Transcript Some applications of graph theory, combinatorics and

Gregory Gutin
Royal Holloway, U. London, UK
and U. Haifa, Israel
Introduction to the min cost
homomorphism problem
for undirected and directed
graphs
Homomorphisms
For a pair of graphs G and H, a mapping
h:V(G) → V(H) is called a homomorphism
if xy ε E(G) implies h(x)h(y) ε E(H) (also
called H-coloring).
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The Homomorphism Problem
Fix a graph H. H-HOM: For an input graph G, check
whether there is a homomorphism of G to H.
Theorem (Hell & Nešetřil, 1990)
Let H be an unditected graph. H-HOM is polynomial
time solvable if H is bipartite or has a loop. If H is not
bipartite and it has no loop, then H-HOM is NP-complete.
Theorem (Bang-Jensen, Hell & MacGillivray, 1988)
Let H be a semicomplete graph. H-HOM is polynomial
time solvable if H has at most one cycle. If H has at least
two cycles, then H-HOM is NP-complete.
The List Homomorphism Problem
Fix a graph H. H-ListHOM: For an input graph G and a list
L(v) for each v ε V(G), check if there is a homomorphism
f of G to H s.t. f(v) ε L(v).
Theorem (Feder, Hell & Huang, 1999)
Let H be an undirected loopless graph. H-ListHOM is
polynomial-time solvable if H is bipartite and the
complement of a circular-arc graph. Otherwise, HListHOM is NP-complete.
Theorem (Gutin,Rafiey,Yeo, 2006) If H is a semicomplete
digraph with at most one cycle, H-ListHOM is
polynomial-time solvable. If H is a SD with at least two
cycles, then H-ListHOM is NP-complete.
The Min Cost Homomorphism Problem
Introduced in Gutin, Rafiey, Yeo and Tso,
2006. Fix H. MinHOM(H): Given a graph
G and a cost ci(u) of mapping u to i for
each u ε V(G), i ε V(H), find if there is a
homomorphism of G to H and if it does,
then find a homomorphism f of G to H of
minimum cost.
cost(f)= ΣuεV(G) cf(u)(u)
Min Cost vs ListHOM
H-ListHOM: G; L(v), v ε V(G)
Special MinHOM(H): ci(v)=0 if i ε L(v) and
ci(v)=1, otherwise. Э H-coloring of cost 0?
Motivation: LORA
• Level of Repair Analysis (LORA): procedure for
defence logistics, optimal provision of repair and
maintenance facilities to minimize overall lifecycle costs
• Complex system with thousands of assemblies,
sub-assemblies, components, etc.
• Has λ ≥2 levels of indenture and with r ≥ 2
repair decisions
• LORA can be reduced to MinHOM(H) for some
bipartite graphs H (Gutin, Rafiey, Yeo, Tso, ‘06)
LORA
• Introduced and studied by Barros (1998)
and Barros and Riley (2001) who designed
branch-and-bound heuristics for LORA
• We showed that LORA is polynomial-time
solvable for some practical cases
Important Polynomial Case of
MinHOM(H) and LORA
• Let HBR=(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 T={Dd,Cd,Cc,Ld,Lc,Lℓ}.
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Other Applications
• General Optimum Cost Chromatic
Partition: H=Kp (many applications)
• Special Cases:
• Optimum Cost Chromatic Partition:
ci(u)=f(i)≥0
• Minsum colorings:, ci(u)=i
Easy Polynomial Cases of
MinHOM(H): H is a di-Ck
Easy Polynomial Cases of
MinHOM(H): H is an extended L
Replacing each vertex of
H by an independent set
of vertices, we get an
extended H.
If MinHOM(L) is polytime
solvable and H is an
extended L, then
MinHOM(H) is polytime
solvable.
E.g. MinHOM(ext-di-Ck)
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Easy NP-hard Case
Let H be a connected undirected graph in which
there are vertices with and without loops. Then
MinHOM(H) is NP-hard. Indeed:
(1) H has an edge ij such that ii is a loop and jj is
not. Set cj(x)=0 and ci(x)=1 for each x in G.
