Accessible Set Systems

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Accessible Set Systems

• Andreas Klappenecker

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Matroid

• Let S be a finite set, and F a nonempty family of
  subsets of S, that is, F? P(S).
• We call (S,F) a matroid if and only if
• M1) If B?F and A ? B, then A?F.
• The family F is called hereditary
• M2) If A,B?F and AltB, then there exists x in
  B\A such that A?x in F
• This is called the exchange property

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Weight Functions

• A matroid (S,F) is called weighted if it equipped
  with a weight function w S-gtR, i.e., all
  weights are positive real numbers.
• If A is a subset of S, then
• w(A) ?a in A w(a).

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Greedy Algorithm for Matroids

• Greedy(M(S,F),w)
• A ?
• Sort S into monotonically decreasing order by
  weight w.
• for each x in S taken in monotonically decreasing
  order do
• if A?x in F then A A?x fi
• od
• return A

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Matroid Terminology

• Let (S,F) be a matroid.
• The elements in F are called independent sets or
  feasible sets.
• A maximal feasible set is called a basis.
• Since we have assume that the weight function is
  positive, the algorithm Greedy always returns a
  basis.

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Matroid Example

• Suppose that Sa,b,c,d.
• Construct the smallest matroid
• (S,F) such that a,b and c,d are contained in
  F.
• F ?, a, b, c, d, a,b, c,d,
• by the hereditary property
• a,c, b,c, a,d, b,d
• by the exchange property

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Conclusion

• Matroids characterize a group of problems for
  which the greedy algorithm yields an optimal
  solution.
• Kruskals minimum spanning tree algorithm fits
  nicely into this framework.

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Prims Algorithm for MST
v9
v1

• We first pick an arbitrary vertex v1 to start
  with.
• Maintain a set S v1.
• Over all edges from v1, find a lightest one. Say
  its (v1,v2).
• S ? S ? v2
• Over all edges from v1,v2 (to V-v1,v2), find
  a lightest one, say (v2,v3).
• S ? S ? v3
• In general, suppose we already have the subset S
  v1,,vi, then over all edges from S to V-S,
  find a lightest one (vj, vi1).
• Update S ? S ? vi1
• Finally we get a tree this tree is a minimum
  spanning tree.

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v8
v3
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v5
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v7
Slide due to Prof. Shengyu Zhang
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Critique

• Prims algorithm is a greedy algorithm that
  always produces optimal solutions.
• S edges of the graph G
• F A T(V,A) is a induced subgraph of G, is
  a tree, and contains the vertex v
• Apparently, Prims algorithm is essentially
  Greedy applied to (S,F). However, the set system
  (S,F) is not a matroid, since it is not
  hereditary! Why?
• Removing an edge from a tree can yield
  disconnected components, not necessarily a tree.

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Goal

• Generalize the theory from matroids to more
  general set systems (so that e.g. Prims MST
  algorithm can be explained).
• Allow weight functions with arbitrary weight.

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Accessible Set Systems

• Let S be a finite set, F a non-empty family of
  subsets of S. Then (S,F) is called a set system.
• The elements in F are called feasible sets.
• A set system (S,F) satisfying the accessibility
  axiom
• If A is a nonempty set in F, then there exists an
  element x in S such that A\x in F.
• is called an accessible set system.
• Greedy algorithms need to be able to construct
  feasible sets by adding one element at the time,
  hence the accessibility axiom is needed.

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Weight Functions

• Let w S -gt R be a weight function (negative
  weights are now allowed!).
• For a subset A of S, define
• w(A) ?a in A w(a).

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Accessible Set Systems

• A maximal set B in an accessible set system (S,F)
  is called a basis.
• Goal Solve the optimization problem BMAX
• maximize w(B) over all bases of M.

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Greedy Algorithm

• Greedy(M(S,F),w)
• A ?
• Sort S into monotonically decreasing order by
  weight w.
• for each x in S taken in monotonically decreasing
  order do
• if A?x in F then A A?x fi
• od
• return A

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Question

• Characterize the accessible set systems such that
  the greedy algorithm yields an optimal solution
  for any weight function w.
• This question was answered by Helman, Moret, and
  Shapiro in 1993, after initial work by Korte and
  Lovasz (Greedoids) and Edmonds, Gale, and Rado
  (Matroids).

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Extensibility Axiom Motivation

• Define ext(A) x in S\A A?x in F .
• Let A be in F such that ext(A)? and B a basis
  properly containing A. This can happen, see
  homework.
• Define
• w(x) 2 if x in A
• w(x) 1 if x in B\A
• w(x) 0 otherwise.
• Then the Greedy algorithm incorrectly returns A
  instead of B.

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Extensibility Axiom

• Extensibility Axiom
• For each basis B and every feasible set A
  properly contained in B, we have
• ext(A) ? B ? ?.
• This axiom is clearly necessary for the
  optimality of the greedy algorithm.

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Matroid Embedding Axiom

• Let M(S,F) be a set system. Define
• clos(M) (S,F), where F A A?B, B in F
• We call clos(M) the hereditary closure of M.
• Matroid embedding axiom
• clos(M) is a matroid.
• This axiom is necessary for the greedy algorithm
  to return optimal sets. Thus, matroid are always
  in the background, even though the accessible set
  system might not be a matroid.

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Congruence Closure Axiom

• Congruence closure axiom
• For every feasible set A and all x,y in ext(A)
  and every subset X in S\(A?ext(A)),
  A?X?x in clos(M) implies A?X?y in clos(M).
• This axiom restricts the future extensions of
  the set system. Note that it is a property in the
  hereditary closure.

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Theorem (Helman, Moret, Shapiro)

• Let M(S,F) be an accessible set system.
• For each weight function w S-gtR the optimal
  solutions to BMAX are the bases of M that are
  generated by Greedy (assuming a suitable ordering
  of the elements with the same weight)
• if and only if
• the accessible set system M satisfies the
  extension axiom, the congruence closure axiom,
  and the matroid embedding axiom.