , vik ∈ VC all the edges of which are colored with the same color Si ; we will call such
a clique a unicolored clique. Under this alternative formulation, min set cover can be viewed
as a particular clique-covering problem where the objective is to determine a minimum size
covering of VC by unicolored cliques.
Consider a set cover S ′ for the initial instance (S, C) and a sub-instance I ′ of (S, C) consisting
of some elements of C and of the subsets of S including these elements. These objects correspond,
in GC , to a vertex-covering by unicolored cliques and the subgraph G ′C of GC defined with respect
to I ′ . Restriction of S ′ in I ′ , can be viewed, with respect to GC as restriction of the initial vertexcovering by unicolored cliques to the vertices of G′C just as described in Section 3. Observe finally
that “unicolored clique” is a hereditary property. So, under this formulation, robust min set
cover exactly fits conditions of Theorem 2.
So, according to any of the formulations used for min set cover, given an instance (S, C)
′
of min set cover, with element-probabilities
Q pi , for any ci ∈ C, and a feasible solution S
P
′
of (S, C), then, E((S, C), S ) = Si ∈S ′ (1 − cj ∈Si (1 − pj )) and can be computed in polynomial
time. The robust version of min set cover amounts to a particular weighted version of the
initial problem where each set Si = {ci1 , . . . , cik } in S is weighted as in (4).
Hence, robust min set cover is indeed a simple weighted version of min set cover,
where one has to determine a set cover minimizing its total weight. In this sense, the problem
dealt seems to be simpler than the majority of the problems captured by Theorem 2 as, for
instance min coloring. This is due to the fact that, dealing with min set cover, there is
a polynomial number of unicolored cliques in GC (the sets of S) candidate to be part of any
solution, while, for min coloring the number of the independent sets that may be part of a
solution is exponential.
3.3
A generic approximation result for the problems fitting conditions of Theorem 2
This section extends an approximation result of [24] for robust min coloring, in order to
capture the whole of problems meeting the conditions of Theorem 2.
Consider such an NPO problem Π, an instance G(V, E) of Π, set n = |V | and consider a
solution S = (V1 , . . . , Vk ) of Π on G. Denote by pmin and pmax the minimum and maximum
vertex-probabilities, respectively. Then, the following bounds hold for E(G, S):
( n
)
n
n X
n
X
X
X
pi pj , kpmin 6 E(G, S) 6 min
max
pi −
pi 6 npmax , k
(6)
i=1
i=1
i=1 j=i+1
Observe first that the rightmost upper bound for E(G, S) in (6) is immediately derived from the
expression for E(G, S) in the statement of Theorem 2.
We now proveQthe leftmost upper bound and lower bound of (6). We first produce a framing
for the term 1 − vi ∈Vj (1 − pi ). For simplicity, assume |Vj | = ℓ and arbitrarily denote vertices
9
in Vj by v1 , . . . , vℓ . Then, by induction in ℓ the following holds:
ℓ
X
i=1
pi −
ℓ
ℓ
X
X
i=1 j=i+1
pi pj 6 1 −
ℓ
Y
i=1
(1 − pi ) 6
ℓ
X
(7)
pi
i=1
Indeed, for P
the left-hand
P side
Pof (7), observe first
Q that it is true for ℓ = 1 and suppose it true for
ℓ = κ, i.e., κi=1 pi − κi=1 κj=i+1 pi pj 6 1 − κi=1 (1 − pi ), or:
κ
Y
i=1
(1 − pi ) 6 1 −
κ
X
pi +
κ
κ X
X
(8)
pi pj
i=1 j=i+1
i=1
Suppose now that ℓ = κ + 1 and multiply both terms of (8) by (1 − p κ+1 ); then:
κ
κ X
κ+1
κ
X
Y
X
pi pj (1 − pκ+1 )
(1 − pi ) 6 1 −
pi +
i=1
i=1
= 1−
= 1−
κ
X
pi +
κ
X
i=1 j=i+1
i=1
κ+1
X
i=1 j=i+1
κ
X
pi +
κ+1
κ+1 X
X
i=1 j=i+1
i=1
pi pj − pκ+1 + pκ+1
pi pj − pκ+1
κ
X
κ
X
i=1
κ
X
i=1 j=i+1
pi − pκ+1
κ
κ X
X
pi pj
i=1 j=i+1
pi pj 6 1 −
κ+1
X
i=1
pi +
κ+1
κ+1 X
X
pi pj
i=1 j=i+1
which proves the left-hand side inequality in (7).
Q
P
For the right-hand side of (7), we show by induction on ℓ that Qℓi=1 (1 − pi ) > 1 − Pℓi=1 pi .
This is clearly true for ℓ = 1. Suppose it also true for any ℓ 6 κ, i.e., κi=1 (1 − pi ) > 1 − κi=1 pi .
