Matroidal Degree-Bounded Minimum Spanning Trees · 2016. 11. 23. · jT \Uj B U for U 2U 2E....

Matroidal Degree-BoundedMinimum Spanning Trees

Rico Zenklusen

Degree-constrained MST problems

min{c(T ) | T spanning tree in G = (V ,E ), satisfying degree constraints}

Types of degree constraints

Degree-bounded MST (the classic):

|T ∩ δ(v)| ≤ Bv ∀v ∈ V .

Cut-constraints:

|T ∩ δ(S)| ≤ BS for S ∈ S ⊆ 2V .

0/1-packing constraints:

|T ∩ U| ≤ BU for U ∈ U ⊆ 2E .

Typical motivations for degree constraints

Technical restrictions (VLSI design, telecommunication networks).

Increase reliability by avoiding overloaded vertices.

Reduce vulnerability against malicious attacks.

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|T ∩ δ(v)| ≤ Bv ∀v ∈ V .

Cut-constraints:

|T ∩ δ(S)| ≤ BS for S ∈ S ⊆ 2V .

|T ∩ U| ≤ BU for U ∈ U ⊆ 2E .

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|T ∩ δ(v)| ≤ Bv ∀v ∈ V .

Cut-constraints:

|T ∩ δ(S)| ≤ BS for S ∈ S ⊆ 2V .

|T ∩ U| ≤ BU for U ∈ U ⊆ 2E .

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|T ∩ δ(v)| ≤ Bv ∀v ∈ V .

Cut-constraints:

|T ∩ δ(S)| ≤ BS for S ∈ S ⊆ 2V .

|T ∩ U| ≤ BU for U ∈ U ⊆ 2E .

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|T ∩ δ(v)| ≤ Bv ∀v ∈ V .

Cut-constraints:

|T ∩ δ(S)| ≤ BS for S ∈ S ⊆ 2V .

|T ∩ U| ≤ BU for U ∈ U ⊆ 2E .

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|T ∩ δ(v)| ≤ Bv ∀v ∈ V .

Cut-constraints:

|T ∩ δ(S)| ≤ BS for S ∈ S ⊆ 2V .

|T ∩ U| ≤ BU for U ∈ U ⊆ 2E .

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Trading slight infeasibility for “optimal” costAt the example of degree-bounded MST

OPT = min{c(T ) | T ∈ T︸︷︷︸all spanning trees

(⊆ 2E

), |T ∩ δ(v)| ≤ Bv ∀v ∈ V }

Even checking feasibility is NP hard(e.g. if Bv = 2 ∀v ∈ V → Hamiltonian path problem.)

Find tree T of cost ≤ LP relaxation (c(T ) ≤ OPTf ), minimizing deg violation.

= min cT x

x ∈ PST = conv({1T | T ∈ T })x(δ(v)) ≤ Bv ∀v ∈ V

x ∈ {0, 1}E

min{c(T ) | T ∈ T , |T ∩ δ(v)| ≤ Bv + k ∀v ∈ V }

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(⊆ 2E

), |T ∩ δ(v)| ≤ Bv ∀v ∈ V }

= min cT x

x ∈ {0, 1}E

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(⊆ 2E

), |T ∩ δ(v)| ≤ Bv ∀v ∈ V }

OPTf = min cT x

/////////////////x ∈ {0, 1}E

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Previous results

Degree-bounded MST

+1 violation (no costs) Furer and Raghavachari (1994)various super-constantviolations with cost ≤ OPTf

Konemann and Ravi (2002, 2003), Chaudhuriet al. (2005), . . .

+2 Goemans (2006)+1 Singh and Lau (2007)

Generalized bounds

Laminar cut bounds:+O(log |V |) Bansal et al. (2010)

Bounds on arbitrary edge sets x(U) ≤ BU ∀U ∈ U ⊆ 2E :+ maxe∈E |{U ∈ U | e ∈ U}| Bansal et al. (2009)

Thinness bounds wrt y ∈ PST : x(δ(S)) ≤ y(δ(S)) ∀∅ 6= S ( V (thin trees):

log |V |log log |V |

)Asadpour et al. (2010)

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Previous results

Degree-bounded MST

+1 violation (no costs) Furer and Raghavachari (1994)various super-constantviolations with cost ≤ OPTf

Konemann and Ravi (2002, 2003), Chaudhuriet al. (2005), . . .

