Level of Repair Analysis and Minimum Cost
Homomorphisms of Graphs
Gregory Gutin1 , Arash Rafiey1 , Anders Yeo1 , and Michael Tso2
1
2
Department of Computer Science
Royal Holloway University of London
Egham, Surrey TW20 OEX, UK
gutin,arash,anders@cs.rhul.ac.uk
School of Mathematics, University of Manchester
P.O. Box 88, Manchester M60 1QD, UK
mike.tso@manchester.ac.uk
This paper is dedicated to the memory of Lillian Barros
Abstract. Level of Repair Analysis (LORA) is a prescribed procedure for defence logistics
support planning. For a complex engineering system containing perhaps thousands of assemblies, sub-assemblies, components, etc. organized into several levels of indenture and with a
number of possible repair decisions, LORA seeks to determine an optimal provision of repair
and maintenance facilities to minimize overall life-cycle costs. For a LORA problem with two
levels of indenture with three possible repair decisions, which is of interest in UK and US
military and which we call LORA-BR, Barros (1998) and Barros and Riley (2001) developed
certain branch-and-bound heuristics. The surprising result of this paper is that LORA-BR
is, in fact, polynomial-time solvable. To obtain this result, we formulate the general LORA
problem as an optimization homomorphism problem on bipartite graphs, and reduce a generalization of LORA-BR, LORA-M, to the maximum weight independent set problem on a
bipartite graph. We prove that the general LORA problem is NP-hard by using an important
result on list homomorphisms of graphs. We introduce the minimum cost graph homomorphism problem, provide partial results and pose an open problem. Finally, we show that
our result for LORA-BR can be applied to prove that an extension of the maximum weight
independent set problem on bipartite graphs is polynomial time solvable.
Keywords: Computational Logistics; Level of Repair Analysis; Independent Sets in Graphs;
Homomorphisms of Graphs
1
Introduction
Level of Repair Analysis (LORA) is a prescribed procedure for defence logistics support
planning (see, e.g., Crabtree and Sandel [10] and the website of the UK MoD Acquisition
Management System at www.ams.mod.uk/ams). For a complex engineering system containing perhaps thousands of assemblies, sub-assemblies, components etc. organized into
ℓ ≥ 2 levels of indenture and with r ≥ 2 possible repair decisions, LORA seeks to determine an optimal provision of repair and maintenance facilities to minimize overall life-cycle
costs.
Barros [4] and Barros and Riley [6] provide a generic integer programming formulation
of the LORA optimization problem for systems with ℓ levels of indenture and r possible repair decisions (including the non-repair option). A special case with ℓ = 2 and r = 3, which
we call LORA-BR, is of particular importance because it corresponds to recommendations
in certain UK and US military standard handbooks, see Barros and Riley [6]. In French
military standards, ℓ = 2 and r = 5. Notice that the actual research of Barros and Riley
was only for LORA-BR [5] for which the corresponding software have been developed.
While Barros [4] solves LORA-BR using a general purpose IP solver, Barros and Riley
[6] outline a specialized branch-and-bound heuristic, which appears to be more efficient
in computational experiments. Their heuristic is based on a relaxation of LORA-BR into
a pair of uncapacitated facility location (UFLP) problems [9, 14]. A branch-and-bound
procedure then employs local search heuristics to satisfy additional side constraints ensuring
consistency between repair decisions for pairs of items nested on adjacent indenture levels.
Since UFLP is NP–hard [9, 13, 14], it could be expected that LORA-BR would also be
intractable. However, the surprising result of this paper is that LORA-BR is polynomially
solvable and this is achieved by reducing its generalization, LORA-M (defined in Section
3), to the maximum weight independent set problem on a bipartite graph.
As it was pointed out above, the case of two levels of indenture is of particular interest
(e.g., in UK, USA and French military). For clarity of exposition, in the rest of this paper
apart from Section 4, we restrict ourselves to two levels of indenture, ℓ = 2, but our
approach can be extended to arbitrary ℓ as demonstrated in Section 4.
We will use the notion of a homomorphism of graphs that generalizes the notion of
coloring (see, e.g., Hell and Nesetril [16]). For a pair of graphs H = (V (H), E(H)) and
B = (V (B), E(B)), a mapping k : V (B)→V (H) such that if xy ∈ E(B) then k(x)k(y) ∈
E(H) is called a homomorphism of B to H. To study the LORA problem, we show how
to formulate it as a problem of finding a homomorphism of minimum cost belonging to a
certain class of homomorphisms of a bipartite graph to a fixed bipartite graph. This allows
us to use a nontrivial result on the list H-homomorphism problem from [11] to easily show
that the general LORA problem with ℓ = 2 is NP–hard. We also prove that LORA-M is
polynomial time solvable.
