Chapter 2' Optimal Trees and Paths

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Chapter 2. Optimal Trees and Paths

• 2.1 Minimum Spanning Trees
• Definitions about Graphs
• graph G (V, E), V (V(G)) set of nodes, E
  (E(G)) set of edges, relation associating with
  each edge a pair of its end nodes.
• simple graph graph with no parallel edges, no
  loops
• complete graph simple graph such that every
  pair of nodes is the set of ends of some edge.
• subgraph H of G V(H) ? V(G), E(H) ? E(G), and
  each e ? E(H) has the same ends in H as in G.
• G \ A, A ? E subgraph H s.t. V(H) V, E(H)
  E \ A.
• G \ B ( G V\B ), B ? V delete B and all
  edges incident with nodes in B. G\B is the
  subgraph induced by V\B.
• spanning subgraph H V(H) V(G)
• Usually use n V, m E.

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• path P from v0 to vk , (v0, vk)-path sequence
  v0, e1, v1, e2, , ek, vk such that the ends of
  ei are vi-1, vi. (walk)
• closed path path with v0 vk.
• edge-simple path e1, , ek distinct.
  (trail)
• simple path v0, , vk distinct. (path)
• circuit edge-simple and closed path, v0, ,
  vk-1 distinct. (cycle)
• length of path P number of edge-terms of P.
• G is connected if every pair of nodes is joined
  by a path.
• v is cut node of connected graph G if G \ v is
  not connected.
• Similar definitions for directed network.

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• Connector Problem
• Given a connected graph G and a positive cost ce
  for each e ? E, find a minimum cost spanning
  connected subgraph of G.

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• Lemma 2.1 An edge e uv of G is an edge of a
  circuit of G if and only if there is a path in G
  \ e from u to v.
• We still have a connected graph if we delete an
  edge of some circuit. So an optimal solution to
  the connector problem will not have any circuits
  since edge costs are positive.
• Def
• forest a graph having no circuit ( acyclic
  graph )
• tree connected forest ( connected acyclic
  graph )
• Minimum spanning tree problem
• Given a connected graph G and a real cost ce for
  each e ? E, find a minimum cost spanning tree of
  G.

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• Prop Following conditions are equivalent in
  defining a tree.
• G is a forest containing n-1 edges.
• Edge-minimal connected graph spanning V.
• Contains a unique path between each pair of nodes
  of V.
• The addition of any one edge not in E creates a
  unique cycle.
• Greedy style MST algorithms
• Kruskals algorithm
• Prims algorithm

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• Kruskals algorithm for MST
• Keep a spanning forest H (V, F) of G with F
  ? initially.
• At each step add to F a least-cost edge e ? F
  such that H remains a forest. (sort the edges
  and use them sequentially)
• Stop when H is a spanning tree.

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• Prims algorithm for MST
• Keep a tree H (V(H), T) with V(H) initially
  r for some r ? V, and T initially ?.
• At each step add to T a least-cost edge e not in
  T such that H remains a tree.
• Stop when H is a spanning tree.

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• Validity of MST algorithms
• Notation Let A ? V.
• ?(A) e ? E e has an end in A and an end in
  V \ A
• ?(A) e ? E both ends of e are in A
  (also denoted as E(A) freq.)
• ?(A) for some A ? V is called a cut of G
• (similar definition for directed graph,
  ?(R) vw vw?E, v?R, w?R for some R?E. (r,
  s)-cut is a cut for which r?R, s?R.)
• Thm 2.3 G (V, E) is connected if and only if
  there is no set A ? V, ? ? A ? V, with ?(A) ?.
• Pf) Show contraposition in both directions.
• ?) If ?(A) ? for some nontrivial A ? V and u
  ? A, v ? A, there is no path from u to v, hence G
  is not connected.
• ?) If G is not connected, there exists such
  set A. ?

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• Thm 2.4 Suppose B ? E and B is extendible to an
  MST. Edge e is a minimum cost edge of some cut D
  with D ? B ?.
• Then B ? e is extendible to an MST.
• Lemma 2.7 Let H (V, T) be a spanning tree of
  G. Let e vw be an edge of G but not H, and let
  f be an edge of a simple path in T from v to w.
  Then the subgraph H (V, (T ? e) \ f ) is a
  spanning tree of G.
• Pf of Them 2.4)
• Let H (V, T) be an MST such that B ? T.
• If e ? T, then done. Suppose e vw ? T. Let P
  be a simple path in T from v to w (e vw)
• ? ? f ? P such that f ? D.
• Then cf ? ce, hence (V, (T ? e) \ f ) is
  also an MST by Lemma 2.7.
• From D ? B ? ? f ? B ? B ? e is
  extendible to an MST. ?