+2 Goemans (2006)+1 Singh and Lau (2007)

Generalized bounds

Laminar cut bounds:+O(log |V |) Bansal et al. (2010)

Bounds on arbitrary edge sets x(U) ≤ BU ∀U ∈ U ⊆ 2E :+ maxe∈E |{U ∈ U | e ∈ U}| Bansal et al. (2009)

Thinness bounds wrt y ∈ PST : x(δ(S)) ≤ y(δ(S)) ∀∅ 6= S ( V (thin trees):

log |V |log log |V |

)Asadpour et al. (2010)

Obtain constant violation for constraints beyond degree-bounded MST?

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Our contributionsMain results

Efficent algorithm with +8 guarantee for matroidal degree constraints.

This is based on extensions to the iterative relaxation framework.

Definition (Matroidal degree constraints)

min{c(T ) | T ∈ T ,T ∩ δ(v) ∈ Iv ∀v ∈ V },

where Mv = (δ(v), Iv ) is a matroid ∀v ∈ V .

Examples

deg-bounded MST (Mv : uniform matroids)

partition constraints (part. matroids)

laminar constraints (lam. matroids)

Definition (≤ k violation of degree constraints)

T ∈ T violates degree constraint at v by ≤ k units if:feasibility can be obtained by ignoring the contribution of ≤ k edges.

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min{c(T ) | T ∈ T ,T ∩ δ(v) ∈ Iv ∀v ∈ V },

Examples

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min{c(T ) | T ∈ T ,T ∩ δ(v) ∈ Iv ∀v ∈ V },

Examples

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min{c(T ) | T ∈ T ,T ∩ δ(v) ∈ Iv ∀v ∈ V },

Examples

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min{c(T ) | T ∈ T ,T ∩ δ(v) ∈ Iv ∀v ∈ V },

Examples

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min{c(T ) | T ∈ T ,T ∩ δ(v) ∈ Iv ∀v ∈ V },

Examples

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min{c(T ) | T ∈ T ,T ∩ δ(v) ∈ Iv ∀v ∈ V },

Examples

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min{c(T ) | T ∈ T ,T ∩ δ(v) ∈ Iv ∀v ∈ V },

Examples

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min{c(T ) | T ∈ T ,T ∩ δ(v) ∈ Iv ∀v ∈ V },

Examples

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Iterative relaxation by Singh and Lau (+1 guarantee)

Step 1

Get basic LP solution x∗.

If sol is integral: stop.

Step 2

Delete 0-edges.

Fix 1-edges.

Step 3 (let spare: z = 1− x∗)

Find v ∈ V s.t. z(δ(v)) < 2.

Delete deg constraint at v .

Back to Step 1.

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Step 1

Step 2

Delete 0-edges.

Fix 1-edges.

Find v ∈ V s.t. z(δ(v)) < 2.

Back to Step 1.

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Step 1

Step 2

Delete 0-edges.

Fix 1-edges.

Find v ∈ V s.t. z(δ(v)) < 2.

Back to Step 1.

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Step 1

Step 2

Delete 0-edges.

Fix 1-edges.

Find v ∈ V s.t. z(δ(v)) < 2.

Back to Step 1.

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Step 1

Step 2

Delete 0-edges.

Fix 1-edges.

Find v ∈ V s.t. z(δ(v)) < 2.

Back to Step 1.

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Step 1

Step 2

Delete 0-edges.

Fix 1-edges.

Find v ∈ V s.t. z(δ(v)) < 2.

Back to Step 1.

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Step 1

Step 2

Delete 0-edges.

Fix 1-edges.

Find v ∈ V s.t. z(δ(v)) < 2.

Back to Step 1.

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Step 1

Step 2

Delete 0-edges.

Fix 1-edges.

Find v ∈ V s.t. z(δ(v)) < 2.

Back to Step 1.

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Step 1

Step 2

Delete 0-edges.

Fix 1-edges.

Find v ∈ V s.t. z(δ(v)) < 2.

Back to Step 1.

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Step 1

Step 2

Delete 0-edges.

Fix 1-edges.

Find v ∈ V s.t. z(δ(v)) < 2.

Back to Step 1.