The formulation of the LORA problem in terms of special homomorphisms leads us
to the introduction of the minimum cost H-homomorphism problem (MCHP): For a fixed
graph H and an input graph G given together with costs cz (u), the cost of mapping a
vertex u ∈ V (G) to z ∈ V (H), verify whether there is a P
homomorphism of G to H, and if
one exists, find such a homomorphism k that minimizes u∈V (G) ck(u) (u). MCHP extends
the well-studied list H-homomorphism problem [16]. We use our results for the LORA
problem to obtain the corresponding results for MCHP. In particular, we show that if H is
a bipartite graph with the complement being an interval graph, then MCHP is polynomial
time solvable. In contrast, if H is not bipartite with the complement being a circular arc
graph, then MCHP is NP–hard.
We also use our results to show that the bipartite case of the critical independent set
problem (defined in Section 6), which generalizes the maximum weight independent set
problem, is polynomial time solvable.
In this paper, all graphs are finite, undirected, and simple (i.e., without loops or multiple
edges). For standard graph-theoretical terminology and notation, see, e.g., Asratian, Denley
and Haggkvist [3] or West [18]. For terminology and results on homomorphisms, see Hell
and Nesetril [16].
The rest of the paper is organized as follows. In Section 2, we provide formulations of
LORA-BR and the general LORA problem with ℓ = 2 in terms of graph homomorphisms.
We prove that the general LORA problem with ℓ = 2 is NP–hard. In Section 3, we show
how to solve a generalization of LORA-BR, LORA-M with ℓ = 2, in polynomial time. In
Section 4, we extend the general LORA problem with ℓ = 2 to the general LORA problem
with arbitrary ℓ ≥ 2 as well extend the main result of Section 3. In Section 5, we introduce
the minimum cost H-homomorphism problem and show that the results of Sections 2 and
3 can be easily extended to it. In the end of the section, we pose an open problem. Finally,
in Section 6 we apply a result from Section 3 to solve the bipartite case of the critical
independent set problem in polynomial time.
2
LORA-BR and General LORA with ℓ = 2
Consider first a special case of LORA with ℓ = 2 and r = 3 following Barros [4] and
Barros and Riley [6] (we will call this special case LORA-BR). We refer to the first level of
indenture in LORA-BR as subsystems s ∈ S and the second level of indenture as modules
m ∈ M. The distribution of modules in subsystems can be given by a bipartite graph
G = (V1 , V2 ; E) with partite sets V1 = S and V2 = M . For arbitrary s ∈ V1 and m ∈ V2 ,
sm ∈ E if and only if module m is in subsystem s. We consider G to be an arbitrary
bipartite graph and denote its vertex set V (V = V1 ∪ V2 ).
There are r = 3 available repair decisions for each level of indenture: ”discard”, ”local
repair” and ”central repair”, labelled respectively D, L, C (subsystems) and d, l, c (modules). To be able to use a decision z ∈ {D, L, C, d, l, c}, we have to pay a fixed cost cz .
Assume also known additive costs (over a system life-cycle) cz (u) of prescribing repair
decision z for subsystem or module u.
We wish to minimize the total cost of choosing a subset of the six repair decisions and
assigning available repair options to the subsystems and modules subject to the following
constraints:
If a module m occurs in subsystem s (i.e., sm ∈ E) we impose the following logical
restrictions on the repair decisions for the pair (s, m) motivated through practical considerations:
R1 : Ds ⇒ dm , R2 : lm ⇒ Ls ,
where Ds , dm denote the decisions to discard subsystem s, module m, respectively, etc.
Notice that even though module m may be common to several subsystems we are required
to prescribe a unique repair decision for that module.
R1 has the interpretation that a decision to discard subsystem s necessarily entails discarding all enclosed modules. R2 is a consequence of R1 and a policy of “no backshipment”
which rules out the local repair option for any module enclosed in a subsystem which is
sent for central repair [6].
Let FBR = (Z1 , Z2 ; T ) be a bipartite graph with partite sets Z1 = {D, C, L} (subsystem repair options) and Z2 = {d, c, l} (module repair options) and with edges T =
{Dd, Cd, Cc, Ld, Lc, Ll}. Let Z = Z1 ∪ Z2 . Observe that any homomorphism k of G to
FBR such that k(V1 ) ⊆ Z1 and k(V2 ) ⊆ Z2 satisfies the rules R1 and R2 . Indeed, let u ∈ V1 ,
v ∈ V2 , uv ∈ E. If k(u) = D then k(v) = d, and if k(v) = l then k(u) = L.
