Transcript The Traveling Salesman Problem

Case Studies:
Bin Packing &
The Traveling Salesman Problem
David S. Johnson
AT&T Labs – Research
© 2010 AT&T Intellectual Property. All rights reserved. AT&T and the AT&T logo are trademarks of AT&T Intellectual Property.
Applications
Packing commercials into station breaks
Packing files onto floppy disks (CDs, DVDs, etc.)
Packing MP3 songs onto CDs
Packing IP packets into frames, SONET time
slots, etc.
• Packing telemetry data into fixed size packets
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Standard Drawback: Bin Packing is NP-complete
NP-Hardness
• Means that optimal packings cannot be constructed in
worst-case polynomial time unless P = NP.
• In which case, all other NP-hard problems can also be
solved in polynomial time:
–
–
–
–
Satisfiability
Clique
Graph coloring
Etc.
• See M.R. Garey & D.S.Johnson, Computers and
Intractability: A Guide to the Theory of NPCompleteness, W.H.Freeman, 1979.
Standard Approach to Coping
with NP-Hardness:
• Approximation Algorithms
– Run quickly (polynomial-time for theory,
low-order polynomial time for practice)
– Obtain solutions that are guaranteed to be
close to optimal

First Fit (FF):
Decreasing,
Put eachBest
item Fit
in the
Decreasing
first bin with
(FFD,BFD):
enough space
Start by reordering the items by non-increasing size.
Best Fit (BF): Put each item in the bin that will hold it with
the least space left over
Worst-Case Bounds
• Theorem [Ullman, 1971][Johnson et al.,1974].
For all lists L,
BF(L), FF(L) ≤ (17/10)OPT(L) + 3.
• Theorem [Johnson, 1973]. For all lists L,
BFD(L), FFD(L) ≤ (11/9)OPT(L) + 4.
(Note 1: 11/9 = 1.222222…)
(Note 2: These bounds are asymptotically tight.)
Lower Bounds: FF and BF
½-
½-
½+
OPT: N bins
½+
½-
FF, BF:
N/2 bins
+
N bins
= 1.5 OPT
Lower Bounds: FF and BF
1/6 - 2
1/6 - 2
1/6 - 2
1/3 + 
1/6 - 2
1/3 + 
1/6 - 2
½+
1/2 + 
1/6 - 2
1/3 + 
1/6 - 2
OPT: N bins
FF, BF:
N/6 bins + N/2 bins + N bins
= 5/3 OPT
Lower Bounds: FF and BF
1/7 + 
1/43 + ,
1/1806 + ,
etc.
1/3 + 
FF, BF = N(1 + 1/2 + 1/6 + 1/42 + 1/1805 + … )
1/2 + 
OPT: N bins
 (1.691..) OPT
Jeff Ullman’s Idea
1/6 + 
1/3 + 
1/2 + 
OPT: N bins
FF, BF = N (1 + 1/2 + 1/5) = (17/10) OPT
Key modification:
Replace some 1/3 +’s & 1/6+’s
with 1/3-i & 1/6+i, and some
with 1/3+i & 1/6-i,
with i increasing from bin to bin.
(Actually yields FF, BF ≥ (17/10)OPT - 2
See References for details.)
Lower Bounds: FFD and BFD
1/4-2
1/4-2
1/4+
1/4-2
½+
1/4+2
1/4-2
1/4+2
3n bins
OPT = 9n
1/4-2
1/4+
1/4-2
½+
1/4+
1/4+2
6n bins
1/4+
6n bins
2n bins
1/4-2
3n bins
FFD, BFD = 11n
Asymptotic Worst-Case Ratios
• RN(A) = max{A(I)/OPT(I): OPT(I) = N}
• R(A) = max{RN(A): N > 0}
– absolute worst-case ratio
• R∞(A) = limsupN∞RN(A)
– asymptotic worst-case ratio
• Theorem: R∞(FF) = R∞(BF) = 17/10.
• Theorem: R∞(FFD) = R∞(BFD) = 11/9.
Why Asymptotics?
• Partition Problem: Given a set X of
numbers xi, can they be partitioned into
two subsets with the sums (xXx)/2?
• This problem is NP-hard, and is
equivalent to the special case of bin
packing in which we ask if OPT = 2.
• Hence, assuming P  NP, no polynomialtime bin packing algorithm can have R(A)
< 3/2.
Online Algorithms
• We have no control over the order in which items
arrive.
• Each time an item arrives, we must assign it to a bin
before knowing anything about what comes later.
• Examples: FF and BF
• Simplest example: Next Fit (NF)
– Start with an empty bin
– If the next item to be packed does not fit in the current bin,
put it in a new bin, which now becomes the “current” bin.
Problem 1: Prove R∞(NF) = 2.
Best Possible Online Algorithms
• Theorem [van Vliet, 1996]. For any online bin packing
algorithm A, R∞(A) ≥ 1.54.
• Theorem [Richey, 1991]. There exist polynomial-time
online algorithms A with R∞(A) ≤ 1.59.
• Drawback: Improving the worst-case behavior of
online algorithms tends to guarantee that their
average-case behavior will not be much better.
• FF and BF are much better on average than they are
in the worst case.
Best Possible Offline Algorithm?
• Theorem [Karmarkar & Karp, 1982]. There is a
polynomial-time (offline) bin packing algorithm KK
that guarantees
KK(L) ≤ OPT(L) + log2(OPT(L)) .
• Corollary. R∞(KK) = 1.
• Drawback: Whereas FFD and BFD can be
implemented to run in time O(NlogN), the best bound
we have on the running time of KK is O(N8log3N).
• Still open: Is there a polynomial-time bin packing
algorithm A that guarantees A(L) ≤ OPT(L) + c for
any fixed constant c?
The Traveling Salesman Problem
Given:
Set of cities {c1,c2,…,cN }.
For each pair of cities {ci,cj}, a distance d(ci,cj).
Find:
Permutation π : {1,2,...,N}  {1,2,...,N} that minimizes
N1
 d(c
i1
π(i)
, cπ(i1) )  d(cπ(N) , cπ(1) )
Implementing Best Fit
• Put the unfilled bins in a binary search tree,
ordered by the size of their unused space
(choosing a data structure that implements
inserts and deletes in time O(logN)).
Bin
Index
Bin with smaller gap
Gap
Size
Bin with larger gap
Implementing Best Fit
• When an item x arrives, the initial “current node” is
the root of the tree.
• While the item x is not yet packed,
– If x fits in the current node’s gap
•
If x fits in the gap of the node’s left child, let that node
become the current node.
• Otherwise, pack x in the current node’s bin, and reinsert that
bin into the tree with its new gap.
– Otherwise,
• If the current node has a right child, let that become the
current node.
• Otherwise, pack x in a new bin, and add that to the tree.
Implementing First Fit
• New tree data structure:
– The bins of the packing are the leaves, in their original order.
– Each node contains the maximum gap in bins under it.
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To Be Continued…
• Next time: Average-Case Behavior
• For now: On to the TSP!