Co-rigid sets in redundantly rigid augmentations Andr as Mih alyk - - PowerPoint PPT Presentation

Co-rigid sets in redundantly rigid augmentations

Andr´ as Mih´ alyk´

• with Tibor Jord´

an and Csaba Kir´ aly

Department of Operations Research, E¨

• tv¨
• s Lor´

and University Budapest, Hungary

Geometric constraint systems: rigidity, flexibility and applications, Lancaster 14th June 2019

Redundant rigid augmentation (k, ℓ)-redundant augmentation Redundantly rigid subgraph (k, ℓ)-redundant subgraph

Introduction

Rigidity of graphs Definition A G = (V , E) graph minimally rigid (or Laman), if i(X) ≤ 2|X| − 3 ∀X ⊂ V , |X| ≥ 2 and |E| = 2|V | − 3. Definition A rigid graph is redundantly rigid, if its every edge is redundant, which is if we leave out any edge, it still remains rigid. Theorem [Jackson and Jord´ an, 2005] A graph of at least 4 vertices is globally rigid ⇔ redundantly rigid and 3-connected.

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Introduction

Rigidity of graphs Definition A G = (V , E) graph minimally rigid (or Laman), if i(X) ≤ 2|X| − 3 ∀X ⊂ V , |X| ≥ 2 and |E| = 2|V | − 3. Definition A rigid graph is redundantly rigid, if its every edge is redundant, which is if we leave out any edge, it still remains rigid. Theorem [Jackson and Jord´ an, 2005] A graph of at least 4 vertices is globally rigid ⇔ redundantly rigid and 3-connected.

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Introduction

Rigidity of graphs Definition A G = (V , E) graph minimally rigid (or Laman), if i(X) ≤ 2|X| − 3 ∀X ⊂ V , |X| ≥ 2 and |E| = 2|V | − 3. Definition A rigid graph is redundantly rigid, if its every edge is redundant, which is if we leave out any edge, it still remains rigid. Theorem [Jackson and Jord´ an, 2005] A graph of at least 4 vertices is globally rigid ⇔ redundantly rigid and 3-connected.

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Problem statement

Problem [Garc´ ıa and Tejel, 2011] Let L be a rigid graph. What is the minimal number of edges we need to add to L that augments it to redundantly rigid? Theorem [Garc´ ıa and Tejel, 2011] This problem is NP-hard, but if L is Laman, the optimal edgeset can be found polynomially. O(n2) algorithm, but no theorem

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Problem statement

Problem [Garc´ ıa and Tejel, 2011] Let L be a rigid graph. What is the minimal number of edges we need to add to L that augments it to redundantly rigid? Theorem [Garc´ ıa and Tejel, 2011] This problem is NP-hard, but if L is Laman, the optimal edgeset can be found polynomially. O(n2) algorithm, but no theorem

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Co-rigid sets

Add a uv edge to a graph G. L(uv) = {ij|G + uv − ij is rigid}. L(uv) is the set of generated edges by uv. Lemma If G is Laman, L(uv) = ∩{L|u, v ∈ L, L Laman subgraph} Definition Let L = (V , E) be a Laman graph. C ⊂ V is called co-rigid, if V − C is rigid. Equivalently, |C| < |V | − 1 and e(C) = 2|C|. If L + H is redundantly rigid, H must touch every co-rigid sets. Lemma Let L be a Laman graph, and H an edgeset so that L + H is redundantly rigid. |H| ≥ {

• |C|

2

• | C is set of disjoint co-rigid sets}.

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Co-rigid sets

Add a uv edge to a graph G. L(uv) = {ij|G + uv − ij is rigid}. L(uv) is the set of generated edges by uv. Lemma If G is Laman, L(uv) = ∩{L|u, v ∈ L, L Laman subgraph} Definition Let L = (V , E) be a Laman graph. C ⊂ V is called co-rigid, if V − C is rigid. Equivalently, |C| < |V | − 1 and e(C) = 2|C|. If L + H is redundantly rigid, H must touch every co-rigid sets. Lemma Let L be a Laman graph, and H an edgeset so that L + H is redundantly rigid. |H| ≥ {

• |C|

2

• | C is set of disjoint co-rigid sets}.

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Co-rigid sets

Add a uv edge to a graph G. L(uv) = {ij|G + uv − ij is rigid}. L(uv) is the set of generated edges by uv. Lemma If G is Laman, L(uv) = ∩{L|u, v ∈ L, L Laman subgraph} Definition Let L = (V , E) be a Laman graph. C ⊂ V is called co-rigid, if V − C is rigid. Equivalently, |C| < |V | − 1 and e(C) = 2|C|. If L + H is redundantly rigid, H must touch every co-rigid sets. Lemma Let L be a Laman graph, and H an edgeset so that L + H is redundantly rigid. |H| ≥ {

• |C|

2

• | C is set of disjoint co-rigid sets}.