(2) Let J be a maximum independent set of G. A
cheapest H-coloring assigns j to each x in J
and i to each x not in J.
(3) MaxIndepSet ≤ MinHOM(H)
(4) The maximum independent set is NP-hard.
Dichotomy for directed Ck with
possible loops
Theorem (Gutin and Kim, submitted)
Let H be a di-Ck (k≥3) with at least one loop.
Then MinHOM(H) is NP-hard.
Proof: Let kk be a loop in H, G input digraph
of order n. To obtain D replace every x in
V(G) by the path x1 x2 … xk-1 and every arc
xy by xk-1 y1. Costs: ci(xi)=0, cj(xi )=(k1)n+1, ck(xi )=1. Observe that h(xi )=k is an
H-coloring of D of cost (k-1)n .
Proof continuation
Let f be a minimum cost H-coloring of D. Then
for each x in G we have: f(xi )=i for all i or f(xi )=k
for all i . Let f(x1)= f(y1 )=1 and xy an arc of G.
Then xk-1y1 is an arc in D, a contradiction since
f(xk-1)=k-1. Thus, I={ x ε V(G): f(x1)=1} is an
independent set in G and cost(f)=(k-1)(n-|I|).
Conversely, if I is indep. in G set f(xi )=i if
x in G and f(xi )=k, otherwise; cost(f)=(k-1)(n-|I|).
Dichotomy
Theorem (Gutin and Kim, submitted)
Let H be a di-Ck (k≥2) with possible loops.
If di-Ck has no loops or k=2 and there are
two loops, then MinHOM(H) is polytime
solvable. Otherwise, MinHOM(H) is NPhard.
Min-Max Ordering for Digraphs
A digraph H=(V,A), an
ordering v1,…,vp and
is Min-Max if vivj ε A
and vrvs ε A imply vavb
ε A for both a =
min{i,r}, b = min{j,s}
and a = max{i,r}, b =
max{j,s}.
MinHOM(H) and Min-Max ordering
Theorem (Gutin, Rafiey, Yeo, 2006)
If a digraph H has a Min-Max ordering of
V(H), then MinHOM(H) is polytime
solvable.
Let TTp be the transitive tournament on
vertices 1,2,…,p (ij arc iff i<j).
Corollary MinHOM(H) is polytime solvable
if H=TTp or TTp- {1p}.
Dichotomy for SMDs
Theorem (Gutin,Rafiey,Yeo,submitted) Let H be
a semicomplete k-partite digraph, k≥3. Then
MinHOM(H) is polytime solvable if H is an
extension of TTk or TTk+1-{(1,k+1)} or di-C3 .
Otherwise, MinHOM(H) is NP-hard.
Theorem (Gutin,Rafiey,Yeo,2006) Let H be a
semicomplete digraph. Then MinHOM(H) is
polytime solvable if H is TTk or di-C3 . Otherwise,
MinHOM(H) is NP-hard.
Min-Max Orderings for Bipartite
Graphs
• A bipartite graph H=(U,W;E), orderings
u1,…,up and w1,…,wq of U and W are MinMax orderings if uiwj ε E and urws ε E imply
uawb ε E for both a = min{i,r}, b = min{j,s}
and a = max{i,r}, b = max{j,s}
•
implies
• Theorem (Spinrad, Brandstadt, Stewart,
1987) A bipartite graph H has Min-Max
orderings iff H is a proper interval bigraph.
Interval Bigraphs
• G=(R,L;E) is an interval bigraph if there
are families {I(u): u ε R} and {J(v): v ε L} of
intervals such that uv ε E iff I(u) intersects
J(v)
• An interval bigraph G=(R,L;E) is proper iff
no interval in either family contains
another interval in the family
Illustration (from LORA)
HBR has Min-Max orderings; HBR is an interval bigraph
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Polynomial Cases
• Corollary (Gutin,Hell,Rafiey,Yeo, 2007)
(a) If a bipartite graph H has Min-Max
orderings, then MinHOM(H) is polytime
solvable; (b) If H is a proper interval
bigraph, then MinHOM(H) is polytime
solvable.