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Step 1

Step 2

Delete 0-edges.

Fix 1-edges.

Find v ∈ V s.t. z(δ(v)) < 2.

Back to Step 1.

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Step 1

Step 2

Delete 0-edges.

Fix 1-edges.

Find v ∈ V s.t. z(δ(v)) < 2.

Back to Step 1.

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Step 1

Step 2

Delete 0-edges.

Fix 1-edges.

Find v ∈ V s.t. z(δ(v)) < 2.

Back to Step 1.

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Step 1

Step 2

Delete 0-edges.

Fix 1-edges.

Find v ∈ V s.t. z(δ(v)) < 2.

Back to Step 1.

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Step 1

Step 2

Delete 0-edges.

Fix 1-edges.

Find v ∈ V s.t. z(δ(v)) < 2.

Back to Step 1.

Key property: x∗ is sparse (in particular | supp(x∗)| ≤ 2|V | − 1).

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Step 1

Step 2

Delete 0-edges.

Fix 1-edges.

Find v ∈ V s.t. z(δ(v)) < 2.

Back to Step 1.

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Step 1

Step 2

Delete 0-edges.

Fix 1-edges.

Find v ∈ V s.t. z(δ(v)) < 2.

Back to Step 1.

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Step 1

Step 2

Delete 0-edges.

Fix 1-edges.

Find v ∈ V s.t. z(δ(v)) < 2.

Back to Step 1.

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Challenges with more general/matroidal constraints

Previous iterative relaxation/rounding approaches are hard to generalize tomatroidal deg constraints (or other generalized constraint).

Some issues with previous iterative relaxation approaches

Not sufficient sparsity to drop full degree constraints at some vertex.

Previous approaches relied on the fact that each edge is only in a constantnumber of linear constraints (belonging to degree constraints).

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High-level goal of our algorithm

Iteratively change constraints to approach matroid intersection problem instead oftargeting ST polytope (which is a matroid base polytope).→ Iteratively “remove” each edge {u, v} either from deg constraint at u or v .

(this is similar in spirit to Goemans’ algorithm, but works iteratively.)

If each edge belong to at most one degree constraints, all matroidal degreeconstraints together form one single matroid.

Resulting optimization problem is matroid intersection and thus integral.

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Summary of further technical contributions

Further contributions on algorithm design level

When removing edges from constraints: old constraint gets replaced by a possiblymore complicated matroidal constraint (s.t. violation is bounded by slack).

We fix tight ST constraints → they help respect degree constraints.

New ideas for the analysis

New argument to prove sparsity that exploits interplay of constraints.

Exploit properties of low-dimensional faces of ST polytope (to deal with caseswhere many lin indep ST constraints are tight).

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An oversimplified sketch of the algorithmH = (W ,F ): current graph, F2 ⊆ F : edges currently in both constraints.Nw : current matroidal deg constraints with corresp. matroid polytope PNw .

Step 1

Get basic solution x∗ to LP:min{cT x | x ∈ PST , x

∣∣δ(w)∈ PNw ∀w ∈W }

If x∗ is integral: stop.

Step 2

Delete 0-edges.

Contract 1-edges.

Fix tight spanning tree constraints.

Step 3: degree constraint adaptation

Find w ∈W s.t. z(δ(w) ∩ F2) ≤ 4.

“Remove” δ(w) ∩ F2 from constraint at w .

Go back to Step 1.

For v ∈ V , only ≤ 1 deg adaptations impacts δ(v) ⇒ violation ≤ 4.

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Step 1

∣∣δ(w)∈ PNw ∀w ∈W }

Step 2

Delete 0-edges.

Contract 1-edges.

Find w ∈W s.t. z(δ(w) ∩ F2) ≤ 4.

Go back to Step 1.

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Step 1

∣∣δ(w)∈ PNw ∀w ∈W }

Step 2

Delete 0-edges.

Contract 1-edges.

Find w ∈W s.t. z(δ(w) ∩ F2) ≤ 4.

Go back to Step 1.

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Step 1

∣∣δ(w)∈ PNw ∀w ∈W }

Step 2

Delete 0-edges.

Contract 1-edges.

Find w ∈W s.t. z(δ(w) ∩ F2) ≤ 4.

Go back to Step 1.