Then, by multiplying both members
by (1 −
κ+1 ), we get that the product
Pκ of this inequality
Pκ
Ppκ+1
obtained is equal to 1 − pκ+1 − i=1 pi + pκ+1 i=1 pi > 1 − i=1 pi , q.e.d.
Taking the sums of the members
of (7) for m = 1 to k, the right-hand side inequality
P
immediately gives E(G, S) 6 ni=1 pP
.
i
P P
We now prove that E(G, S) > ni=1 pi − ni=1 nj=i+1 pi pj , i.e., the leftmost lower bound
claimed in (6). From the left-hand side of (7), we get:
n
n X
n
ℓ
ℓ
k X
n
ℓ
ℓ
ℓ
k
X
X
X
X
X
X
X
X
X
pi pj
(9)
pi −
pi pj >
pi −
pi pj =
pi −
m=1
i=1
i=1 j=i+1
i=1
m=1 i=1 j=i+1
Observe that, from the first inequality of (7), we have:
k
k
ℓ
ℓ
ℓ
X
X
X
X
X
pi pj 6
pi −
m=1
i=1
m=1
i=1 j=i+1
1−
i=1 j=i+1
i=1
ℓ
Y
i=1
(1 − pi )
!
(10)
The righthand side of (10) is exactly E(G, S). Putting this together with (9), the leftmost lower
bound for E(G, S) in (6) is proved.
Q
Finally, in order to derive the rightmost lower bound in (6), observe that vi ∈Sj (1 − pi ) 6
Q
(1 − pmin )|Sj | 6 1 − pmin , i.e., 1 − vi ∈Sj (1 − pi ) > pmin . Summing for j = 1 to k, we get the
bound claimed.
We are ready now to study an approximation algorithm for the whole class of problems
meeting Theorem 2. Fix a vertex-probability p′ , assume that there exists a ρ-approximation
polynomial time algorithm A for Π, and run the following algorithm, called RA for robust Π:
10
1. partition the vertices of G into three subsets: the first, V 1 including the vertices with
probabilities at most 1/n, the second, V2 , including the vertices with probabilities in the
interval [1/n, p′ ] and the third, V3 , including the vertices with probabilities greater than p′ ;
2. feasibly solve Π in G[V1 ] and G[V2 ] separately;
3. run A in G[V3 ];
4. take the union of the solutions computed in steps 2 and 3 as solution for G.
Theorem 3. If A achieves approximation ratio ρ for Π, then RA approximately solves in polyno√
mial time the robust version of Π within ratio O( ρn).
Proof. Denote by S ∗ = (V1∗ , . . . , Vk∗∗ ) an optimal anticipatory solution and by S = (V̂1 , . . . , V̂k )
∗ ,...,V ∗
the approximate solution computed in step 4 and, respectively by S i∗ = (V1,i
|S ∗ |,i ) and
i
Si = (V̂1,i , . . . , V̂|Si |,i ), the optimal and approximate solutions in G[Vi ], i = 1, 2, 3. Denote
by S ∗ [V1 ], S ∗ [V2 ] and S ∗ [V3 ] the restrictions of S ∗ in G[V1 ], G[V2 ] and G[V3 ], respectively. Denote
finally by ni , the orders of G[Vi ], for i = 1, 2, 3, respectively. The proof is based upon the following
claims:
1. any feasible polynomial time approximation algorithm for robust Π achieves in G[V 1 ]
approximation ratio bounded above by 2;
2. any feasible polynomial time approximation algorithm for robust Π achieves in G[V 2 ]
approximation ratio bounded above by O(np′ );
3. assuming that A achieves approximation ratio ρ for Π, algorithm RA, when running in G[V 3 ],
achieves approximation ratio bounded above by ρ/p′ for robust Π.
Pn1
∗
For Claim P
1, using
(6) for S1 and S1∗ , we get: E(G[V1 ], S1 ) 6
i=1 pi and E(G[V1 ], S1 ) >
P
P
n1
n1
n1
i=1
i=1 pi −
j=i+1 pi pj . Combining them, we derive:
E (G [V1 ] , S1∗ )
E (G [V1 ] , S1 )
> 1−
n1
n1 P
P
pi pj
i=1 j=i+1
n1
P
= 1−
pi
n
P1
pi
i=1
2
> 1−
i=1
2
pi
+
n1
P
−
pi
n1
P
i=1
p2i
i=1
i=1
n1
P
2
n1
P
p2i
i=1
n1
P
2
pi
> 1−
n1
P
pi
i=1
(11)
2
i=1
Since pi ’s are smaller than 1/n and n1 6 n, the right-hand side of (11) is at least as large as 1/2.
Hence, the approximation ratio of any algorithm for Π in G[V1 ] is at most 2, and the proof of
Claim 1 is complete.