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Structure of minimal co-rigid sets

Lemma [Jord´ an, 2014] Let L be a Laman graph and C be the family of inclusion-wise minimal co-rigid sets of L. The sets of C are pairwise disjoint or ∃{u, v} that C ∩ {u, v} = ∅ ∀C ∈ C. Moreover, this u and v are not neighbouring. If there exists such a {u, v} pair, we can augment to redundantly rigid with one edge between u and v. Suppose that the sets of C are pairwise disjoint. Representative vertices of the minimal co-rigid sets: i1, ..., i|C| Lemma L(i1i2) ⊃ C1 ∪ N(C1) ∪ C2 ∪ N(C2)

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Structure of minimal co-rigid sets

Lemma [Jord´ an, 2014] Let L be a Laman graph and C be the family of inclusion-wise minimal co-rigid sets of L. The sets of C are pairwise disjoint or ∃{u, v} that C ∩ {u, v} = ∅ ∀C ∈ C. Moreover, this u and v are not neighbouring. If there exists such a {u, v} pair, we can augment to redundantly rigid with one edge between u and v. Suppose that the sets of C are pairwise disjoint. Representative vertices of the minimal co-rigid sets: i1, ..., i|C| Lemma L(i1i2) ⊃ C1 ∪ N(C1) ∪ C2 ∪ N(C2)

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Lemma Let L be a Laman graph with minimal co-rigid sets C. For any connected graph H on the representative vertices, L + H is redundantly rigid. Lemma (reduction) Let L be a Laman graph and H an edge set on its representative vertices for which L + H is redundantly rigid. If there exists a vertex v with dH(v) ≥ 3, there is an edge set H′ on the same vertices for which |H′| < |H| and L + H′ is redundant too.

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Lemma Let L be a Laman graph with minimal co-rigid sets C. For any connected graph H on the representative vertices, L + H is redundantly rigid. Lemma (reduction) Let L be a Laman graph and H an edge set on its representative vertices for which L + H is redundantly rigid. If there exists a vertex v with dH(v) ≥ 3, there is an edge set H′ on the same vertices for which |H′| < |H| and L + H′ is redundant too.

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Lemma Let L be a Laman graph with minimal co-rigid sets C. For any connected graph H on the representative vertices, L + H is redundantly rigid. Lemma (reduction) Let L be a Laman graph and H an edge set on its representative vertices for which L + H is redundantly rigid. If there exists a vertex v with dH(v) ≥ 3, there is an edge set H′ on the same vertices for which |H′| < |H| and L + H′ is redundant too.

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Redundantly rigid augmentation

Theorem Let L be a Laman graph. min{|H| |H is an edgeset, L + H is redundantly rigid} = = max{

• |C|

2

• | C is a set of pairwise disjoint co-rigid sets}.

There exists a polynomial algorithm that can find such an edgeset. Algorithm

• Check if L can be augmented to redundant with one edge
• Chose a representative set i1, ..., i|C| arbitrarily
• Let H be the star on i1, ..., i|C| with i1 as center
• Use the reduction step to decrease the number of edges in H
• L + H is redundantly rigid consisting of
• |C|

2

• edges

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Redundantly rigid augmentation

Theorem Let L be a Laman graph. min{|H| |H is an edgeset, L + H is redundantly rigid} = = max{

• |C|

2

• | C is a set of pairwise disjoint co-rigid sets}.

There exists a polynomial algorithm that can find such an edgeset. Algorithm

• Check if L can be augmented to redundant with one edge
• Chose a representative set i1, ..., i|C| arbitrarily
• Let H be the star on i1, ..., i|C| with i1 as center
• Use the reduction step to decrease the number of edges in H
• L + H is redundantly rigid consisting of
• |C|

2

• edges

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(k, ℓ)-redundancy

Definitions We call a G = (V , E) graph

• (k, ℓ)-sparse, if i(X) ≤ k|X| − ℓ, ∀X ⊂ V
• (k, ℓ)-tight, if it is(k, ℓ)-sparse, and |E| = k|V | − ℓ
• (k, ℓ)-rigid, if it contains (k, ℓ)-tight spanning subgraph
• (k, ℓ)-redundant, if omitting any edge, G remains (k, ℓ)-rigid

Suppose that 1 ≤ k and ℓ ≤ 3

2k.