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Step 1

∣∣δ(w)∈ PNw ∀w ∈W }

Step 2

Delete 0-edges.

Contract 1-edges.

Find w ∈W s.t. z(δ(w) ∩ F2) ≤ 4.

Go back to Step 1.

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Step 1

∣∣δ(w)∈ PNw ∀w ∈W }

Step 2

Delete 0-edges.

Contract 1-edges.

Find w ∈W s.t. z(δ(w) ∩ F2) ≤ 4.

Go back to Step 1.

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Step 1

∣∣δ(w)∈ PNw ∀w ∈W }

Step 2

Delete 0-edges.

Contract 1-edges.

Find w ∈W s.t. z(δ(w) ∩ F2) ≤ 4.

Go back to Step 1.

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Step 1

∣∣δ(w)∈ PNw ∀w ∈W }

Step 2

Delete 0-edges.

Contract 1-edges.

Find w ∈W s.t. z(δ(w) ∩ F2) ≤ 4.

Go back to Step 1.

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Step 1

∣∣δ(w)∈ PNw ∀w ∈W }

Step 2

Delete 0-edges.

Contract 1-edges.

Find w ∈W s.t. z(δ(w) ∩ F2) ≤ 4.

Go back to Step 1.

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Step 1

∣∣δ(w)∈ PNw ∀w ∈W }

Step 2

Delete 0-edges.

Contract 1-edges.

Find w ∈W s.t. z(δ(w) ∩ F2) ≤ 4.

Go back to Step 1.

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Step 1

∣∣δ(w)∈ PNw ∀w ∈W }

Step 2

Delete 0-edges.

Contract 1-edges.

Find w ∈W s.t. z(δ(w) ∩ F2) ≤ 4.

Go back to Step 1.

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Step 1

∣∣δ(w)∈ PNw ∀w ∈W }

Step 2

Delete 0-edges.

Contract 1-edges.

Find w ∈W s.t. z(δ(w) ∩ F2) ≤ 4.

Go back to Step 1.

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Step 1

∣∣δ(w)∈ PNw ∀w ∈W }

Step 2

Delete 0-edges.

Contract 1-edges.

Find w ∈W s.t. z(δ(w) ∩ F2) ≤ 4.

Go back to Step 1.

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Step 1

∣∣δ(w)∈ PNw ∀w ∈W }

Step 2

Delete 0-edges.

Contract 1-edges.

Find w ∈W s.t. z(δ(w) ∩ F2) ≤ 4.

Go back to Step 1.

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Step 1

∣∣δ(w)∈ PNw ∀w ∈W }

Step 2

Delete 0-edges.

Contract 1-edges.

Find w ∈W s.t. z(δ(w) ∩ F2) ≤ 4.

Go back to Step 1.

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Step 1

∣∣δ(w)∈ PNw ∀w ∈W }

Step 2

Delete 0-edges.

Contract 1-edges.

Find w ∈W s.t. z(δ(w) ∩ F2) ≤ 4.

Go back to Step 1.

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Step 1

∣∣δ(w)∈ PNw ∀w ∈W }

Step 2

Delete 0-edges.

Contract 1-edges.

Find w ∈W s.t. z(δ(w) ∩ F2) ≤ 4.

Go back to Step 1.

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Step 1

∣∣δ(w)∈ PNw ∀w ∈W }

Step 2

Delete 0-edges.

Contract 1-edges.

Find w ∈W s.t. z(δ(w) ∩ F2) ≤ 4.

Go back to Step 1.

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Step 1

∣∣δ(w)∈ PNw ∀w ∈W }

Step 2

Delete 0-edges.

Contract 1-edges.

Find w ∈W s.t. z(δ(w) ∩ F2) ≤ 4.

Go back to Step 1.

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Step 1

∣∣δ(w)∈ PNw ∀w ∈W }

Step 2

Delete 0-edges.

Contract 1-edges.

Find w ∈W s.t. z(δ(w) ∩ F2) ≤ 4.

Go back to Step 1.

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Step 1

∣∣δ(w)∈ PNw ∀w ∈W }

Step 2

Delete 0-edges.

Contract 1-edges.

Find w ∈W s.t. z(δ(w) ∩ F2) ≤ 4.

Go back to Step 1.

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Step 1

∣∣δ(w)∈ PNw ∀w ∈W }

Step 2

Delete 0-edges.