Q
We deal now with Claim 2. Here, for any vi , pi > 1/n. Consequently, 1 − vi ∈V ∗ (1 − pi ) >
j,2
∗ |
|Vj,2
∗ |/n)
(|Vj,2
∗ |(|V ∗ |
(|Vj,2
j,2
1)/2n2 )
−
where the last inequality is an easy
−
1 − (1 − (1/n))
>
application of the left-hand side of (7) with pi = 1/n for any vertex vi . Furthermore:
∗
∗ −1
∗
∗
∗
∗ −1
∗
Vj,2
Vj,2
Vj,2
Vj,2
Vj,2
Vj,2
Vj,2
n+1
−
=
>
(12)
1−
>
n
2n2
n
2n
n
2n
2n
11
Summing inequality (12) for j = 1, . . . , |S2∗ |, we get E(G[V2 ], S2∗ ) > n2 /2n, where n2 is the order
of G[V2 ]. On the other hand, using the leftmost upper bound in (6) we get E(G[V 2 ], S2 ) 6 n2 p′ .
The bounds for E(G[V2 ], S2∗ ) and E(G[V2 ], S2 ) immediately derive approximation ratio at most
2np′ = O(np′ ) and the proof of Claim 2 is complete.
We now turn to Claim 3. Using the rightmost lower bound of (6), E(G[V 3 ], S3∗ ) > |S3∗ |p′ .
On the other hand, by the rightmost upper bound of (6), E(G[V 3 ], S3 ) 6 |S3 |. So, assuming
that A achieves ratio ρ for Π, step 3 achieves ratio |S3 |/|S3∗ |p′ for G[V3 ], that turns out to a ratio
bounded above by ρ/p′ , completing so the proof of Claim 3.
We prove that, for any k ∈ {1, 2, 3}: E(G, S ∗ ) > E(G[Vk ], S ∗ [Vk ]) > E(G[Vk ], Sk∗ ). Remark
that S ∗ [Vk ] is a particular feasible solution for G[Vk ]; hence: E(G[Vk ], S ∗ [Vk ]) > E(G[Vk ], Sk∗ ).
In order to prove the first inequality, fix Q
a k and consider a component,
say V j∗ of S ∗ . Then,
Q
the contribution of Vj∗ in S ∗ [Vk ] is: 1 − vi ∈V ∗ ∩Vk (1 − pi ) 6 1 − vi ∈V ∗ (1 − pi ), which is its
j
j
contribution in S ∗ . Iterating this argument for all the colors in C ∗ [Vk ], the claim follows.
Algorithm RA solves separately each G[Vk ], k ∈ {1, 2, 3} and returns as solution S the union
of the colors used. Hence, E(G, S) = E(G[V1 ], S1 ) + E(G[V2 ], S2 ) + E(G[V3 ], S3 ). On the other
hand, E(G, S ∗ ) is at least as large as any of E(G[Vk ], Sk∗ ), k ∈ {1, 2, 3}. So, the ratio of the
algorithm in G is at most the sum of the ratios proved by Claims 1, 2 and 3, i.e., at most
O(2 + np′ + (ρ/p′ )).
Remark that the ratio claimed in Claim 2 is increasing with p′ , while
pthe one in Claim 3 is
decreasing with p′ . Equality of expressions np′ and ρ/p′ holds for p′ = ρ/n. In this case the
√
value of the ratio obtained is O( ρn), and the proof of the theorem is now complete.
4
Solutions are subsets of the initial edge-set
We deal in this section with problems for which solutions are sets of edges. Notice that whenever
a vertex is absent from some subset V ′ ⊆ V , the edges incident to it are also absent from G[V ′ ].
So, our assumption is that, given a solution (in terms of a set of edges) S, and a set V ′ ⊆ V
inducing a subgraph G[V ′ ] = G′ (V ′ , E ′ ) of G, the set S ∩ E ′ is feasible for Π in G′ . The main
result for this case, is the following theorem.
Theorem 4. Consider a graph-problem Π verifying the following assumptions: (1) an instance
of Π is an edge- (or arc-) valued graph G(V, E, ~ℓ); (2) any solution of Π on any instance G is a
subset of E; (3) for any solution S and any subset V ′ ⊆ V , denoting by G′ (V ′ , E ′ ) the subgraph
of G induced by V ′ , P
the set S∩E ′ is feasible; (4) the value of any solution S ⊆ E of Π is defined by:
m(G, S) = w(S) = (vi ,vj )∈S ℓ(vi , vj ), where ℓ(vi , vj ) is the valuation of the edge (or arc) (vi , vj )
P
of G. Then, the functional of robust Π is expressed as: E(G, S) = (vi ,vj )∈S ℓ(vi , vj )pi pj and
can be computed in polynomial time. Furthermore, dealing with their respective computational
complexities, robust Π and Π are equivalent.