Problem [Kir´ aly, 2015] Let T be a (k, ℓ)-tight graph. What is the minimal number of edges we need to add to T that augments it to (k, ℓ)-redundant?

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(k, ℓ)-redundancy

Definitions We call a G = (V , E) graph

• (k, ℓ)-sparse, if i(X) ≤ k|X| − ℓ, ∀X ⊂ V
• (k, ℓ)-tight, if it is(k, ℓ)-sparse, and |E| = k|V | − ℓ
• (k, ℓ)-rigid, if it contains (k, ℓ)-tight spanning subgraph
• (k, ℓ)-redundant, if omitting any edge, G remains (k, ℓ)-rigid

Suppose that 1 ≤ k and ℓ ≤ 3

2k.

Problem [Kir´ aly, 2015] Let T be a (k, ℓ)-tight graph. What is the minimal number of edges we need to add to T that augments it to (k, ℓ)-redundant?

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Co-tight sets

Definition Let G = (V , E) be a (k, ℓ)-tight graph. C ⊂ V is called (k, ℓ)-co-tight, if G − C is (k, ℓ)-rigid. Equivalently C is (k, ℓ)-co-tight, if |C| < |V | − 1 and e(C) = k|C|. Remark Let c be the maximum cardinality of the pairwise disjoint (k, ℓ)-co-tight sets. The optimal augmentation must contain at least

• c

2

• edges.

Lemma Let G = (V , E) be a (k, ℓ)-tight graph. The inclusion-wise minimal co-tight sets of G are pairwise disjoint or there are two vertices that cover all of them.

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Co-tight sets

Definition Let G = (V , E) be a (k, ℓ)-tight graph. C ⊂ V is called (k, ℓ)-co-tight, if G − C is (k, ℓ)-rigid. Equivalently C is (k, ℓ)-co-tight, if |C| < |V | − 1 and e(C) = k|C|. Remark Let c be the maximum cardinality of the pairwise disjoint (k, ℓ)-co-tight sets. The optimal augmentation must contain at least

• c

2

• edges.

Lemma Let G = (V , E) be a (k, ℓ)-tight graph. The inclusion-wise minimal co-tight sets of G are pairwise disjoint or there are two vertices that cover all of them.

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(k, ℓ)-redundant augmentation

Theorem Let T be a (k, ℓ)-tight graph. min{|H| |H is an edgeset, T + H is (k, ℓ)-redundant} = = max{

• |C|

2

• | C is a set of pairwise disjoint (k, ℓ)-co-tight sets}.

There exists a polynomial algorithm that can find such an edgeset. Algorithm

• Check if T can be augmented with one edge
• Chose a representative set i1, ..., i|C| arbitrarily
• Let H be the star on i1, ..., i|C| with i1 as center
• Use reduction steps to decrease the number of edges in H
• T + H is (k, ℓ)-redundant consisting of
• |C|

2

• edges

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(k, ℓ)-redundant augmentation

Theorem Let T be a (k, ℓ)-tight graph. min{|H| |H is an edgeset, T + H is (k, ℓ)-redundant} = = max{

• |C|

2

• | C is a set of pairwise disjoint (k, ℓ)-co-tight sets}.

There exists a polynomial algorithm that can find such an edgeset. Algorithm

• Check if T can be augmented with one edge
• Chose a representative set i1, ..., i|C| arbitrarily
• Let H be the star on i1, ..., i|C| with i1 as center
• Use reduction steps to decrease the number of edges in H
• T + H is (k, ℓ)-redundant consisting of
• |C|

2

• edges

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Further remarks

Lemma Let H be an optimal augmenting edgeset. If |H| ≥ 2, then we can choose H to be disjoint from E. Augmenting (k, ℓ)-rigid graphs If k ≥ ℓ we can augment a (k, ℓ)-rigid graph to (k, ℓ)-redundant

• ptimally.

If ℓ = 3

2k, this problem is NP-hard.

k < ℓ < 3

2k ??

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Further remarks

Lemma Let H be an optimal augmenting edgeset. If |H| ≥ 2, then we can choose H to be disjoint from E. Augmenting (k, ℓ)-rigid graphs If k ≥ ℓ we can augment a (k, ℓ)-rigid graph to (k, ℓ)-redundant

• ptimally.

If ℓ = 3

2k, this problem is NP-hard.

k < ℓ < 3

2k ??

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Minimal cost redundantly rigid subgraph

Problem Let c be a metric cost function on Kn. What is the minimal cost redundantly rigid spanning subgraph of Kn? Remark In 1 dimension this is the minimal cost 2-edge-connected subgraph problem. Lemma This problem is NP-hard.