Let Li ⊆ Zi , i = 1, 2. We call a homomorphism k of G to FBR an (L1 , L2 )-homomorphism
of G to FBR if k(u) ∈ Li for each u ∈ Vi , i = 1, 2. Now LORA-BR can be formulated as
the following graph-theoretical problem: We are given a bipartite graph G = (V1 , V2 ; E),
V = V1 ∪ V2 , and we consider homomorphisms k of G to FBR . (If no homomorphisms of
G to FBR , then the problem has no feasible solution.) Mapping of u ∈ V to z ∈ Z (i.e.,
k(u) = z) incurs a real cost cz (u). The use of a vertex z ∈ Z in a homomorphism k (i.e.,
k −1 (z) 6= ∅) incurs a real cost cz . We wish to choose subsets Li ⊆ Zi , i = 1, 2, and find an
(L1 , L2 )-homomorphism k of G to FBR that minimize
X
u∈V
ck(u) (u) +
X
cz .
(1)
z∈L1 ∪L2
We call the expression in (1) the cost of k.
The graph-theoretical formulation of LORA-BR can be naturally extended as follows:
The above problem with FBR replaced by an arbitrary fixed bipartite graph F = (Z1 , Z2 ; T )
is called the general LORA problem with ℓ = 2. Let Z = Z1 ∪ Z2 . Notice that the general
LORA problem with ℓ = 2 extends the generic formulation of the LORA problem with ℓ = 2
given in [6]. The formulation of the general LORA problem (with arbitrary ℓ) provided in
Section 4 extends the generic formulation of the LORA problem (with arbitrary ℓ) given
in [6].
To prove that the general LORA problem with ℓ = 2 is NP–hard, we will use an
important result on the list H-homomorphism problem defined below. Suppose that we
are given a pair of graphs H and B and a list Λ(v) ⊆ V (H) for each v ∈ V (B). A
homomorphism f : V (B)→V (H) such that f (v) ∈ Λ(v) for each v ∈ V (B) is called a
Λ-homomorphism. For a fixed H, the list H-homomorphism problem asks whether there
exists a Λ-homomorphism f of B to H for an input graph B with lists Λ.
A graph P = (V (P ), E(P )) is a circular arc graph if there is a family of arcs Av ,
v ∈ V (P ), on a fixed circle, such that xy ∈ E(P ) if and only if Ax and Ay intersect. Feder,
Hell and Huang [11] obtained the following important result.
Theorem 1. If H is a bipartite graph with the complement being a circular arc graph, then
the list H-homomorphism problem is polynomial time solvable. Otherwise, the problem is
NP–complete.
Observe that, if H is bipartite, we may restrict inputs B of the list H-homomorphism
problem to bipartite graphs since there is no homomorphism of a non-bipartite graph to H.
Brightwell [7] found the first proof that the general LORA problem with ℓ = 2 is NP–hard.
Since his proof does not use Theorem 1, our proof turns out to be shorter and it covers
much wider family of graphs than that of Brightwell.
Theorem 2. The general LORA problem with ℓ = 2 and with each cz = 0, and each cost
cz (u) in {0, 1} is NP–hard provided the complement of F is not a circular arc graph.
Proof: Let F be a bipartite graph and assume that the complement of F is not a circular
arc graph (see Theorem 1). Let a bipartite graph G and lists Λ be an input of the list
F -homomorphism problem. Define costs cz (u) for each z ∈ V (F ) and u ∈ V (G) as follows:
cz (u) = 0 if z ∈ Λ(u) and cz (u) = 1, otherwise. We put cz = 0 for each z ∈ V (F ). In other
words, the use of each vertex z ∈ V (F ) in homomorphisms of G to H is free. In this case,
in the general LORA problem with ℓ = 2, we can always put L1 ∪ L2 = V (F ).
Let G1 , G2 , . . . , Gg be components of G and let F1 , F2 , . . . , Ff be components of F .
Let Z1j , Z2j be partite sets of Fj for every j = 1, 2, . . . , f. Observe that there exists a
Λ-homomorphism of G to F if and only if for each i = 1, 2, . . . , g there is a j(i) ∈
{1, 2, . . . , f } such that there exists a Λ-homomorphism of Gi to Fj(i) . However, there is
j(i)
j(i)
a Λ-homomorphism of Gi to Fj(i) if and only if the minimum cost of either a (Z1 , Z2 )j(i)
j(i)
homomorphism of Gi to Fj(i) or a (Z2 , Z1 )-homomorphism of Gi to Fj(i) is equal to
0 (with the costs defined above). Thus, we have a polynomial time Turing-reduction [13]
from the NP–complete list H-homomorphism problem to the general LORA problem with
ℓ = 2. Hence, by the definition of the NP–hardness (see Section 5.1 in [13]), the general
LORA problem with ℓ = 2 is NP–hard.