′
′
′
Proof. Set S ′ =
PS ∩ E ; by the assumptions of the theorem, S is feasible for G . Furthermore,
′
′
ℓ(vi , vj )1{(vi ,vj )∈E ′ } . Then, using (1):
m(G , S ) =
(vi ,vj )∈S
E(G, S) =
X
V ′ ⊆V
=
m G′ , S ′ Pr V ′ =
X
(vi ,vj )∈S
ℓ (vi , vj )
X
V ′ ⊆V
X
X
V ′ ⊆V (vi ,vj )∈S
1{(vi ,vj )∈E ′ } Pr V ′
ℓ (vi , vj ) 1{(vi ,vj )∈E ′ } Pr V ′
(13)
Any edge (or arc) (vi , vj ) ∈ E belongs to G′ = G[V ′ ], if and only if both of its endpoints belong
to V ′ . Let Vij = V \ {vi , vj } and Vij′ = {V ′ ⊆ V : V ′ = {vi } ∪ {vj } ∪ V ′′ , V ′′ ⊆ Vij }, the set of
12
all the subsets of V containing both vi and vj . Using also the fact that presence-probabilities of
the vertices of V are independent, we get:
X
X
X
Pr {vi } ∪ {vj } ∪ V ′′
1{(vi ,vj )∈E ′ } Pr V ′ =
Pr V ′ =
V ′ ⊆V
V ′′ ⊆Vij
′
V ′ ∈Vij
=
X
V ′′ ⊆Vij
X
Pr V ′′ = pi pj
pi pj Pr V ′′ = pi pj
(14)
V ′′ ⊆Vij
Combination of (13) and (14) immediately leads to the expression claimed for the functional.
It is easy to see that this functional can be computed in time quadratic with n. Furthermore, computation of an optimal anticipatory solution for robust Π in G obviously amounts
to a computation of an optimal solution for Π in an edge- (or arc-) valued graph G(V, E, ~ℓ′ )
where, for any (vi , vj ) ∈ E, ℓ′ (vi , vj ) = ℓ(vi , vj )pi pj . Consequently, by this observation and by
assumption (4), Π and robust Π have the same complexity.
The reasons for which the functional derived in Theorem 4 becomes polynomial are quite
analogous to the ones in Theorem 1. Since an edge that does not belong to the anticipatory
solution S will never be part of any solution in any subgraph G ′ (V ′ , E ′ ) of G, the computation
of the functional amounts to the specification, for any G′ , of the set S ∩ E ′ . This is equivalent to
first determining, for any edge e ∈ S, all the subgraphs containing e and next to a summation
of the probabilities of these subgraphs.This sum equals to the product of the probabilities of the
endpoints of e.
Let us note that, as in Section 2, Theorem 4 can be used for getting generic approximation
results for robust Π. Since this problem is a particular weighted version of Π, one immediately
concludes that if Π is approximable within approximation ratio ρ, so is robust Π.
Corollary 4. Under the hypotheses of Theorem 4, whenever Π and robust Π are NP-hard,
they are equi-approximable.
Corollary 5. Consider a robust combinatorial optimization problem Π verifying assumptions (1)
through (4)Pof Theorem 4 with ~ℓ = ~1. Then, the functional of robust Π is expressed as:
E(G, S) = (vi ,vj )∈S pi pj and can be computed in polynomial time and robust Π is equivalent
to an edge- (or arc-) valued version of Π where the values of an edge is the product of the
probabilities of its endpoints.
4.1
Application of Theorem 4 to robust max matching
In max matching, the objective is, given a graph G(V, E) to determine a maximum-size matching, i.e., a maximum-size subset of E such that its edges are pairwise disjoint, in the sense that
they have no common endpoint.
Clearly, max matching in both edge-valued and non-valued graphs, fits conditions of Theorem 4 and Corollary 5, respectively. Moreover, since max weighted matching is polynomial,
both robust max weighted matching and robust max matching are polynomial also.
4.2
Application of Theorem 4 to robust max cut
We deal here with max cut in both edge-valued and unitary edge-valued graphs. Consider a
graph G(V, E). In max cut (resp. max weighted cut) we wish to determine a maximum
cardinality (resp., maximum weight) cut, that is to partition V into two subsets V 1 and V2
such that a maximum number of edges (resp., maximum-weight set of edges) have one of their
endpoints in V1 and the other one in V2 .
13
1
2
1
4
5
3
7
8
10
9
11
5
6
13
V1
2
3
7
8
10
9
13
12
12
V1
V2
(a) A graph G with a cut S
(thick edges)
V2
(b) Some “surviving” subgraph and the “surviving”
solution
Figure 1: An example for robust max cut.
We can represent an anticipatory cut S as a set of edges in such a way that whenever
(vi , vj ) ∈ S, vi ∈ V1 and vj ∈ V2 . For example, in Figure 1(a), where for simplicity values of edges are not mentioned, the cut partitions V in subsets V 1 = {1, 3, 4, 7, 10, 13} and
V2 = {2, 5, 6, 8, 9, 11, 12} and the anticipatory cut S (thick edges) can then be written as
S = {(1, 2), (3, 6), (4, 2), (4, 5), (4, 6), . . . , (13, 11)} (where edges are ranged in lexicographic order). In Figure 1(b), we present graph’s and cut’s states assuming that vertices 4, 6 and 11 are
absent. The solution S ′ considered misses in all edges of S having at least one endpoint among
{4, 6, 11} but it obviously remains a feasible cut for the surviving graph.