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Minimal cost redundantly rigid subgraph

Problem Let c be a metric cost function on Kn. What is the minimal cost redundantly rigid spanning subgraph of Kn? Remark In 1 dimension this is the minimal cost 2-edge-connected subgraph problem. Lemma This problem is NP-hard.

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Minimal cost redundantly rigid subgraph

Problem Let c be a metric cost function on Kn. What is the minimal cost redundantly rigid spanning subgraph of Kn? Remark In 1 dimension this is the minimal cost 2-edge-connected subgraph problem. Lemma This problem is NP-hard.

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Approximation

Algorithm

• Compute a minimal cost Laman L
• Check if L can be augmented to redundantly rigid with 1 edge
• If not: Choose representative vertices: i1, ..., i|C|
• Compute H, a minimal cost spanning tree on i1, ..., i|C|
• L + H is redundantly rigid

Theorem [Jord´ an and M. (2019)] This algorithm gives a 2-approximation to the minimal cost redundantly rigid subgraph problem in case of metric costs.

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Remarks

Proof L + H is clearly redundant. Let T be the minimal cost spanning tree on V . c(H) ≤ 2c(T). c(L) < OPT and c(T) ≤ 1

2OPT.

Claim Similar algorithm holds for global rigidity too. Remark We can decrease the cost, if we use the reduction step on H. Conjecture This algorithm is a 1.5-approximation.

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Remarks

Proof L + H is clearly redundant. Let T be the minimal cost spanning tree on V . c(H) ≤ 2c(T). c(L) < OPT and c(T) ≤ 1

2OPT.

Claim Similar algorithm holds for global rigidity too. Remark We can decrease the cost, if we use the reduction step on H. Conjecture This algorithm is a 1.5-approximation.

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Minimal cost (k, ℓ)-redundant subgraph

Problem Given a metric cost function c on Kn, find a minimal cost (k, ℓ)-redundant spanning subgraph of Kn. For fix (k, ℓ), k > 1 there is a 1 +

2 k−1 approximation. For k = 1 it

gives a 3-approximation. Algorithm

• Compute a minimal cost (k, ℓ)-tight T
• Check if T can be augmented with 1 edge
• Representative vertices: i1, ..., i|C|
• H is minimal cost spanning tree on i1, ..., i|C|
• T + H is (k, ℓ)-redundant

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Minimal cost (k, ℓ)-redundant subgraph

Problem Given a metric cost function c on Kn, find a minimal cost (k, ℓ)-redundant spanning subgraph of Kn. For fix (k, ℓ), k > 1 there is a 1 +

2 k−1 approximation. For k = 1 it

gives a 3-approximation. Algorithm

• Compute a minimal cost (k, ℓ)-tight T
• Check if T can be augmented with 1 edge
• Representative vertices: i1, ..., i|C|
• H is minimal cost spanning tree on i1, ..., i|C|
• T + H is (k, ℓ)-redundant

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Minimal cost (k, ℓ)-redundant subgraph

Problem Given a metric cost function c on Kn, find a minimal cost (k, ℓ)-redundant spanning subgraph of Kn. For fix (k, ℓ), k > 1 there is a 1 +

2 k−1 approximation. For k = 1 it

gives a 3-approximation. Algorithm

• Compute a minimal cost (k, ℓ)-tight T
• Check if T can be augmented with 1 edge
• Representative vertices: i1, ..., i|C|
• H is minimal cost spanning tree on i1, ..., i|C|
• T + H is (k, ℓ)-redundant

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Open Questions

Globally rigid augmentation Given a Laman graph L what is the minimal size edgeset that augments it to globally rigid? Redundant rigidity of sparse graphs Given a sparse graph L what is the minimal size edgeset that augments it to redundantly rigid? Redundantly rigid subgraph Minimal cost redundantly rigid subgraph without metric assumptions?

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Open Questions

Globally rigid augmentation Given a Laman graph L what is the minimal size edgeset that augments it to globally rigid? Redundant rigidity of sparse graphs Given a sparse graph L what is the minimal size edgeset that augments it to redundantly rigid? Redundantly rigid subgraph Minimal cost redundantly rigid subgraph without metric assumptions?

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Open Questions

Globally rigid augmentation Given a Laman graph L what is the minimal size edgeset that augments it to globally rigid? Redundant rigidity of sparse graphs Given a sparse graph L what is the minimal size edgeset that augments it to redundantly rigid? Redundantly rigid subgraph Minimal cost redundantly rigid subgraph without metric assumptions?

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Thank you for your attention!

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