⊓
⊔
It is well-known [15] (see also [16]) that for a fixed graph H, the problem to verify
whether there exists a homomorphism of an input graph G into H is NP–complete if H is
non-bipartite and polynomial time solvable if H is bipartite. Thus, the obvious extension
of the general LORA problem to non-bipartite graphs F is NP–hard.
3
LORA-M with ℓ = 2
Let B = (W1 , W2 ; E) be a bipartite graph. For a vertex z ∈ W1 ∪ W2 , let N (z) be the
set of vertices adjacent to z. Orderings x1 , x2 , . . . , x|W1 | and y1 , y2 , . . . , y|W2 | of vertices of
W1 and W2 , respectively, are called monotone if N (xi ) ⊆ N (xi+1 ) and N (yj ) ⊆ N (yj+1 )
for each i = 1, 2, . . . , |W1 | − 1 and j = 1, 2, . . . , |W2 | − 1. A bipartite graph B is called
monotone if it has monotone orderings of its partite sets. Observe that if x1 , x2 , . . . , x|W1 |
and y1 , y2 , . . . , y|W2 | are monotone orderings, then xp yq ∈ E implies that xs yt ∈ E for each
s ≥ p and t ≥ q.
Notice that the bipartite graph FBR corresponding to the rules R1 and R2 of LORA-BR
is monotone (consider orderings D, C, L and l, c, d), so are the bipartite graphs corresponding to R1 and R2 separately (there might be a situation when one of the rules is not used).
Interestingly, monotone bipartite graphs form a family of so-called convex bipartite graphs;
several families of convex bipartite graphs have been found useful in various applications,
see [3].
Let B = (W1 , W2 ; E) be a bipartite graph, let n = |W1 | + |W2 | and let m = |E|.
One can test whether B is monotone in time O(m + n) as follows. Order vertices of W1
and W2 separately according to their degrees deg(z), x1 , x2 , . . . , x|W1 | and y1 , y2 , . . . , y|W2 | ,
such that deg(xi ) ≤ deg(xi+1 ) and deg(yj ) ≤ deg(yj+1 ) for each i = 1, 2, . . . , |W1 | − 1
and j = 1, 2, . . . , |W2 | − 1. Observe that B is monotone if and only if these orderings are
monotone. We can use counting sort (see Chapter 9 of [8]) to get the orderings according
to degrees in time O(n). The remaining computations can be carried out in time O(m).
The general LORA problem restricted to fixed monotone bipartite graphs F = (Z1 , Z2 ; T )
is called LORA-M. We assume that we have monotone orderings x1 , x2 , . . . , x|Z1 | and
y1 , y2 , . . . , y|Z2 | of Z1 and Z2 , respectively.
We reduce LORA-M to the maximal weight independent set problem on bipartite
graphs. Recall that a vertex set I of a graph is independent if there is no edge between
vertices of I.
In the next theorem, we will consider a bipartite graph B with partite sets W1 , W2 and
nonnegative vertex weights p(u), u ∈ V (B), and the following (s, t)-network N (B): add
new vertices s and t to B, append all arcs su of capacity p(u), vt of capacity p(v) for all
u ∈ W1 and v ∈ W2 , and orient every edge xy of B, where x ∈ W1 , from x to y (these arcs
are of capacity ∞). For results on flows and cuts in networks see [8].
Theorem 3. If (S, T ) is a minimum cut in N (B), s ∈ S, then (S ∩ W1 ) ∪ (T ∩ W2 ) is a
maximum weight independent set in B. One can find a maximum weight independent set
√
in B in time O(n21 m + n1 m), where n1 = |U1 | and m = |E(B)|.
The structural part of Theorem 3 is well-known, cf. Frahling and Faigle [12] (a similar
result is described in [17]). The complexity claim follows from the fact that one can find a
√
minimum cut in N (B) in time O(n21 m + n1 m) by first finding a maximum flow by the
bipartite preflow-push algorithm of Ahuja et al. [2] and then finding a minimum cut (e.g.,
by finding vertices reachable from s in the residual network using depth-first search).
Let us return to LORA-M and formulate it as a maximization problem. Choose sets
Li ⊆ Zi , i = 1, 2. Let u ∈ Vi and set lists Λ(u) = Li , i = 1, 2. Recall that x1 , x2 , . . . , x|Z1 | and
y1 , y2 , . . . , y|Z2 | are monotone orderings of Z1 and Z2 . Assume that u ∈ V1 , xp , xq ∈ Λ(u),
p < q and cxp (u) > cxq (u). Observe that since cxp (u) > cxq (u) and F is monotone, an
optimal (L1 , L2 )-homomorphism k will not map u to xp . Thus, we may reduce the list Λ(u)
of possible images of u by deleting xp . Certainly, we may reduce all Λ(v), v ∈ V1 , such that
if xr , xs ∈ Λ(v) and r < s, then cxr (v) ≤ cxs (v). We call such a list Λ(v) reduced. Similarly,
one defines the reduced list of a vertex in V2 .
