Robust Assignments
with Vulnerable Nodes
David Adjiashvili
Viktor Bindewald
Dennis Michaels
Technische Universität Dortmund
9-13, January 2017, Aussois
This work is supported by DFG through RTG 1855 “Discrete Optimization of Technical
Systems under Uncertainty”
1. Problem definition and applications
2. Hardness and algorithmic results
Approximation for the weighted version
Approximation for the unweighted version
3. Robust assignments with vulnerable edges
Hardness and algorithmic results
The simplest non-trivial case
Problem definition and
applications
Motivation
×
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Robust Assignment Problem
Certain problem
• given G = (U ∪˙ W , E ), costs c ∈ RW
≥0
• nodes in U are called jobs and those in W machines
• a set of machines X ⊆ W is called an assignment if all
jobs can be performed using machines in X
• i.e. the induced graph G [U ∪˙ X ] contains a U-perfect
matching (a set of non-adjacent edges covering U)
• objective: choose a minimum-cost assignment
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Robust Assignment Problem
Robust version
• G = (U ∪˙ W , E ), costs c ∈ RW
≥0 , |U| < |W |
• Scenario set F = {f1 , . . . , fk } ⊆ W (vulnerable machines)
• If scenario fi emerges, then machine fi is deleted from G
min c(X )
s.t. ∀f ∈ F : X \ {f } contains an assignment
X ⊆W
(RAP)
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Robust Assignment Problem
Interpreted as a game:
1. the adversary announces a list of machines he wants to
sabotage
2. the decision maker buys a set of machines
3. the adversary destroys one machine from the list
4. the decision maker selects an assignment from the
remaining machines he bought and performs the jobs
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Example (unit weights)
1
1
2
2
a
3
a
3
b
4
b
4
c
5
c
5
6
F = {1, 2, 4, 6}
6
optimal solution
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Classification and potential areas of application
• RAP is a redundancy-based robust problem
(example: finding k-connected spanning subgraphs)
• more precisely, it is a bulk-robust problem ([Adjiashvili,
Stiller, Zenklusen ’15])
The concept of bulk-robustness models long-term decision
problems where “infrastructure” must be fixed in advance.
In our setting,
• staff scheduling / rostering
• hiring / subcontracting
• reservation systems
• manufacturing process flexibility
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Hardness and algorithmic results
How hard is the problem?
Theorem
Set Cover can be restated equivalently as RAP, preserving the
cost.
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Set Cover Problem
Recap:
• Given a tuple ({1, . . . , n}, C), where C = {S1 , S2 , . . . , Sl }
with Sj ⊆ {1, . . . , n}.
• Find a minimum-cardinality sub-collection C 0 ⊆ C such
S
that S∈C 0 S = {1, . . . , n}.
Theorem [Feige ’98]
For any d < 1, there is no d · log n –approximation for the Set
Cover problem, unless NP ⊆ DTIME(nlog log n ).
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Hardness of RAP
Set Cover → RAP
({1, . . . , n}, {S1 , S2 , . . . , Sl }) → bipartite graph on 2n + l nodes
w1
u1
w2
u2
wS1 cw = 1
wS2 cw = 1
wSl cw = 1
wn
un
W2 = F
U
W1
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Asymptotically tight algorithm for RAP
Theorem
RAP admits a (log |U| + 2)-approximation.
Approach:
Select a cheap matching and complete it to a solution.
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(log |U| + 2)-Approximation for RAP
Require: Feas. instance: G = (U ∪˙ W , E ), c ∈ RW
≥0 , F ⊆ W
1: for each w ∈ W : for each e ∈ δ(w ) : de := cw
2: M ← minimum-cost U-perfect matching w.r.t. d
3: W M ← V (M) ∩ W
4: UF ← {jobs matched to a vulnerable machine in M}
5: for each w ∈ W \ W M do
6:
Rw ← {u ∈ UF | ∃ M-alt. u-w -path in G [V (M)] + w }
7: S ← apx. sln. to SC instance (UF , {Rw | w ∈ W \ W M })
˙
8: return W M ∪{w
∈ W \ W M | Rw ∈ S}
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(log |U| + 2)-Approximation for RAP
Proof:
• c(ALG) = c(W M ) + c(S)
• c(W M ) < c(OPT)
• c(S) ≤ (log |UF | + 1) · c(OPT(Set Cover)) ([Chvatal ’79])
• c(OPT(Set Cover)) ≤ c(OPT) and |UF | ≤ |U|
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Approximation for minimum-cardinality RAP
We now focus on the variant of RAP with unit weights, i.e.
cw = 1, ∀w ∈ W (card-RAP).
Theorem
card-RAP with F = W admits a 1.75-approximation algorithm.
Approach:
Greedily augment a U-perfect matching.
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1.75-Approximation for card-RAP
Require: Feas. card-RAP-instance: G = (U ∪˙ W , E ), F = W .
1: M ← any U-perfect matching in G
2: X ← V (M) ∩ W (machines from matching M)
3: for each machine node w ∈ W \ X do
4:
Rw ← {u ∈ U | ∃ M-altern. u-w -path in G [V (M)]+w }
5: UF (X ) ← U
6: while |UF (X )| > 0 do
7:
w ← arg max{|UF (X ) ∩ Rw )| | w ∈ W \ X }
8:
UF (X ) ← UF (X ) \ Rw
9:
X ← X ∪ {w }
10: return X
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1.75-Approximation for card-RAP
Proof idea:
• an iteration is called good, if |UF (X )| is decreased by at
least 2
• g :⇔ total decrease of |UF (X )| obtained by good
iterations
• |ALG| ≤ 2n −
g
2
• |OPT| ≥ max{n, 2(n − g )}
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Hardness of approximation for card-RAP
Theorem
There is no PTAS for card-RAP, unless P = NP. This is also
true for F = W .
Proof idea
• Vertex Cover in subcubic graphs (VC 3) is APX-hard
([Alimonti, Kann ’97])
• Restate VC 3 in terms of a set cover problem,
• use an extended version of the prior presented
reduction
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Robust assignments with
vulnerable edges
Edge-Robust Assignment Problem
Certain problem
• Given G = (U ∪˙ W , E ) with |U| = |W |, costs c ∈ RE≥0
• Find a minimum-cost perfect matching M ⊆ E of G
Robust version (E-RAP)
• Scenario set F = {f1 , . . . , fk } ⊆ E (vulnerable edges)
• If scenario fi emerges, then edge fi is deleted from G
min c(X )
s.t. ∀f ∈ F : X \ {f } contains a perf. matching
X ⊆E
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Example (unit weights)
a
x
a
x
b
y
b
y
c
z
c
z
d
w
d
w
F = {{a, z}, {d, z}, {d, w }}
optimal solution
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How hard is the problem?
Theorem
Set Cover can be restated equivalently as E-RAP, preserving
the cost.
For details see ”Robust Assignments via Ear Decompositions and
Randomized Rounding” [Adjiashvili, B., Michaels ’16]
Theorem
E-RAP admits a randomized O(log n)-approximation algorithm.
Approach:
Obtain a matching-covered subgraph via randomized rounding.
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Technical ingredients I
Definition
A graph G = (V , E ) is called matching-covered if each of its
edges is part of some perfect matching of G
Why is it useful?
Up to technicalities, inclusion-wise minimal feasible solutions
to E-RAP are matching-covered subgraphs and vice versa.
