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On the Hardness Of TSP with On the Hardness Of TSP with Neighborhoods and related ProblemsNeighborhoods and related Problems

(some slides borrowed from Dana Moshkovitz)(some slides borrowed from Dana Moshkovitz)

O. Schwartz & S. Safra

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Desire: Desire: A Tour Around the World A Tour Around the World

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The Problem: The Problem: Traveling Costs MoneyTraveling Costs Money

1795$

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But I want to do so muchBut I want to do so much

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The Group-TSP (G-TSP)The Group-TSP (G-TSP)

A Minimal cost tour, butAll goals are accomplished.

TSP with NeighborhoodsOne of a Set TSPErrand Scheduling

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The G-TSPThe G-TSP

Generalizes:TSPHitting Set

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G-TSP - The Euclidean VariantG-TSP - The Euclidean Variant

TSP – PTAS [Aro96, Mit96]

Hitting Set – hardness factor log n [Fei98]

Which is it more like ?

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ApproximationsApproximations

[AH94] – Constant for well behaved regions.

[MM95],[GL99] – O(log n) for more generalized cases.

[DM01] – PTAS for unit disk.

[dBGK+02] – Constant for Convex fat objects.

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Group Steiner Tree (G-ST)Group Steiner Tree (G-ST)

Say you have a network, with links between some components, each with different capabilities (fast computing, printing, backup, internet access, etc).

Each link can be protected against monitoring, at a different cost.

The goal is to have all capabilities accessible through protected lines (at least for some nodes on the net) .

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The G-STThe G-ST

A minimal cost tree, butAll capabilities are accessible.

Class Steiner ProblemTree Cover ProblemOne of a Set Steiner Problem

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The G-STThe G-ST

Generalizes:Steiner Tree - Each location

contains a single distinguished goal.

Hitting Set - The graph is complete and all edges are of weight 1.

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G-ST - The Euclidean VariantG-ST - The Euclidean Variant

ST – PTAS [Aro96, Mit96]

Hitting Set – hardness factor - log n [Fei98]

Which is it more like ?

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Some Parameters of the Geometric Some Parameters of the Geometric VariantVariant

Dimension of the Domain

Is each region connected ?

Are regions Pairwise Disjoint ?

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Mitchell’s Open Problems [Mit00]Mitchell’s Open Problems [Mit00]

[21] Is there an O(1)-approximation for the group Steiner problem on a set of points in the Euclidean plane ?

[27] Does the TSP with connected neighborhoods problem have a polynomial-time O(1)-approximation algorithm ? What if neighborhoods are not connected sets (e.g. if neighborhoods are discrete sets of points) ?

[30] Give an efficient approximation algorithm

for watchman routes in polyhedral domain.

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Previous Result [dBGKPrevious Result [dBGK++02]02]

G-TSP in the plane cannot be approximated to within

unless P = NP

Holds for connected sets, but not necessarily for pairwise disjoint sets.

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Our ResultsOur Results

G-TSP and G-ST

Dimension2-D3-Dd

Pairwise Disjoint Sets

YesNoYesNoYesNo

Connected sets-2 -

Unconnected sets

Resolving [Mit00, o.p. 21 and 27]

Improving [dBGK+02dBGK+02]

And resolving [Mit00, o.p. 30] regarding WT & WP

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gap- G-TSP-[gap- G-TSP-[aa,, b b]]

YES - There exists a solution of size at most b.

NO - The size of every solution is at least a.

Otherwise – Don’t care.

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From Gap to InapproximabilityFrom Gap to Inapproximability

If we can show it’s NP-hard to distinguish between two far off cases,

then it’s also hard to even approximate the solution.

the size of the min-Traversal is extremely small

the size of the min-Traversal is

tremendously big

Similarly for G-ST

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gap- G-TSP-[gap- G-TSP-[aa, , bb]]

If gap- G-TSP-[a, b] is NP-hard

then (for any > 0)

approximate G-TSP to within

is NP-hard

a

b

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Gap Preserving ReductionsGap Preserving Reductions

Gap-VC Gap-G-ST

•YES

•don’t care

•NO

• YES

• don’t care

• NO

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Hyper-GraphsHyper-Graphs

A hyper-graph G=(V,E), is a set of vertices V and a set of edges E, where each edge is a subset of V.