For a vertex u ∈ V , we can get the reduced list Λ(u) in time O(1) by the following simple
procedure (the running time is constant since F is fixed). To simplicity the description,
assume that u ∈ V1 . The input is Λ(u) := L1 = {xp(1) , xp(2) , . . . , xp(t) }, p(1) < p(2) < · · · <
p(t). We start from xp(t) . We compare cxp(t) (u) with cxp(t−1) (u), cxp(t−2) (u),. . . and find the
maximal i such that cxp(i) (u) ≤ cxp(t) (u). We delete from Λ(u) all xp(i+1) , xp(i+2) , . . . , xp(t−1) .
We compare cxp(i) (u) with cxp(i−1) (u), cxp(i−2) (u),. . . and continue as above. Thus, we can
obtain the reduced lists Λ(v), v ∈ V , in time O(|V |).
In the reminder of this section, we will use the following notation for the reduced lists:
Λ(u) = {zp(1) , zp(2) , . . . , zp(|Λ(u)|) }, where p(1) < p(2) < · · · < p(|Λ(u)|) and z = x if u ∈ V1
and z = y, otherwise.
Recall that a homomorphism k of G to F is a Λ-homomorphism if k(u) ∈ Λ(u) for
each u ∈ V. Observe that LORA-M is equivalent to the problem of choosing sets Li ⊆ Zi ,
i = 1, 2 and finding a Λ-homomorphism k of G to F that minimize the cost of k, where
Λ(u) is the reduced list for u ∈ V.
Now we replace the costs by weights. Let M be the maximum of all costs in LORA-M
(i.e., cz (u)’s and cz ’s). For each pair of vertices z ∈ Zi and u ∈ Vi , i = 1, 2, let wz (u) =
M − cz (u) and for each vertex z ∈ Z let wz = M − cz . Notice that, by the definition, all
the weights are nonnegative. Let k be a Λ-homomorphism of G to F. The weight of k is
defined as
X
X
wz .
(2)
wk(u) (u) +
u∈V
z∈L1 ∪L2
Observe that LORA-M is equivalent to the problem of choosing sets Li ⊆ Zi , i = 1, 2 and
finding a Λ-homomorphism k of G to F that maximize the weight of k, where Λ(u) is the
reduced list for u ∈ V.
We now prove the following main result of the paper.
Theorem 4. For fixed subsets Li , i = 1, 2, LORA-M with ℓ = 2 can be solved in time
√
O(n21 m + n1 m + n), where n1 = |V1 |, n = |V | and m = |E|.
Proof: Recall that all our graphs have no loops. If F is edgeless, then there is no homomorphism of G to F. Thus, we may assume that x|U1 | y|U2 | ∈ T. Since Li , i = 1, 2, are fixed,
for simplicity, we will assume that all weights wij = 0 in (2). Let Λ(u) be the reduced list
for each u ∈ V (we have shown how to find these lists in time O(n)).
Let W
P be a constant larger than max{wj (u) : u ∈ V, j ∈ Λ(u)}. Construct a new graph
H with u∈V |Λ(u)| vertices:
V (H) = {uz : u ∈ V, z ∈ Λ(u)}.
Let an edge ux vy be in H if uv ∈ E and xy 6∈ T . Let u ∈ V . For every j ∈ {1, 2, . . . , |Λ(u)|},
let the weight w(uzp(j) ) be equal to wzp(j) (u) + W , if j = |Λ(u)|, and equal to wzp(j) (u) −
wzp(j+1) (u), otherwise. Since each list Λ(u) is reduced, the weights of the vertices of H are
nonnegative.
Clearly, if we replace, in G, a vertex u ∈ V by |Λ(u)| independent copies such that
there is an edge between a copy of u and a copy of v if and only if uv ∈ E, then we obtain
a supergraph G∗ of H. Since G is bipartite, so is G∗ and, thus, H.
Observe that, by monotonicity of F , if uxp(i) , uxp(j) , vyp(f ) , vyp(g) are vertices of H, j ≥ i,
g ≥ f and uxp(i) vyp(f ) 6∈ E(H), then uxp(j) vyp(g) 6∈ E(H) as well. We call this property of H
index-antimonotonicity.