Hence, both weighted and cardinality robust max cut meet the conditions of Theorem 4
and Corollary 5, respectively. Consequently, max cut being NP-hard, robust max weighted
cut and robust max cut are also NP-hard.
5
When things become complicated
In this section we tackle edge-weighted graph-problems where feasible solutions are connected
sets of edges (for example, paths, trees, cycles, etc.) but we assume that, given a solution S and
a set V ′ ⊆ V inducing a subgraph G[V ′ ] = G′ (V ′ , E ′ ) of G, the set S ∩ E ′ is not always feasible
for G′ .
Formally, consider a problem Π where a feasible solution is a connected set S of edges.
Consider also that vertices in S are ordered in some appropriate order. Assume that operation
S ∩ E ′ results in a set of connected subsets C1 , C2 , . . . , Ck of S but that S ′′ = ∪ki=1 Ci is not
connected (i.e., S ′′ does not constitute a feasible solution for Π). Assume also that connected
subsets C1 , C2 , . . . , Ck are also ranged in this order (always following some appropriate ordering
implied by the one of S).
We consider a kind of “completion” of S ′′ by additional edges linking, for i = 1, . . . , k − 1, the
last vertex (in the ordering considered for S) of Ci with the first vertex of Ci+1 . In other words,
given S (representing a connected set of edges), we apply the following algorithm, denoted by A
in the sequel:
1. range the vertices of S following some appropriate order;
14
2. compute S ∩ E ′ ; let C1 , C2 , . . . , Ck be the resulting connected components of S ∩ E ′ ;
3. for i = 1, . . . , k − 1, use an edge to link the last vertex of Ci with the first vertex of Ci+1 ;
4. output S ′ the solution so computed.
Obviously, in order that step 3 of A is able to link components Ci and Ci+1 , an edge must
exist between the vertices implied; otherwise, A is definitely unfeasible. So, in order to assure
feasibility, we make, for the rest of the section the basic assumption that the input graph for the
problems tackled is complete.
In what follows, we denote by V [S ′ ] the set of vertices in S ′ and set G′′ (V [S ′ ], E ′′ ) = G[V [S ′ ]].
Also, we denote by [vi , vj ] the set {vi+1 , vi+2 , . . . , vj−1 } (i < j in the ordering assumed for S)
such that: (a) for any ℓ = i, i + 1, . . . , j − 1, (vℓ , vℓ+1 ) ∈ S (i.e., [vi , vj ] is the set of vertices in the
path linking vi to vj in S, where vi and vj themselves are not encountered3 ) and (b) vi and vj
belong to consecutive4 connected subsets Cm and Cm+1 , for some m < k.
Theorem 5. Consider a robust combinatorial optimization problem robust Π verifying the
following assumptions: (i) instances of Π are edge-valued complete graphs (Kn , ~ℓ) = G(V, E, ~ℓ);
furthermore, in the robust version of Π any vertex vi ∈ V has a presence-probability pi ; (ii) a
solution of Π is a subset S of E verifying some connectivity property; (iii) given an anticipatory
solution S (the vertices of which are ranged in some appropriate order), algorithm A computes
a feasible solutionPS ′ , for any subgraph G′ (V ′ , E ′ , ~ℓ) = G[V ′ ] of G (obviously, G′ is complete);
ℓ(v , v ). Then, E(G, S) is computable in polynomial time and is
(iv) m(G, S) =
(vi ,vj )∈S
Q
P
P i j
expressed by: E(G, S) = (vi ,vj )∈S ℓ(vi , vj )pi pj + (vi ,vj )∈E ′′ \S ℓ(vi , vj )pi pj vl ∈[vi ,vj ] (1 − pl ).
Proof. Denote by C[E ′ ], the set of edges added to S ′′ during the execution of step 3 of A.