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• See text Thm 2.5, Thm 2.6 for correctness of the
  Kruskal and Prims algorithms.
• Also see the text for running time of the
  algorithms.
• Prims alg. O(n2)
• Kruskals alg. O(mlogm) (sorting m edges)

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MST and LP

• Notation set A, vector p ? RA and any B ?
  A, let
• p(B) ? ? ( pj j ? B ).
• IP formulation of MST problem
• zIP min cTx (2.1)
• x( ?(S) ) ? S - 1, for all S, ? ? S ?
  V (2.2)
• x(E) V - 1 (2.3)
• xe ? 0, 1, for all e ? E
• ( xe ? 0 and integer, for all e ? E
  ) (2.4)
• ( n-1 edges and no cycles, subtour elimination
  formulation. )
• Dont be alarmed by the number of constraints.

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• Linear programming relaxation drop integrality
  requirements
• zLP min cTx (2.1)
• x( ?(S) ) ? S - 1, for all S, ? ? S ?
  V (2.2)
• x(E) V - 1 (2.3)
• xe ? 0, for all e ? E (2.4)
• feasible solutions to IP ? feasible
  solutions to LP relaxation
• Hence zLP ? zIP , i.e. zLP provides a lower
  bound on optimal IP value.
• Upper bound on zIP usually obtained by finding a
  feasible solution to IP.
• If lower bound upper bound, then we have found
  optimal value.
• For MST problem, zLP zIP.

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• We know that there exists an extreme point
  optimal solution of a linear programming problem
  if the LP has finite optimal value.
• (With the assumption that the underlying
  polyhedron has an extreme point.)
• So, if all extreme points of the feasible
  solution set of LP are integer vectors, we can
  find integer optimal solution by solving the LP
  relaxation.
• Results to be proven later If the LP relaxation
  has an integer optimal solution for any cost
  vector c, all the extreme points of the feasible
  solution set of the LP are integer vectors.
• Suppose we have two types of correct formulations
  for the same problem (in minimization form).
  Consider the LP relaxations and corresponding
  polyhedra P1 and P2 with P1 ? P2.
• Then zP2 ? zP1 ? zIP , hence LP relaxation over
  P1 provides more tight lower bound, which can be
  quite helpful in solving the IP problem by
  branch-and-bound.

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Idea of Branch-and-Bound Algorithm

• Solution set is non-convex. Difficult to solve.
• Idea divide and conquer.
• Divide feasible solutions set into many disjoint
  sets. Find the best solution for each subset.
  Then choose the best one. To divide the feasible
  solutions set, we add linear inequalities to the
  problem. (e.g. x1 ? 1, x2 ? 0, )
• But the divided problem is still integer
  programming problem. So we need a means to find
  the best integer solution for each integer
  programming problem.
• Linear programming relaxation

(IP)
(LPR)
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• Let zIP be the optimal value of (IP) and zLP be
  the optimal value of (LPR)
• If x is a feasible solution to (IP) ? x
  feasible solution to (LPR)
• Hence set of feasible solutions to (IP) ?
  set of feasible solutions to (LPR) and
  objective function is the same.
• ? zLP ? zIP
• Also a feasible solution to (IP) provides an
  upper bound on zIP.
• Hence if we solve (LPR) and obtain optimal
  solution which happens to be integral, then the
  solution provides an upper bound which is the
  same as the lower bound. It implies that the
  solution is optimal to the integer programming
  problem.
• If the obtained solution is not integral, we
  only have lower bound on optimal value. Then we
  divide the feasible solution set and repeat the
  procedure again for each divided problem.
• Note that other schemes are possible to obtain
  lower bound on zIP, other than the linear
  programming relaxation. (e.g., Lagrangian
  relaxation, dual problems, )

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• Results of solving LP relaxation.
• unbounded ? integer program unbounded
• infeasible ? integer program infeasible
• optimal solution which is integer ? it is
  optimal to integer program
• optimal solution not integer ? only obtain
  lower bound. Need to branch
• How to divide the solution set in case of 4.
• Suppose x optimal solution to LP relaxation
  and
• Then consider 2 sets
• Any feasible solution to integer program is
  contained in one of (a), (b). So we do not miss
  any feasible solution. Then we solve LP
  relaxation of (a), (b) again. (Search procedure
  with tree structure)

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• Useful tool to facilitate the search
• Suppose x is the best solution currently known
  and its objective value is z.
• If the LP relaxation of a subproblem gives
  optimal value z such that z ? z, we know
  that there does not exist an integer solution in
  this subproblem which gives objective value
  smaller than z.
• Hence we do not need to explore the subproblem
  any further and prune the subproblem (node).
• Obtaining a good (large) lower bound in each
  subproblem is important to reduce the number of
  subproblems examined. Modern approach adds more
  valid inequalities to tighten the feasible region
  (cutting plane approach).