Assume that there exists a Λ-homomorphism k of G to F . Let k(u) = zp(iu ) . Then the
set {uzp(iu ) : u ∈ V } is independent in H. Moreover, by index-antimonotonicity of H,
S = ∪u∈V {uzp(j) : iu ≤ j ≤ |Λ(u)|}
(3)
is an independent set in H. Observe that S contains S ′ = {uzp(Λ(u)) : u ∈ V } and the
weight of S is equal to that of the homomorphism plus W × |V | (we use telescopic sums).
Assume that a maximum weight independent set S in H contains S ′ . Then map each
u ∈ V to k(u) = zp(iu ) such that iu = min{j : uzp(j) ∈ S}. By maximality, S is of the
form (3) or, due to index-antimonotonicity of H, S may be extended to (3) by adding some
vertices of zero weight. Observe that the weight of S is equal to that of the homomorphism
plus W × |V |. If a maximum weight independent set S in H does not contain S ′ , then S ′
is not an independent set in H (since the weight of S ′ is larger than the weight of S) and,
thus, there is no Λ-homomorphism of G to F .
Thus, there is an Λ-homomorphism of G to F if and only if a maximum weight independent set in H contains S ′ . If there is an Λ-homomorphism of G to F , then this
homomorphism corresponds to a maximum weight independent set S in H. It remains to
observe that we may apply Theorem 3 to find a maximum weight independent set of H. ⊓
⊔
There are less than a = 2|Z1 |+|Z2 | choices of nonempty L1 and L2 . Since F is fixed, a is
a constant. Thus, we obtain the following:
√
Theorem 5. LORA-M with ℓ = 2 can be solved in time O(n21 m + n1 m + n), where n1 ,
n and m are defined in Theorem 4.
4
General LORA Problem and LORA-M
Let ℓ ≥ 2 be a constant. An ℓ-partition X1 , X2 , . . . , Xℓ of a set X is a collection of subsets
of X such that Xi ∩ Xj = ∅ for each i 6= j and X1 ∪ X2 ∪ · · · ∪ Xℓ = X. An ℓ-partition
X1 , X2 , . . . , Xℓ of the vertex set X of a graph H is called layered if, for each edge xy
of H, there exists an index i such that one vertex of xy is in Xi and the other is in
Xi+1 . Observe that a graph H with a layered ℓ-partition is bipartite with partite sets
∪{Xi : 1 ≤ i ≤ ℓ, i ≡ 1 (mod 2)} and ∪{Xi : 1 ≤ i ≤ ℓ, i ≡ 0 (mod 2)}.
Let G = (V, E) be a graph with a layered ℓ-partition V1 , V2 , . . . , Vℓ of V . Let F = (U, T )
be a fixed graph with a layered ℓ-partition U1 , U2 , . . . , Uℓ of U . Let Li ⊆ Ui , i = 1, 2, . . . , ℓ.
We call a homomorphism k of G to F an (L1 , L2 , . . . , Lℓ )-homomorphism of G to H if
k(u) ∈ Li for each u ∈ Vi , i = 1, 2, . . . , ℓ.
We formulate the general LORA problem as follows: We are given a graph G as above
and we consider homomorphisms k of G to F . Mapping u ∈ V to z ∈ U (i.e., k(u) = z)
incurs a real cost cz (u). The use of a vertex z ∈ U in a homomorphism k (i.e., k −1 (z) 6= ∅)
incurs a real cost cz . We wish to choose subsets Li ⊆ Ui , i = 1, 2, . . . , ℓ, and find an
(L1 , L2 , . . . , Lℓ )-homomorphism k of G to F that minimizes
X
X
ck(u) (u) +
cz ,
(4)
u∈V
z∈L
where L = ∪ℓi=1 Li . Notice that the graph F is fixed and is not part of the input.
By Theorem 2, the general LORA problem is NP–hard (even the general LORA problem
in which all costs cz (u) = 0 for u ∈ Vi , i ≥ 3, is NP–hard). To define (the general) LORA-M
for ℓ ≥ 2, let us define ℓ-monotone graphs. Let F = (U, T ) be a fixed graph with a layered
i
ℓ-partition U1 , U2 , . . . , Uℓ ; F is called ℓ-monotone if there is an ordering z1i , z2i , . . . , z|U
of
i|
vertices of Ui for each i = 1, 2, . . . , ℓ such that the subgraph F [Uj ∪ Uj+1 ] of F induced
j
j+1
by Uj ∪ Uj+1 is monotone with z1j , z2j , . . . , z|U
and z1j+1 , z2j+1 , . . . , z|U
being monotone
j|
j+1 |
orderings for each j = 1, 2, . . . , ℓ − 1. LORA-M is the general LORA problem with F being
ℓ-monotone. Similarly to Theorem 5, one can prove the following:
√
Theorem 6. LORA-M with fixed ℓ ≥ 2 can be solved in time O(n21 m + n1 m + n), where
n1 is the number of vertices in the smaller partite set of input graph G, n = |V (G)| and
m = |E(G)|.