Contract 1-edges.

Find w ∈W s.t. z(δ(w) ∩ F2) ≤ 4.

Go back to Step 1.

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Step 1

∣∣δ(w)∈ PNw ∀w ∈W }

Step 2

Delete 0-edges.

Contract 1-edges.

Find w ∈W s.t. z(δ(w) ∩ F2) ≤ 4.

Go back to Step 1.

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Step 1

∣∣δ(w)∈ PNw ∀w ∈W }

Step 2

Delete 0-edges.

Contract 1-edges.

Find w ∈W s.t. z(δ(w) ∩ F2) ≤ 4.

Go back to Step 1.

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Step 1

∣∣δ(w)∈ PNw ∀w ∈W }

Step 2

Delete 0-edges.

Contract 1-edges.

Find w ∈W s.t. z(δ(w) ∩ F2) ≤ 4.

Go back to Step 1.

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Step 1

∣∣δ(w)∈ PNw ∀w ∈W }

Step 2

Delete 0-edges.

Contract 1-edges.

Find w ∈W s.t. z(δ(w) ∩ F2) ≤ 4.

Go back to Step 1.

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Step 1

∣∣δ(w)∈ PNw ∀w ∈W }

Step 2

Delete 0-edges.

Contract 1-edges.

Find w ∈W s.t. z(δ(w) ∩ F2) ≤ 4.

Go back to Step 1.

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Step 1

∣∣δ(w)∈ PNw ∀w ∈W }

Step 2

Delete 0-edges.

Contract 1-edges.

Find w ∈W s.t. z(δ(w) ∩ F2) ≤ 4.

Go back to Step 1.

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Step 1

∣∣δ(w)∈ PNw ∀w ∈W }

Step 2

Delete 0-edges.

Contract 1-edges.

Find w ∈W s.t. z(δ(w) ∩ F2) ≤ 4.

Go back to Step 1.

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Updating degree constraints

Contraction of 1-edge

Contract f in Nw1 and Nw2 .

Nw1,2 is disjoint union of Nw1 and Nw2 .

Deletion of 0-edge

Delete f from Nw1 and Nw2 .

Removing edges from deg constr. (a bit more involved)

Update is done such that:i) T ∈ T satisfies N ′w ⇒ T violates Nw by ≤ dz(U)e.ii) Current LP sol remains feasible.

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Proving sparsity to show ∃ adaptation stepδD(w) ⊆ δ(w): edges not yet removed from Nw .

If k linearly indep degree constraints of PNw are tight wrt x∗ ⇒ x∗(δD(w)) ≥ k.

Total # of linearly indep tight deg constraints at start of algo.:

|D| ≤∑

w∈W x∗(δD(w)) =∑

w∈W x∗(δ(w)) = 2x∗(F )x∗∈PST= 2(|W | − 1).

Total # of linearly indep & tight ST constraints: |L|uncrossing

≤ |W | − 1.

| supp(x∗)| = |F | ≤ |L|+ |D| ≤ 3(|W | − 1).

∃ adaptation step in first iteration∑w∈W

z(δD(w)) =∑w∈W

z(δ(w)) =∑w∈W

(|δ(w)| − x∗(δ(w)))

≤ 2|F | −∑w∈W

x∗(δ(w)) = 2|F | − 2x∗(F ) ≤ 4(|W | − 1).

In later iterations, this averaging argument does not work anymore

For some nodes w ∈W we will have δD(w) = ∅.We improve sparsity with 2nd type of deg adaptation (→ another +4 in violation).

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|D| ≤∑

w∈W x∗(δD(w)) =∑

w∈W x∗(δ(w)) = 2x∗(F )x∗∈PST= 2(|W | − 1).

≤ |W | − 1.

| supp(x∗)| = |F | ≤ |L|+ |D| ≤ 3(|W | − 1).

z(δD(w)) =∑w∈W

z(δ(w)) =∑w∈W

(|δ(w)| − x∗(δ(w)))

≤ 2|F | −∑w∈W

x∗(δ(w)) = 2|F | − 2x∗(F ) ≤ 4(|W | − 1).

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|D| ≤∑

w∈W x∗(δD(w)) =∑

w∈W x∗(δ(w)) = 2x∗(F )x∗∈PST= 2(|W | − 1).