We call it a k-hyper-graph if each edge is of size k.

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VERTEX-COVER in Hyper-GraphsVERTEX-COVER in Hyper-Graphs

Instance: a hyper-graph G.

Problem: find a set UV of minimal size s.t. for any (v 1 ,… , v k)E, at least one of the vertices v 1 ,… , v k is in U.

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How hard is Vertex Cover ?How hard is Vertex Cover ?

Theorems:

[Tre01] For sufficiently large k, Gap-k-hyper-graph-VC-[1-, k-19 ] is NP-hard

[DGKR02] Gap-k-hyper-graph-VC-[1-, (k-1- )-1] is NP-hard ( for k > 4 )

[DGKR02] Gap-hyper-graph-VC-[1-, O(log-1/3n)] is intractable unless NP µ TIME (nO(log log n))

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Main ResultMain Result

Thm: G-ST in the plane is hard to approximate to within any constant factor.

Proof: By reduction from Gap-Hyper-Graph-Vertex-Cover.

We’ll show that for any k, Gap-ST-[ ] is NP-hard

19-

2κ1 ,-ε

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The Construction: The Construction: XX

n

1

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CompletenessCompleteness

n

Claim: If every vertex cover of G is of size at least (1-)n then every solution T for X is of size at least (1-)n-1.

Proof: Trivial.

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SoundnessSoundness

3

tn

n

Lemma:

If there is a vertex cover of G of size at most

then there is a solution T for X of size at most .

1

tn

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ProofProof: A Natural Tree T: A Natural Tree TNN(U)(U)

t

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ProofProof: A Natural Tree T: A Natural Tree TNN(U)(U)

t

3( ) 2

2N

n tT U

nn

t t

n

t

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Therefore, from the NP-hardness of [Tre01]Gap-k-hyper-graph-VC-[ ]

we deduce that Gap-ST-[ ] is NP-hard

Hence, (as k is arbitrary large), G-ST in the plane cannot be approximated to within any constant factor, unless P=NP.

▪

19-

2, 1 3k - ε

-191-ε,k

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Using A Stronger Complexity AssumptionUsing A Stronger Complexity Assumption

[DGKR02] Gap-hyper-graph-VC-[ ] is intractable unless NP µ TIME (nO(log log n))

we deduce that Gap-ST-[ ] in the plane is intractable unless NP µ TIME (nO(log log n))

Hence, G-ST in the plane cannot be approximated

to within unless NP µ TIME (nO(log log

n)).

▪

1

63lo, g1-ε n

1

3lo, g1-ε n

1

6(log )O n

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G-TSPG-TSP

Corollary 1: G-TSP cannot be approximated to within any constant factor unless P=NP.

Corollary 2: G-TSP cannot be approximated to within unless NP µ TIME (nO(log log n)).

1

6(log )O n

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G-TSPG-TSP

Proof: any efficient -approximation for G-TSP , yields an efficient 2-approximation for G-ST

(by removing an edge), as

T*G-TSP · 2T*

G-ST

▪

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How about log n ?How about log n ?

Why not use the ln n hardness of [Fei98] ?

(to obtain a factor of log½n)

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How hard is Vertex Cover ?How hard is Vertex Cover ?

Theorems:

[Tre01] For sufficiently large k, Gap-k-hyper-graph-VC-[1-, k-19 ] is NP-hard

[DGKR02] Gap-k-hyper-graph-VC-[1-, (k-1- )-1] is NP-hard ( for k > 4 )

[DGKR02] Gap-hyper-graph-VC-[1-, O(log-1/3n)] is intractable unless NP µ TIME (nO(log log n))

We need this (almost) perfect CompletenessCompleteness!

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Gap LocationGap Location

Theorems:

[Fei98] Gap-hyper-graph-VC-[t ln n, t ] is intractable unless NP µ TIME (nO(log log n))

Where t<1

What’s the problem ?

n

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If the two properties are jointIf the two properties are joint

Conjecture:

Gap-hyper-graph-VC-[1- , log-1n ] is intractable unless NP µ TIME (nO(log log n))

Corollary:

G-TSP and G-ST cannot be approximated to within log½n, unless NP µ TIME (nO(log log n))

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Other resultsOther results

Applying it to connected sets, dimension 3 and above.