5
Minimum Cost H-Homomorphism Problem
This paper provides a motivation to study the following minimum cost H-homomorphism
problem (MCHP): For a fixed graph H and an input graph G given together with costs
cz (u), the cost of mapping a vertex u ∈ V (G) to z ∈ V (H), verify whether there is a
homomorphism
of G to H, and if one exists, find such a homomorphism k that minimizes
P
u∈V (G) ck(u) (u).
An argument similar to that in the proof of Theorem 2 shows that MCHP problem
generalizes the list H-homomorphism problem and that if H is not bipartite with the
complement being circular arc graph, then MCHP is NP–hard.
Theorem 7. If H = (U1 , U2 , ; T ) is a monotone bipartite graph, then MCHP can be solved
√
in time O(n2 m + nm + n), where n is the number of vertices in the input graph G and
m is the number of edges in G.
√
Proof: Let t(n, m) = O(n2 m + nm + n). Since H is bipartite (and loopless), if there is
a homomorphism of G to H, then G is bipartite. So we may assume that G = (V1 , V2 ; E)
is bipartite.
Assume that G and H are connected. Then for each homomorphism k of G to H,
we have either k(Vi ) ⊆ Ui or k(Vi ) ⊆ U3−i for every i = 1, 2. Thus, to find an optimal
homomorphism of G to H, it suffices to compute an optimal (U1 , U2 )-homomorphism and
optimal (U2 , U1 )-homomorphism and compare their costs. By Theorem 4, the total running
time for finding the two optimal homomorphisms is t(n, m).
If H is disconnected, then by the definition of monotonicity, H consists of isolated
vertices and at most one component H ′ , which is not an isolated vertex. The case when all
components of H are isolated vertices is trivial, so we may assume that H ′ does exist.
Assume that G consists of components G1 , G2 , . . . , Gb . Observe that every homomorphism k of G to H consists of b ’independent’ homomorphisms ki : Gi →H. In fact, if Gi has
more than one vertex that ki maps Gi into H ′ and, by the above, we can find an optimal
homomorphism of Gi to H ′ in time t(ni , mi ), where ni = |V (Gi )| and mi = |E(Gi )|. If Gi
is a vertex v, ki may map it to any vertex of H and, in an optimal ki it maps Gi into z with
minimum cz (u), z ∈ U1 ∪ U2 . The running time to
such a vertex z is t(1, 0) = O(1).
Pfind
b
To complete our proof, it suffices to observe that i=1 t(ni , mi ) = t(n, m).
⊓
⊔
The following theorem allows us to relate the NP-hardness and polynomial solvable
cases above. Recall that a graph P = (V (P ), E(P )) is an interval graph if there is a family
of intervals Iv , v ∈ V (P ), of the real line, such that xy ∈ E(P ) if and only if Ix and Iy
intersect. The clique covering number of a graph B is the minimum number of complete
subgraphs of B covering V (B).
Theorem 8. A graph H is a monotone bipartite graph if and only if its complement H̄ is
an interval graph with clique covering number two.
Proof: First assume that H is a monotone bipartite graph with partite sets {v1 , v2 , . . . , vk }
and {w1 , w2 , . . . , wl }. By the definition of a bipartite monotone graph we may assume that
vi wj ∈ E(H) implies that vi′ wj ′ ∈ E(H) for all i′ ≥ i and j ′ ≥ j. Let m(j) be defined as
the least index such that vm(j) wj ∈ E(H). Now consider the following intervals:
si = [i, k + 1]
tj = [0, m(j) − 12 ]
for all i = 1, 2, . . . , k
for all j = 1, 2, . . . , l
Let B be the interval graph obtained from the above intervals, such that V (B) = S ∪ T ,
where S = {s1 , s2 , . . . , sk } and T = {t1 , t2 , . . . , tl } and there is an edge between two vertices
if and only if the corresponding intervals intersect. Note that both S and T form a clique
in B. Furthermore si tj ∈ E(B) if and only if i < m(j), which happens if and only if
vi wj 6∈ E(H). Therefore B = H̄, and we have completed one direction.
So assume that H̄ is an interval graph with clique covering number two. Let [si , ti ],
i = 1, 2, . . . , k, denote the intervals corresponding to one of the cliques in the clique cover
of size two and let [s′i , t′i ], i = 1, 2, . . . , l, denote the intervals corresponding to the other
clique in the clique cover. Let T denote the minimum value of all ti and let T ′ denote the
minimum value of all t′i . Without loss of generality we may assume that T ≤ T ′ . Again
without loss of generality we may assume that t1 ≥ t2 ≥ . . . ≥ tk and s′1 ≤ s′2 ≤ . . . ≤ s′l .