Obviously, S ′ = S ′′ ∪ C[E ′ ]; also, if an edge belongs to C[E ′ ], then it necessarily belongs to E[S],
the set of edges in E induced by the endpoints of the edges in S. By assumptions (i) to (iii), S ′
is a feasible set of edges. Furthermore:
X
X
m G′ , S ′ =
ℓ (vi , vj ) 1{(vi ,vj )∈S ′ } =
ℓ (vi , vj ) 1{(vi ,vj )∈S ′′ ∪C[E ′ ]}
(15)
(vi ,vj )∈E
(vi ,vj )∈E
By construction, any element of C[E ′ ] is an edge (or arc) the initial endpoint of which corresponds
to the terminal endpoint of a connected subset Ci of S and the terminal endpoint of which
corresponds to the initial endpoint of the “next” connected subset C i+1 of S. Then, for any
subgraph G′ of G, the following two assertions hold: (1) S ′ ⊆ E ′′ , and (2) any edge that does
not belong to E ′′ , will never be part of any feasible solution. Indeed, for such an edge, at least
one of its endpoints does not belong to V [S ′ ]; so, C[E ′ ] ⊆ E ′′ . We so have from (15):
X
X
m G′ , S ′ =
ℓ (vi , vj ) 1{(vi ,vj )∈S ′′ ∪C[E ′ ]} =
ℓ (vi , vj ) 1{(vi ,vj )∈S ′′ ∪C[E ′ ]}
(vi ,vj )∈E
=
X
(vi ,vj
=
(vi ,vj )∈E ′′
ℓ (vi , vj ) 1{(vi ,vj )∈S ′′ } +
(vi ,vj
)∈E ′′
X
ℓ (vi , vj ) 1{(vi ,vj )∈E ′ } +
It is assumed that if [vi , vj ] = ∅, then
4
With respect to the order C1 , . . . , Ck .
ℓ (vi , vj ) 1{(vi ,vj )∈C[E ′ ]}
)∈E ′′
X
(vi ,vj )∈E ′′ \S
(vi ,vj )∈S
3
X
Q
vl ∈[vi ,vj ] (1
− pl ) = 0.
15
ℓ (vi , vj ) 1{(vi ,vj )∈C[E ′ ]}
(16)
Using (16), we get:
E (G, S) =
X
V ′ ⊆V
=
X
X
(vi ,vj )∈E ′′ \S
(vi ,vj )∈S
ℓ (vi , vj )
(vi ,vj )∈S
+
ℓ (vi , vj ) 1{(vi ,vj )∈E ′ } +
X
X
X
V ′ ⊆V
1{(vi ,vj )∈E ′ } Pr V ′
ℓ (vi , vj )
X
V ′ ⊆V
(vi ,vj )∈E ′′ \S
ℓ (vi , vj ) 1{(vi ,vj )∈C[E ′ ]} Pr V ′
1{(vi ,vj )∈C[E ′ ]} Pr V ′
(17)
As in the proof of Theorem 4, the first term of (17) can be simplified as follows:
X
X
X
ℓ (vi , vj )
1{(vi ,vj )∈E ′ } Pr V ′ =
ℓ (vi , vj ) pi pj
(18)
Using (18) in (17), we get:
X
E (G, S) =
ℓ (vi , vj ) pi pj +
(19)
(vi ,vj )∈S
(vi ,vj )∈S
V ′ ⊆V
(vi ,vj )∈S
X
(vi ,vj
)∈E ′′ \S
ℓ (vi , vj )
X
V
′ ⊆V
1{(vi ,vj )∈C[E ′ ]} Pr V ′
We now deal with the second term of (19) that, in this form seems to be exponential. Consider
some edge (vi , vj ) added during step 3 in order to “patch”, say connected components C l and Cl+1
of the anticipatory solution S. Since (vi , vj ) ∈
/ S, there exists in S a sequence µ = [vi , vj ] of
consecutive edges (or arcs) linking vi to vj . Assume that this sequence is listed by its vertices and
that neither vi , nor vj belong to µ. Edge (vi , vj ) ∈ E ′′ \ S ′ is added to S ′ just because all the vertices in µ are absent. In other words, inclusion (vi , vj ) ∈ C[E ′ ] holds for any subgraph G′ (V ′ , E ′ ),
where V ′ ∈ Uij′ with: Uij′ = {V ′ ⊆ V : vi ∈ V ′ , vj ∈ V ′ , any vertex of µ = [vi , vj ] is absent}.
Consequently, the second sum in the second term of (19) can be written as:
X
Y
X
(1 − pl )
(20)
1{(vi ,vj )∈C[E ′ ]} Pr V ′ =
Pr V ′ = pi pj
V ′ ⊆V
′
V ′ ∈Uij
vl ∈[vi ,vj ]
Combination of (17), (19) and (20) derives the expression claimed for the functional. It is easy
to see that computation of a single term in the second sum of the functional requires O(n)
computations (at most n + 1 multiplications). Since we may do this for at most O(n 2 ) times
(the edges in E), it ensues that the whole complexity of functional’s computation is of O(n 3 ),
that concludes the proof of the theorem.
The fact that E(G, S) is polynomial is partly due to the same reasons as in Theorems 1
and 4. Furthermore, the order of the additional edges-choices in step 3 of A is also crucial for this
efficience. Indeed, this order is such that one can say a priori under which conditions an edge
(or arc) (vi , vj ) will be added in S ′ . These conditions carry over, the presence or the absence of
the edges initially lying between vi and vj in S.