≤ |W | − 1.

| supp(x∗)| = |F | ≤ |L|+ |D| ≤ 3(|W | − 1).

z(δD(w)) =∑w∈W

z(δ(w)) =∑w∈W

(|δ(w)| − x∗(δ(w)))

≤ 2|F | −∑w∈W

x∗(δ(w)) = 2|F | − 2x∗(F ) ≤ 4(|W | − 1).

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|D| ≤∑

w∈W x∗(δD(w)) =∑

w∈W x∗(δ(w)) = 2x∗(F )x∗∈PST= 2(|W | − 1).

≤ |W | − 1.

| supp(x∗)| = |F | ≤ |L|+ |D| ≤ 3(|W | − 1).

z(δD(w)) =∑w∈W

z(δ(w)) =∑w∈W

(|δ(w)| − x∗(δ(w)))

≤ 2|F | −∑w∈W

x∗(δ(w)) = 2|F | − 2x∗(F ) ≤ 4(|W | − 1).

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Conclusions

Even for very general degree constraints (matroidal degree constraints), a tree ofcost ≤ OPT can be obtained with a constant additive degree violation.

Targeting matroid intersection instead of single matroid seems like an interestingplan in iterative relaxation framework.

Extensions to other problems?

Obtaining an additive violation < 8?

Constant multiplicative errors for special families of cut constraints?

Constant-thin spanning trees (implies constant factor approx for ATSP)?

Thank you!

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Conclusions

Even for very general degree constraints (matroidal degree constraints), a tree ofcost ≤ OPT can be obtained with a constant additive degree violation.

Targeting matroid intersection instead of single matroid seems like an interestingplan in iterative relaxation framework.

Extensions to other problems?

Obtaining an additive violation < 8?

Constant multiplicative errors for special families of cut constraints?

Constant-thin spanning trees (implies constant factor approx for ATSP)?

Thank you!

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References I

Asadpour, A., Goemans, M. X., Madry, A., Oveis Gharan, S., and Saberi, A. (2010).An O(log n/ log log n)-approximation algrithm for the asymmetric travelingsalesman problem. In Proceedings of the 20th Annual ACM -SIAM Symposium onDiscrete Algorithms (SODA).

Bansal, N., Khandekar, R., Konemann, J., Nagarajan, V., and Peis, B. (2010). Ongeneralizations of network design problems with degree bounds. In Proceedings ofInteger Programming and Combinatorial Optimization (IPCO), pages 110–123.

Bansal, N., Khandekar, R., and Nagarajan, V. (2009). Additive guarantees fordegree-bounded directed network design. SIAM Journal on Computing,39(4):1413–1431.

Chaudhuri, K., Rao, S., Riesenfeld, S., and Talwar, K. (2005). What would edmondsdo? augmenting paths and witnesses for degree-bounded MSTs. In Chekuri, C.,Jansen, K., Rolim, J. D. P., and Trevisan, L., editors, Approximation,Randomization and Combinatorial Optimization, volume 3624 of Lecture Notes inComputer Science, pages 26–39. Springer Berlin / Heidelberg. 10.1007/11538462 3.

Furer, M. and Raghavachari, B. (1994). Approximating the minimum-degree SteinerTree to within one of optimal. Journal of Algorithms, 17(3):409–423.

References IIGoemans, M. X. (2006). Minimum bounded degree spanning trees. In Proceedings of

the 47th IEEE Symposium on Foundations of Computer Science (FOCS), pages273–282.

Konemann, J. and Ravi, R. (2002). A matter of degree: Improved approximationalgorithms for degree-bounded minimum spanning trees. SIAM Journal onComputing, 31:1783–1793.

Konemann, J. and Ravi, R. (2003). Primal-dual meets local search: approximatingMST’s with nonuniform degree bounds. In Proceedings of the 35th Annual ACMSymposium on Theory of Computing (STOC), pages 389–395.

Singh, M. and Lau, L. C. (2007). Approximating minimum bounded degree spanningtrees to within one of optimal. In Proceedings of the 39th Annual ACM Symposiumon Theory of Computing (STOC), pages 661–670.

Matroidal Degree-Bounded Minimum Spanning Trees · 2016. 11. 23. · jT \Uj B U for U 2U 2E....

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