Assume that [si , ti ] and [s′j , t′j ] do not intersect. Suppose that t′j < si , which implies
that tk < si contradicting the fact that [sk , tk ] and [si , ti ] intersect. Therefore we must have
ti < s′j , which implies that [sa , ta ] and [s′b , t′b ] do not intersect for any a ≥ i and b ≥ j.
Therefore H̄ is the complement of a monotone bipartite graph.
⊓
⊔
The last two theorems imply the following:
Theorem 9. If H is a bipartite graph and its complement is an interval graph, then MCHP
√
can be solved in time O(n2 m + nm + n), where n is the number of vertices in an input
graph G and m is the number of edges in G.
Let P5 be the path with 5 vertices. The graph P5 is not a monotone bipartite graph,
but its complement is a circular arc graph. Thus, there remains a gap between the set
of graphs H for which we showed that the problem is NP-hard and for which we proved
that it is tractable. It would be interesting to close the gap. We considered some directed
extension of the 2-SAT approach of [11], but they did not appear to be useful.
6
LORA-BR and Critical Independent Set Problem
Let Q be an arbitrary graph. For a set X ⊆ V (Q), let N (X) = ∪x∈X {y ∈ V (Q) : xy ∈
E(Q)}. Let p, q be a pair of functions from V (Q) to the set of nonnegative reals. In the
critical independent set problem (CISP) we seek A maximizing
X
X
{
p(a) −
q(c) : A is an independent vertex set in Q}.
a∈A
c∈N (A)
Clearly, CISP is NP–hard as the maximum weight independent set problem on arbitrary
graphs is CISP with q(u) = 0 for each u ∈ V (Q). Ageev [1] proved that CISP is polynomial
time solvable if p(u) = q(u) for each u ∈ V (Q). This generalized the corresponding result
of Zhang [19] for p(u) = q(u) = 1 for each u ∈ V (Q). We will show that CISP can be solved
in polynomial time on bipartite graphs for arbitrary functions p and q.
Theorem 10. CISP on a bipartite graph G = (V1 , V2 ; E), V = V1 ∪ V2 , can be solved in
√
time O(n21 m + n1 m + n), where n1 = |V1 |, n = |V | and m = |E|.
Proof: Observe that LORA-BR with fixed lists L1 = V1 , L2 = V2 may be reformulated
as follows: Given a bipartite graph G = (V1 , V2 , E) and three weights wi (v), i = 1, 2, 3,
for each vertex v ∈ V , we color every vertex of G in one of the colors 1,2,3 such that if
a vertex is colored 1, then all its neighbors must be colored 3. Assigning a color i to a
vertex v contributes weight wi (v) to the total weight of the coloring. We seek a coloring of
maximum total weight.
Observe that if w1 (u) < w2 (u) for some u ∈ V, then there is an optimal coloring for
which u is not colored 1. Thus, we may set w1 (u) := w2 (u) and keep a record, say (u, 1, 2),
that indicates that if, in an optimal coloring that we found u is colored 1, we recolor it 2.
Similar arguments allow us to assume that w1 (u) ≥ w2 (u) ≥ w3 (u) for each u ∈ V.
Consider an optimal coloring, in which A is the set of vertices assigned color 1. Then A is
independent, all vertices of N (A) must have color 3 and all vertices of B = V (G)−A−N (A)
may have color 2. The total weight of the coloring is
X
X
X
X
X
X
w1 (a) +
w3 (c) +
w2 (b) =
w2 (d) −
w2,3 (c) +
w1,2 (a),
a∈A
c∈N (A)
b∈B
d∈V
c∈N (A)
a∈A
where w2,3 (c) = w2 (c) − w3 (c), w1,2 (a) = w1 (a) − w2 (a).
Choose weight functions w1 , w2 , w
3 as follows: w1 (u) = p(u) + q(u), w2 (u) = q(u),
P
w3 (u) = 0 for each u ∈ V (G). Since d∈V w2 (d) is a constant, we observe that CISP on
G (and functions p and q) can be reduced to LORA-BR with fixed L1 = V1 , L2 = V2 . It
remains to apply Theorem 4.
⊓
⊔
Acknowledgements We’d like to thank Graham Brightwell, David Cohen and Martin
Green for valuable discussions on the topic of the paper. Research of the first three authors
was partially supported by the Leverhulme Trust. Research of Gutin and Rafiey was supported in part by the IST Programme of the European Community, under the PASCAL
Network of Excellence, IST-2002-506778.
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