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• Procedure to solve the LP relaxation (with many
  constraints)
• Cutting plane approach

Solve LP relaxation with small number of
constraints (e.g., w/o subtour elim. constr.)
If x satisfies all constr., stop. O/w find a
violated constr. (separation problem)
Solve LP after adding the violated constraint.
Y
? violated constr?
N
Stop
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• Alternative formulation for MST and LP relaxation
• zLP min cTx (2.1)
• x(?(S) ) ? 1, for all S, ? ? S ?
  V (2.2)
• x(E) V - 1 (2.3)
• xe ? 0 and integer, for all e ? E
  (2.4)
• ( n-1 edges and connected, cutset formulation.
  )
• The polyhedron of the LP relaxation of subtour
  elimination formulation is properly contained in
  the polyhedron for cutset formulation. Hence LP
  relaxation of the subtour elim. formulation
  provides stronger bound.
• ( see handout )
• Moreover, the extreme points of the polyhedron
  for the LP relaxation of the subtour elim.
  formulation are integer vectors.

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• General questions for a problem
• Extreme points of LP relaxation all integer
  vectors?
• If not, can we make the polyhedron have only
  integer extreme points by adding some
  inequalities to the LP relaxation? How much
  efforts?
• If not, can we approximate the integer
  polyhedron using only part of the inequalities
  needed to describe it? (obtain good lower bound)
  How much efforts?

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• Thm 2.8 Let x0 be the characteristic vector of
  an MST with respect to costs ce. Then x0 is an
  optimal solution of (2.1).
• Pf) Consider a form equivalent to (2.1).
• Let A ? E, ?(A) number of components in
  subgraph (V, A) of G.
• zLP min cTx (2.1)
• x( ?(S) ) ? S - 1, for all S, ? ? S ?
  V (2.2)
• (P1) x(E) V - 1 (2.3)
• xe ? 0, for all e ? E (2.4)
• zLP min cTx (2.5)
• x( A ) ? V - ?(A), for all A ? E (2.6)
• (P2) x(E) V - 1 (2.7)
• xe ? 0, for all e ? E (2.8)
• Let S1, , Sk be the node sets of the
  components in (V, A)
• Interpret V - ?(A) as ?i 1k ( Si - 1)
  V - k.

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• ( Given A ? E, we say that A is independent if
  there is no cycle in A. (2.6) means that the
  cardinality of a maximal independent set in A
  should not exceed V - k . More in matroid. )
• (2.6) ? (2.2) (i.e. x ? P2 ? x ? P1 )
• Let A ?(S). Then
• x(A) x( ?(S) ) ? V - ?( ?(S) ) ? V -
  V\S - 1 S -1 since ?( ?(S) ) ? V\S 1.
  
• (2.2) ? (2.6) (i.e. x ? P1 ? x ? P2 )
• Take nonnegative linear combination of (2.2)
  and x ? 0.
• Let A ? E, S1, , Sk node sets of the
  components in (V, A)
• Add lhs and rhs of (2.2) respectively for S1,
  , Sk .
• ?i 1k x( ?(Si ) ) ? ?i 1k ( Si - 1)
  V - k.
• Add x ? 0 for the edges in ?(Si ) but not in
  A to eliminate them from lhs.
• ? x( A ) ? V - k

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• Let x0 be the characteristic vector generated by
  Kruskals algorithm. (We do not need to assume
  that it is optimal by now)
• We show that x0 is optimal to (2.5) by providing
  a dual feasible solution satisfying complementary
  slackness conditions with x0.
• Consider the form max (- cTx), then dual is
• min ? A ? E ( V - ?(A) ) yA (2.9)
• s.t. ? ( yA e ? A ) ? - ce, for all e ?
  E (2.10)
• yA ? 0, for all A ? E, yE
  free (2.11)
• Let e1, , em be the sorted sequence in
  Kruskals algorithm. ( ce1 ? ce2 ? ? cem ).
  Let Ri e1, , ei , 1 ? i ? m.
• Construct dual solution as follows.
• yA0 0, if A is not one of Ri
• yRi0 ce(i1) ce(i) , 1 ? i ? m-1 (
  ? yRi0 ? 0 )
• yRm0 - ce(m)
• Hence yA0 ? 0 ? A ? E.