The case of sets of constant number of points.

O(log1/6 n) for Minimum Watchman Tour & Minimum Watchman Path.

2- for GTSP nd GST with Connctd sts in th pln.

Dimension d – a hardness factor of

and toward a factor of , which generalizes to

.

Open problems…

1 1

3logd

dO n

1

logd

dO n

1logO n

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Open ProblemsOpen Problems

Is Gap-hyper-graph-VC-[1- , log-1n ] intractable unless NP µ TIME (nO(log log n)) ?

Can we do better than 2- for connected sets in the plane ?Can we do anything for connected, pairwise disjoint sets on the plane ?

Can we avoid the square root loss ?

Does higher dimension impel an increase in complexity ?

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2D unconnected to 3D 2D unconnected to 3D connectedconnected

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Minimum Watchman Tour and Minimum Watchman Tour and PathPath

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Triangular Grid – For a better Triangular Grid – For a better ConstantConstant

1

3

4

2n

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G-TSP and G-ST – Connected sets in the G-TSP and G-ST – Connected sets in the planeplane

Theorem:G-TSP and G-ST cannot be approximated to within 2-, unless P=NP

Proof: By reduction from Hyper-Graph-Vertex-Cover.

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The constructionThe construction

d

r

F = E

G = (V,E) G’

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The constructionThe construction

l

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Making it connectedMaking it connected

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From a vertex cover From a vertex cover UU to to a natural traversal a natural traversal TTNN(U)(U)

|TN(U)| 2d|U| + 2pr

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From a vertex cover From a vertex cover UU to to a natural Steiner tree a natural Steiner tree TTNN(U)(U)

|TN(U)| d|U| + 2pr

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Natural is the BestNatural is the Best

Lemma:

For some parameter d(r), and for sufficiently large n and l, the shortest traversal (tree) is the natural traversal (tree) of a minimal vertex-cover.

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Natural is the BestNatural is the Best

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Natural is the BestNatural is the Best

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Natural is the BestNatural is the Best

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Natural is the BestNatural is the Best

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Natural is the BestNatural is the Best

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Natural is the BestNatural is the Best

|T| ≥ |T’| ≥ |TN(U)|

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Natural is the BestNatural is the Best

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Natural is the BestNatural is the Best

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Natural is the BestNatural is the Best

|T| ≥ |T’| - ≥ |TN(U)| -

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Maximizing the Gap RatioMaximizing the Gap Ratio

|TN(UYES)| 2d|UYES| + 2pr |TN(UNO)| 2d|UNO| + 2pr

We want d as large as possible !

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Maximizing d – for G-STMaximizing d – for G-ST

D ≥ d

2(ρ+d)sin(p/n) ≥ d

2(ρ+d)p/n + ≥ d

2pρ/n + ’ ≥ d

ρ d

Dp/n

|TN(U)| 2 pr |U|/n + 2 pr

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Maximizing d – for G-TSPMaximizing d – for G-TSP

D ≥ 2d +

2(ρ+d)sin(p/n) ≥ 2d +

pρ/n + ’ ≥ dρ d

Dp/n

|TN(U)| 2 pr |U|/n + 2 pr

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G-TSP and G-ST in the PlaneG-TSP and G-ST in the Plane

If Gap-k-Hyper-Graph-Vertex-Cover-[A,B] is NP-hard, then (for any > 0)

Gap-k-G-TSP-[1+A- ,1+B+ ] is NP-hardGap-k-G-ST-[1+A- ,1+B+ ] is NP-hard

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G-TSP and G-ST in the PlaneG-TSP and G-ST in the Plane

[Tre01] For sufficiently large k, Gap-k-hyper-graph-VC-[1-, k-19 ] is NP-hard

Therefore (for any > 0),

Gap-G-TSP-[2-, 1+] is NP-hard and

Gap-G-ST-[2-, 1+] is NP-hard.

even if each set is connected

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