Unfortunately, in the opposite of Theorems 1 and 4, Theorem 5 does not derive a “good” characterization for the optimal anticipatory solutions of the problems meeting the assumptions (i)
to (iv). In particular, the form of the functional does not imply solution of some well-defined
weighted version of Π (the deterministic support of robust Π). Indeed due to the second term
of the expression for E(G, S) in Theorem 5, the “costs” assigned to the edges depend on the
structure of the anticipatory solution chosen.
In what follows, we outline some problems dealing with assumptions of Theorem 5. In
particular, we tackle cases where feasible solutions are either cycles or trees.
16
5.1
Application of Theorem 5 when the anticipatory solution is a cycle
In this section, we consider min tsp and its robust version. Given a complete graph on n vertices,
denoted by Kn , with positive distances on its edges, min tsp consists of minimizing the cost of a
Hamiltonian cycle (i.e., of an ordering hv1 , v2 , . . . , vn i of V such that vn v1 ∈ E and, for 1 6 i < n,
vi vi+1 ∈ E), the cost of such a cycle being the sum of the distances of its edges. We shall represent
any Hamiltonian
Pcycle T (called also a tour in what follows) as the set of its edges; its value
is m(Kn , T ) = e∈T ℓ(vi , vj ). Moreover, we arbitrarily number the vertices of Kn in the order
that they are visited in T ; so, we can set T = {(v1 , v2 ), . . . , (vi , vi+1 ), . . . , (vn−1 , vn ), (vn , v1 )}.
Consider an anticipatory tour T in an edge-valued complete graph K n and a set of absent
vertices. Then, application of step 2 of A may result in a set {P1 , P2 , . . . , Pk } of paths5 , ranged
in the order vertices have been visited in T , that is not feasible for min tsp in the surviving
graph. In order to render this set feasible, one can link (modulo k) the last vertex of the path P i
to the first vertex of Pi+1 ; this is always possible since the initial graph is complete.
v1
v2
v1
v8
v3
v7
v4
v6
v5
v2
v7
v5
(b) The tour T ′ computed
by A
(a) An anticipatory tour
Figure 2: An example of application of algorithm A for robust min tsp.
For example, in Figure 2(a), an anticipatory cycle T , derived from a (symmetric) K 8 is
shown. In Figure 2(b), we consider that vertices v3 , v4 , v6 and v8 are absent. In a first time,
application of Step 2 of A results in a path-set {{(v1 , v2 )}, {v5 }, {v7 }}. In a second time, we will
link vertex v2 to v5 (using dotted edge (v2 , v5 )) and vertex v5 to v7 (by dotted edge (v5 , v7 )).
This creates a Hamiltonian path linking all the surviving vertices of the initial K 8 . Finally, we
link vertex v7 to v1 (by the dotted edge (v7 , v1 )). We so build a new tour feasibly visiting all the
present vertices of the remaining graph.
It is easy to see that for the case we deal with, all conditions of Theorem 5 are verified.
Consequently, its application for the case of robust min tsp gives for E(K n , T ) the expression
claimed in the theorem. We so recover the result of [13] about robust min tsp. The anticipatory
solution minimizing the functional cannot be characterized tightly by means of Theorem 5, since
the expression for E(Kn , T ) depends on the particular anticipatory tour T considered and by
the way this particular tour will be completed in the surviving instance.
5.2
Application of Theorem 5 when the anticipatory solution is a tree
Let us now consider min spanning tree. Given an edge-valued graph G(V, E, ~ℓ),Pmin spanning
tree consists of determining a tree T spanning V and minimizing quantity
e∈T ℓ(e). For
reasons that will be understood later (see also [6]) we restrict ourselves to complete graphs. As
5
These paths may be sets of edges, or simple edges, or even isolated vertices, any such vertex considered as a
path.
17
previously inP
Section 5.1, we consider a tree by the set of its edges. For any tree T its value is
m(Kn , T ) = e∈T ℓ(vi , vj ).
Note that for robust min tsp in Section 5.1, its solution induces an implicit and natural
ordering of the edges. This is not the case here since various orderings can be considered. We
consider the one given by a kind of inverse depth-first-search (dfs) that goes as follows:
• starting from leaf numbered by 1 we first number its incident edge, then one of the edges
intersecting it and so on, until we arrive to a second leaf;
• we then return back to the last vertex of the path incident to an edge not numbered yet
and we continue the numbering from this edge until a third leaf is encountered;
• we continue in this way until all the edges of T are visited.
This ordering is performed in O(n) for a tree of order n (recall that such a tree has n − 1 edges).
4
4
3
5
2
6
1
13
14
8
3
9
7
10
6
12
1
9
7
10
11
14
8
12
′
(b) The solution T derived from application
of algorithm A on T
(a) The ordering of the nodes of an anticipatory solution T
Figure 3: When anticipatory solution is a tree.