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• Check dual feasibility of y0.
• For e ei ( i-th edge considered in Ks alg.)
• ? ( yA0 e ? A) ? j im yRj0 ? j im-1
  ( ce(i1) ce(i) ) ce(m) - ce(i) - ce
• y0 dual feasible and part of CS conditions
  satisfied
• ( xe0 gt 0 ? corresponding dual constraint
  at equality )
• Need to show yA0 gt 0 ? corresponding primal
  constr. active at x0.
• We know A Ri for some i has yA0 gt 0. Consider
  a Ri with yRi0 gt 0 and suppose (2.6) does not
  hold at equality for this Ri.
• Then ? some edge in Ri whose addition to T ? Ri
  decreases the number of components in T ? Ri .
• Such edge would have been added to T by Ks
  algorithm. ?
• Proof here also shows alternatively that Ks
  algorithm gives an optimal solution since we did
  not assume that the solution obtained by Ks
  algorithm is optimal in the proof.

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• Thm 2.8 can be used to examine the strength of
  the bound obtained by 1-tree relaxation
  (Lagrangian relaxation) for traveling salesman
  problem.
• Why consider IP (LP) formulation for MST although
  ? efficient algorithms for it?
• Other variations of MST
• degree constrained spanning tree degree of each
  node should be ? k for all v ? V. (NP-hard)
• shortest total path length spanning tree sum of
  path lengths between every pair u, v is minimum.
  (NP-hard)
• Steiner tree problem Given G (V, E), T ? V,
  costs ce ? 0, e ? E.
• Find the min cost tree spanning T. (NP-hard)
• .

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Steiner tree problem

• Given G (V, E), T ? V, costs ce ? 0, e ? E.
  Find the min cost tree spanning T (called
  terminal nodes).
• Formulation
• zUD min cTx
• x(?(W) ) ? 1, for all W ?V, W ? T ?
  ?, T,
• 0 ? xe ? 1, for all e ? E
• x integer

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• Consider directed version of the Steiner tree
  problem (Steiner arborescence problem)
• We duplicate each edge uv by two antiparallel
  arcs (u, v), (v, u) with same costs cuv,
  obtaining directed graph D (V, A).
• Choose some r ? T, which we call the root. A
  Steiner arborescence (rooted at r) is a set of
  arcs S ? A such that (V(S), S) is a directed
  subtree that contains a directed path from r to t
  for all t ? T \ r.
• Note the one-to-one correspondence between a
  Steiner tree and its directed version (Steiner
  arborescence), hence can find a optimal Steiner
  arborescence and recover the optimal Steiner
  tree.

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• Example of a Steiner arborescence

Terminal nodes
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• Formulation for Steiner arborescence problem
• zD min cTy
• y(?(W) ) ? 1, for all W ?V, r?W,
  (V\W) ? T ? ?,
• 0 ? ya ? 1, for all a ? A
• y integer
• , where ?(W) ? (u, v)?A u?W, v ? V \ W
• Any ?(W) in the formulation is a (r, t)-cut for
  some t ? T.
• Suppose we solve the LP relaxation by cutting
  plane algorithm. Given a current solution y, we
  assign ya, a ? A as arc capacities, and find
  minimum (r, t)-cut for all t ? T for separation.
• It can be shown that the LP relaxation of the
  directed version gives stronger lower bound than
  the undirected version although it has twice as
  many variables. Computationally successful.
• There are other formulations and algorithms
  (e.g., dynamic programming) for Steiner tree
  problems, too.

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Models for Connectivity of Graphs

• Motivation Design network that can survive link
  and/or node failure. MST is the cheapest network
  structure that provides connectivity, but failure
  of an edge or node results in disruption of
  communication. We want to achieve the
  survivability of network in case of failure at
  minimum cost.
• Problem Given undirected G (V, E), costs ce, e
  ? E, find subgraph N (V, F), F ? E, that
  satisfies certain connectivity requirements at
  minimum cost.

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• Connectivity usually defined in terms of edge cut
  and node (vertex) cut.
• Def k-edge cut is an edge cut ( ?(S) for some S
  ? V) of k elements.
• Edge connectivity ( ?(G) ) of G is minimum k
  for which G has a k-edge cut.
• G is called k-edge connected if ?(G) ? k.
• Def A node cut of G is V ? V such that G\V is
  disconnected.
• k-node cut is a node cut of k elements.
• Node connectivity ( ?(G) ) of G is minimum k
  for which G has a k-node cut.
• G is called k-node connected if ?(G) ? k.

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• