Robust Assignments with Vulnerable Nodes David Adjiashvili Viktor - - PowerPoint PPT Presentation

Robust Assignments with Vulnerable Nodes

David Adjiashvili Viktor Bindewald Dennis Michaels

Technische Universit¨ at 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)

a b c 1 2 3 4 5 6 1 2 3 4 5 6 F = {1, 2, 4, 6} a b c 1 2 3 4 5 6 3 4 6 1

• ptimal 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′ ⊆ C such

that

S∈C′ 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 wn w2 w1 un u2 u1 wSl wS2 wS1 W2 = F U W1 cw = 1 cw = 1 cw = 1

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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 b c d x y z w F = {{a, z}, {d, z}, {d, w}} a b c d x y z w

• ptimal 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 ∈ PMATCH(G), for each f ∈ F, x−f

f

= 0, for each f ∈ F, y ≥ x−f , for each f ∈ F, x−f ∈ {0, 1}E, for each f ∈ F, y ∈ {0, 1}E

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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

5:

Decompose x−f into p. matchings: x−f = k

i=1 λiχM−f

i

6:

Select ¯ M ∈ {M−f

i

| i ∈ [k]} with probability λi

7:

Augment X using edges from ¯ M connecting distinct con- nected 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′ ⊆ H. An odd ear of H relative to H′ is an odd path P in H connecting nodes u, v in H′ such that only u and v are in H′. 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. 5 3 1 6 4 2 P0 P1 P2

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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 3: X ← E(P0) ∪

{E(Pi): |E(Pi)| > 1, i = 1, . . . , r}

• 4: return X

5 3 1 6 4 2

• if F = E, then the

approximation guarantee is 1.5

• if F ⊆ E, it is 3

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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 b c d x y z 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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