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On the Hardness Of TSP with Neighborhoods and related Problems

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1. On the Hardness Of TSP with Neighborhoods and related Problems ... the size of the min-Traversal is tremendously big. Similarly for G-ST. 19. gap- G-TSP-[a, b] ... – PowerPoint PPT presentation

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Title: On the Hardness Of TSP with Neighborhoods and related Problems


1
On the Hardness Of TSP with Neighborhoods and
related Problems
O. Schwartz S. Safra
(some slides borrowed from Dana Moshkovitz)
2
Desire A Tour Around the World
3
The Problem Traveling Costs Money
1795
4
But I want to do so much
5
The Group-TSP (G-TSP)
  • A Minimal cost tour, but
  • All goals are accomplished.
  • TSP with Neighborhoods
  • One of a Set TSP
  • Errand Scheduling

6
The G-TSP
  • Generalizes
  • TSP
  • Hitting Set

7
G-TSP - The Euclidean Variant
  • TSP PTAS Aro96, Mit96
  • Hitting Set hardness factor log n Fei98
  • Which is it more like ?

8
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.

9
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) .
10
The G-ST
  • A minimal cost tree, but
  • All capabilities are accessible.
  • Class Steiner Problem
  • Tree Cover Problem
  • One of a Set Steiner Problem

11
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.

12
G-ST - The Euclidean Variant
  • ST PTAS Aro96, Mit96
  • Hitting Set hardness factor - log n Fei98
  • Which is it more like ?

13
Some Parameters of the Geometric Variant
  • Dimension of the Domain
  • Is each region connected ?
  • Are regions Pairwise Disjoint ?

14
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.

15
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.

16
Our Results
Improving dBGK02
And resolving Mit00, o.p. 30 regarding WT WP
Resolving Mit00, o.p. 21 and 27
17
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.

18
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
19
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

20
Gap Preserving Reductions
Gap-VC
Gap-G-ST
  • YES
  • YES
  • dont care
  • dont care
  • NO
  • NO

21
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.

22
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.

23
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))

24
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

25
The Construction X
1
26
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.
27
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 .
28
Proof A Natural Tree TN(U)
29
Proof A Natural Tree TN(U)
30
  • 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.
  • ?

31
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)).
  • ?

32
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)).

33
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
  • ?

34
How about log n ?
  • Why not use the ln n hardness of Fei98 ?
  • (to obtain a factor of log½n)

35
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!
36
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 ?

37
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))
38
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

39
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 ?

40
2D unconnected to 3D connected
41
Minimum Watchman Tour and Path
42
Triangular Grid For a better Constant
1
43
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.
44
The construction
G (V,E)
G
d
?
F E
45
The construction
l
46
Making it connected
47
From a vertex cover U to a natural traversal
TN(U)
  • TN(U) ? 2dU 2??

48
From a vertex cover U to a natural Steiner tree
TN(U)
TN(U) ? dU 2??
49
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.
50
Natural is the Best
51
Natural is the Best
52
Natural is the Best
53
Natural is the Best
54
Natural is the Best
55
Natural is the Best
  • T T TN(U)

56
Natural is the Best
57
Natural is the Best
58
Natural is the Best
  • T T - ? TN(U) - ?

59
Maximizing the Gap Ratio
TN(UNO) ? 2dUNO 2??
TN(UYES) ? 2dUYES 2??
We want d as large as possible !
60
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 ??
61
Maximizing d for G-TSP
D 2d ? ? 2(?d)sin(?/n) 2d ? ? ??/n ?
d
TN(U) ? 2 ?? U/n 2 ??
62
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
63
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 ?
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