NP-hardness
• Key Remark: If MinHOM(H’) is NP-hard
and H’ is an induced subgraph of H, then
MinHOM(H) is NP-hard as well.
Forbidden Subgraphs
• Theorem (Hell & Huang, 2004) A bipartite graph is
not a proper interval bigraph iff it has an induced
subgraph Cn , n≥6, or a bipartite claw, or a
bipartite net, or a bipartite tent.
Dichotomy
• Feder, Hell & Huang, 1999: Cn -ListHOM (n≥6) is
NP-hard.
• MinHOM(H) is NP-hard if H is a bipartite claw,
net, or tent (reduction from max independent set
in 3-partite graphs with fixed partite sets).
• Theorem (Gutin,Hell,Rafiey,Yeo,2007) Let H be
an undirected graph. If every component of H is
a proper interval bigraph or a reflexive interval
graph, then MinHOM(H) is polytime solvable.
Otherwise, MinHOM(H) is NP-hard.
Digraph with Possible Loops
• L is a digraph on vertices 1,2,…,k.
Replacing i by S1 we get L[S1, S2 ,…, Sk].
• An undirected graph US(L) is obtained
from L by deleting all arcs xy for which yx
is not an arc and replacing all remaining
arcs by edges.
• R:
Dichotomy for Semicomplete
Digraphs with Possible Loops
Theorem (Kim & Gutin, submitted) Let H is
a semicomplete digraph wpl. Let H=
TTk[S1, S2 ,…, Sk] where each Si is either a
single vertex without a loop, or a reflexive
semicomplete digraph which does not
contain R as an induced subdigraph and
for which US(Si ) is a connected proper
interval graph. Then, MinHOM(H) is
polytime solvable. Otherwise, MinHOM(H)
is NP-hard.
k-Min-Max Ordering
• A collection V1,…,Vk of subsets of a set V is called
a k-partition of V if V=V1 U … U Vk, and Vi ∩ Vj = ø
provided i ≠ j.
• Let H=(V,A) be a loopless digraph and let k ≥ 2 be an
integer; H has a k-Min-Max ordering if there is k-partition
of V into V1,…,Vk and there is an ordering v1(i),…, vm(i)(i)
of Vi for each i such that
(a) Every arc of H is an arc from Vi to Vi+1 for some i
(b) v1(i),…, vm(i)(i) v1(i+1),…, vm(i+1)(i+1) is a Min-Max
ordering of the subdigraph of H induced by V=Vi U Vi+1
for each i.
k-Min-Max Ordering Theorem
Theorem (Gutin, Rafiey, Yeo, submitted) If
a digraph H has a k-Min-Max ordering for
some k, then MinHOM(H) is polytime
solvable.
Proof: A reduction to the min cut problem.
Dichotomy for SBDs
Theorem (Gutin, Rafiey, Yeo, submitted)
Let H be a semicomplete digraph. If H is an
extension of di-C4 or H has a 2-Min-Max
ordering, then MinHOM(H) is polytime solvable.
Otherwise, MinHOM(H) is NP-hard.
Corollary (Gutin, Rafiey, Yeo, submitted)
Let H be a bipartite tournament. If H is an
extension of di-C4 or H is acyclic, then
MinHOM(H) is polytime solvable. Otherwise,
MinHOM(H) is NP-hard.
Further Research
• P: Dichotomy for other classes of digraphs
• P: Dichotomy for acyclic multipartite
tournaments with possible loops?
• Q: Existence of dichotomy for all
digraphs?
• For ListHOM, Bulatov proved the
existence of dichotomy (no
characterization)
Thank you!
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