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Technical ingredients II
We use the following IP formulation for E-RAP:
(Given G = (U ∪˙ W , E ), F = E , c ∈ RE≥0 )
min c > y
s.t.
x −f
xf−f
y
−f
x
y
∈
=
≥
∈
∈
PMATCH (G ),
0,
x −f ,
{0, 1}E ,
{0, 1}E
for
for
for
for
each
each
each
each
f
f
f
f
∈ F,
∈ F,
∈ F,
∈ F,
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O(log n)-Approximation for RAP
Require: G = (U ∪˙ W , E ), F = E and c ∈ RE≥0 .
1: (x, y ) ← optimal solution of the LP relaxation
2: X ← ∅
3: while X is infeasible do
4:
Select f ∈ F such that X \ {f } has no perf. matching
P
−f
5:
Decompose x −f into p. matchings: x −f = ki=1 λi χMi
6:
Select M̄ ∈ {Mi−f | i ∈ [k]} with probability λi
7:
Augment X using edges from M̄ connecting distinct connected components in (V [G ], X )
8: return X
Can be adopted to F ( E .
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O(1)-Approximation for minimum-cardinal. E-RAP
We now focus on the variant of E-RAP with unit weights, i.e.
ce = 1, ∀e ∈ E (card-E-RAP).
Theorem
card-E-RAP admits a O(1)-approximation algorithm.
Approach:
Exploit ear decompositions of matching-covered graphs.
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Bipartite ear decomposition
Definition
Consider a bipartite graph H and a
subgraph H 0 ⊆ H.
An odd ear of H relative to H 0 is an odd
path P in H connecting nodes u, v in H 0
such that only u and v are in H 0 .
Bipartite Ear Decomposition
If H can be constructed by starting with
an edge via odd ear additions, then H
has a Bipartite Ear Decomposition.
We write H = P0 + P1 + · · · + Pr .
1
3
5
P0
P2
P1
2
4
6
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A characterization of bipartite m.-c. graphs
Theorem [Lovasz, Plummer ’86]
A bipartite graph H is matching-covered if and only if H has a
Bipartite Ear Decomposition.
Observation
Removing ears of length one from a Bipartite Ear
Decomposition yields a matching-covered graph.
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O(1)–Approximation algorithm for card-E-RAP
Require: G = (U ∪˙ W , E ) and F ⊆ E
1: Remove all dispensable edges from G
2: Compute an ear
decomposition G = P0 + P1 + · · · +Pr
S
3: X ← E (P0 ) ∪
{E (Pi ) : |E (Pi )| > 1, i = 1, . . . , r }
4: return X
1
3
2
4
• if F = E , then the
approximation guarantee
is 1.5
• if F ⊆ E , it is 3
5
6
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Hardness of approximation for card-E-RAP
Theorem
There is no PTAS for card-E-RAP, unless P = NP. This is also
true for F = E .
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The simplest non-trivial case of E-RAP
Setting:
G = (U ∪˙ W , E ), F = {f1 , f2 }, unit costs, i.e. ce = 1, ∀e ∈ E .
Observation
A minimal feasible solution is
a) either a perfect matching
not using f1 or f2 ,
b) or is union of an (even)
cycle C containing f1 and
f2 , and a matching that
covers all nodes not in C .
a
x
b
y
c
z
d
w
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The simplest non-trivial case of card-E-RAP
Theorem
card-E-RAP is NP-hard with only two vulnerable edges.
Proof idea.
Reduction from a special variant of the NP-complete
Path-with-Forbidden-Pairs-Problem:
• Given a graph G = (V , E ), s, t ∈ V , and k node pairs
• Is there a s-t-path avoiding at least one node of each
pair?
(Problem [GT54] in [Garey, Johnson ’79])
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The simplest non-trivial case of card-E-RAP
Proof idea (cont’d):
Intermediately we use the Shortest-Nice-Path-Problem:
• Given a bipartite graph and two nodes s and t
• Find a shortest s-t-path such that the remaining nodes
are perfectly matchable
Remark
To the best of our knowledge, this is the first example of an
NP-hard robust counterpart of a polytime optimization
problem with a constant number of vulnerable resources.
Thank you for your attention!
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