Covering Trains by Stations or The power of Data Reduction

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Covering Trains by Stationsor The power of Data
Reduction

• Karsten Weihe, ALEX98, 1998
• Presented by Yantao Song

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Overview

• Problem description
• Data Reduction
• Computational study and experiment results

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Problem

• Given a set of trains, select a set of stations
  such that every train meets at least one of these
  stations and the number of selected trains is
  minimum.

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Formal Problem Description

• Given an undirected graph G(V, E), paths p1,
  p2pn in G, and a partition VV1? V2? ? Vm of V
  into m disjoint vertex classes.
• A PCV (path-cover by vertices) is a subset
  such that every path pi meets at least one
  vertex in V .
• The problem is to find a PCV V of minimum size
  V.
• More specifically, among all PCVs of minimum
  size, V should maximize the vector ( V n V1,
  V n V2, , V n Vm) lexicographically.
• This problem is NP-Hard.

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• Path pl is an ordered sequence (v1l, , vnll ) of
  vertices such that vil, vi1l ? E for i 1, ,
  nl 1.
• Vertices and edges may occur more than once in
  the same path.
• If an edge occurs more than once, it may occur
  several times with the same direction, or
  opposite direction.
• Its possible that two paths are exactly equal,
  or exact reverse of another path.
• Without losing generality, we can assume that
  every edge belongs to some paths.

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Papers background

• The data in this paper comes from the central
  German train railroad company.
• Paths are the trains in the time schedules.
• V is the union of all stations met by at least
  one of the trains.
• We have one edge v,w ? E iff v, w are directly
  connected vertices by at least one train path.
• Purpose find a minimum number of stations.
• It may be desirable to prefer some stations over
  other stations. So we have to maximize the
  vector ( V n V1, V n V2, , V n Vm)
  lexicographically as described above.

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Data Reduction

• For a vertex v? V , P(v) denotes the set of all
  paths pi meeting v.
• For a path pi, V(pi) denotes the ordinary set of
  vertices met by pi, which is unordered and dont
  allow repetitions of vertices.

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Vertexs dominance and equivalence

• Dominance Let i, j?1,,m, v?Vi w?Vj , if iltj
  and P(v)P(w) or igtj and P(w) P(v), then we
  say that v dominate w.
• Equivalence if P(v)P(w) and ij, v and w is
  equivalent.

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Paths dominance and equivalence

• Dominance Let i, j?1,,k, pi pj , if V(pi)
  V(pj), then we say that pi dominate pj.
• Equivalence if V(pi) V(pj), pi , pj is
  equivalent.

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Procedure of reducing vertex

• Remove v from V, and all edges incident to v from
  E.
• If u, w?V are incident to v, there is a path pi
  which contains u-v-w or w-v-u as a subpath, then
  an edge u, w should be added into E.
• All occurrences of v in paths are removed.

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Procedures of reducing a path

• Remove pi from path set.
• Every edge e?E which doesnt belong to any path
  afterward is removed.
• Every vertex v?V whose P(v) is empty afterwards
  is removed.

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• If the vertex/path is dominated by or equivalent
  to some other vertices/paths. Then its feasible
  to be reduced.
• At the early stage of reduction, use
  non-exhaustive vertex reduction at the end of
  reduction, use exhaustive reduction.
• After reducing, we can get an irreducible core.
• An optimal solution to an irreducible core is
  also an optimal solution to the original
  instance.
• Then we use the brute-force approach to solve the
  problem.

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Computational Study

• Experiments based on real world data of Europe
  train network.

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Train classes

• Class 0 high-speed trains
• Class 1 other international or long-distance
  trains
• Class 2 regional trains
• Class 3 local trains
• Class 4 other trains

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Before Reduction
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After reduction
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• For some instances which consist of trivial
  connected components, even we can get solution
  for problem only by data reduction.
• For the other cases, the number of non-trivial
  connected components and size of these components
  are essential to complexity of the problem.

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Conclusion and Discussion

• In this case, the size of problem can be reduced
  to 10 of original size.The reduction algorithm
  is very efficient for this case.
• This is an extreme case, cant be extended to all
  cases with so high efficiency. But it give us an
  case that even the problem is NP-Hard, but we
  still can solve it in affordable time for some
  real world cases.