Loading...

PPT – On the Hardness Of TSP with Neighborhoods and related Problems PowerPoint presentation | free to download - id: 12de3e-MDdmN

The Adobe Flash plugin is needed to view this content

On the Hardness Of TSP with Neighborhoods and

related Problems

O. Schwartz S. Safra

(some slides borrowed from Dana Moshkovitz)

Desire A Tour Around the World

The Problem Traveling Costs Money

1795

But I want to do so much

The Group-TSP (G-TSP)

- A Minimal cost tour, but
- All goals are accomplished.
- TSP with Neighborhoods
- One of a Set TSP
- Errand Scheduling

The G-TSP

- Generalizes
- TSP
- Hitting Set

G-TSP - The Euclidean Variant

- TSP PTAS Aro96, Mit96
- Hitting Set hardness factor log n Fei98
- Which is it more like ?

Approximations

- AH94 Constant for well behaved regions.
- MM95,GL99 O(log n) for more generalized

cases. - DM01 PTAS for unit disk.
- dBGK02 Constant for Convex fat objects.

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) .

The G-ST

- A minimal cost tree, but
- All capabilities are accessible.
- Class Steiner Problem
- Tree Cover Problem
- One of a Set Steiner Problem

The 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.

G-ST - The Euclidean Variant

- ST PTAS Aro96, Mit96
- Hitting Set hardness factor - log n Fei98
- Which is it more like ?

Some Parameters of the Geometric Variant

- Dimension of the Domain
- Is each region connected ?
- Are regions Pairwise Disjoint ?

Mitchells 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.

Previous Result dBGK02

- G-TSP in the plane cannot be approximated to

within - unless P NP
- Holds for connected sets, but not necessarily for

pairwise disjoint sets.

Our Results

Improving dBGK02

And resolving Mit00, o.p. 30 regarding WT WP

Resolving Mit00, o.p. 21 and 27

gap- G-TSP-a, b

- YES - There exists a solution of size at most b.
- NO - The size of every solution is at least a.
- Otherwise Dont care.

From Gap to Inapproximability

- If we can show its NP-hard to distinguish

between two far off cases, - then its 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

gap- G-TSP-a, b

- If gap- G-TSP-a, b is NP-hard
- then (for any ? gt 0)
- approximate G-TSP to within
- is NP-hard

Gap Preserving Reductions

Gap-VC

Gap-G-ST

- YES

- YES

- dont care

- dont care

- NO

- NO

Hyper-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.

VERTEX-COVER in Hyper-Graphs

- Instance a hyper-graph G.
- Problem find a set U?V 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.

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 gt 4 ) - DGKR02 Gap-hyper-graph-VC-1-?, O(log-1/3n) is

intractable unless NP µ TIME (nO(log log n))

Main Result

- Thm G-ST in the plane is hard to approximate to

within any constant factor. - Proof By reduction from Gap-Hyper-Graph-Vertex-C

over. - Well show that for any k,
- Gap-ST- is NP-hard

The Construction X

1

Completeness

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.

Soundness

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 .

Proof A Natural Tree TN(U)

Proof A Natural Tree TN(U)

- 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 PNP. - ?

Using 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)). - ?

G-TSP

- Corollary 1 G-TSP cannot be approximated to

within any constant factor unless PNP. - Corollary 2 G-TSP cannot be approximated to

within unless NP µ TIME (nO(log log n)).

G-TSP

- Proof any efficient ?-approximation for G-TSP ,

yields an efficient 2?-approximation for G-ST - (by removing an edge), as
- TG-TSP 2TG-ST
- ?

How about log n ?

- Why not use the ln n hardness of Fei98 ?
- (to obtain a factor of log½n)

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 gt 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 Completeness!

Gap Location

- Theorems
- Fei98 Gap-hyper-graph-VC-t ln n, t is

intractable unless NP µ TIME (nO(log log n)) - Where tlt1
- Whats the problem ?

If 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))

Other 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 G-TSP and G-ST with Connected sets in the

plane. - Dimension d a hardness factor ofand toward a

factor of , which generalizes to

. - Open problems

Open 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 ?

2D unconnected to 3D connected

Minimum Watchman Tour and Path

Triangular Grid For a better Constant

1

G-TSP and G-ST Connected sets in the plane

TheoremG-TSP and G-ST cannot be approximated to

within 2-?, unless PNP Proof By reduction from

Hyper-Graph-Vertex-Cover.

The construction

G (V,E)

G

d

?

F E

The construction

l

Making it connected

From a vertex cover U to a natural traversal

TN(U)

- TN(U) ? 2dU 2??

From a vertex cover U to a natural Steiner tree

TN(U)

TN(U) ? dU 2??

Natural is the Best

LemmaFor some parameter d(?), and for

sufficiently large n and l, the shortest

traversal (tree) is the natural traversal (tree)

of a minimal vertex-cover.

Natural is the Best

Natural is the Best

Natural is the Best

Natural is the Best

Natural is the Best

Natural is the Best

- T T TN(U)

Natural is the Best

Natural is the Best

Natural is the Best

- T T - ? TN(U) - ?

Maximizing the Gap Ratio

TN(UNO) ? 2dUNO 2??

TN(UYES) ? 2dUYES 2??

We want d as large as possible !

Maximizing d for G-ST

D d ? 2(?d)sin(?/n) d ? 2(?d)?/n ? d ?

2??/n ? d

TN(U) ? 2 ?? U/n 2 ??

Maximizing d for G-TSP

D 2d ? ? 2(?d)sin(?/n) 2d ? ? ??/n ?

d

TN(U) ? 2 ?? U/n 2 ??

G-TSP and G-ST in the Plane

If Gap-k-Hyper-Graph-Vertex-Cover-A,B is

NP-hard, then (for any ? gt 0) Gap-k-G-TSP-1A-

?,1B ? is NP-hardGap-k-G-ST-1A- ?,1B ?

is NP-hard

G-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 ? gt

0), Gap-G-TSP-2-?, 1? is NP-hard

and Gap-G-ST-2-?, 1? is NP-hard. even if

each set is connected ?