For example, consider the tree of Figure 3(a) and assume that it is a minimum spanning tree
of some graph on 14 vertices. Starting from the leftmost leaf (numbered by 1) we visit edges
(1, 2), (2, 3), . . . , (6, 7) in this order; node 7 is another leaf. We then return to node 5 that is the
last node incident to an edge not yet visited, and we visit edges (5, 8), (8, 9)(9, 10), (10, 11). We
then return to node numbered by 9 and visit edge (9, 12). We return back to node numbered
by 4 and visit edges (4, 13), (13, 14) in this order. The ordering just obtained, partitions the
edges of the tree into a number of edge-disjoint paths P1 , P2 , . . . For instance, dealing with
Figure 3(a), the depicted tree T is partitioned into 4 paths: P1 = {(1, 2), (2, 3), . . . , (6, 7)},
P2 = {(5, 8), (8, 9)(9, 10), (10, 11)}, P3 = {(9, 12)} and P4 = {(4, 13), (13, 14)}.
Suppose now that some vertices are absent from the initial graph G. Also, let us refer to
the nodes by their numbering in the ordering just described. Then, step 2 of A will produce a
non connected set of edges (forming paths, any of them being a subset of some P i ); denote by
{P1′ , P2′ , . . . , Pk′ } the set of paths so-obtained. Order them paths following the order their edges
appear in the inverse dfs paths of T . For any l = 1, . . . , k, we link the last vertex, say i of path P l′
′ . Since the initial graph is assumed complete, such an
to the first vertex, say j, of the path Pl+1
edge always exists.
18
′ .
We have now to explicit the path [i, j] associated with the edge (i, j) connecting P l′ and Pl+1
Starting from T the edges of which are ordered as described, one can, w.l.o.g., renumber the
vertices of the graph in such a way that T can be rewritten as T = {(1, 2), (2, 3), . . . , (n − k, n)}.
Then, T can be seen as a sequence of vertices some of them appearing more than once, i.e.,
T = (11 , 21 , 31 , . . . , j1 , i2 , (j + 1)1 , . . . , (n − k)q , n1 ), where i < j and ic represents the c-th time
node i is encountered in T . Based upon this representation, one can reconstruct T in the following
way: for any pair (ic , jc′ ) of consecutive vertices, edge (i, j) belongs to T if and only if i < j.
Note that a leaf appears only once in the list.
Application of step 2 of A implies that, whenever an absent vertex v is not a leaf, its removal
disconnects T . In order to reconnect it, we must link the vertex preceding it in the sequence to
the one succeeding it. By the particular form of the sequence considered, any edge (i, j) that
algorithm A is likely to add in order to reconnect T is such that i < j; the corresponding path [i, j]
(i.e., the list of vertices that have to be absent in order that (i, j) is added), is the portion of
the list between il and j1 , where il is the last occurrence of i before the first occurrence j1 of
vertex j.
Let us revisit the example of Figure 3(a). The sequence associated with the tree is T =
(11 , . . . , 71 , 52 , 81 , . . . , 111 , 92 , 121 , 42 , 131 , 141 ). Assume now that vertices 2, 5, 11 and 13 disappear from the initial graph. Application of step 2 of A results in the subsequence T ′ =
(11 , 31 , 41 , 61 , 71 , 81 , 91 , 101 , 92 , 121 , 42 , 141 ). The first connected component (path) in this sequence is vertex 1 itself, and the second one is represented by nodes 3 and 4 (3 1 and 41 in T ′ );
we so add edge (1, 3) in order to connect these two components. The third component is represented by vertices 6 and 7 (61 and 71 in T ′ ); edge (4, 6) is consequently added. The fourth
connected component is induced by vertices 8, 9, 10 and 12 (81 , 91 , 101 , 92 , and 121 in T ′ );
hence, edge (7, 8) added. The fifth connected component in the sequence is vertex 4 itself (4 2
in T ′ ) but edge (12, 4) is not added because 12 > 4. Finally, the sixth connected component,
vertex 14 (141 in T ′ ), entails introduction of edge (4, 14) that completes the modification of T by
algorithm A. Figure 3(b), where slotted edges represent edges added during execution of step 3
of A, illustrates what has been just discussed.
Denoting also by T the sequence of vertices representing the anticipatory spanning tree T
and denoting any vertex of the initial graph by its numbering in the inverse dfs explained
above, E(Kn , T ) can be expressed as claimed by Theorem 5.
6
Final remarks
We have drawn a framework for the classification of robust problems under the a priori optimization paradigm. What seems to be of interest in this classification is that when restriction
of the initial solution to the “present” subgraph is feasible, then the complexity of determining the optimal anticipatory solution for the problems tackled, amounts to the complexity of
solving some weighted version of the deterministic problem, where the weights depend on the
vertex-probabilities. These weights do not depend on particular characteristics of the anticipatory solution considered, thing that allows a compact characterization of optimal anticipatory
solution. On the contrary, when more-than-one-stage algorithms are needed for building solutions, then the observation above is no more valid. In this case, one also recovers some weighted
version of the original problem, but the weights on the data cannot be assigned independently
of the structure of a particular anticipatory solution.
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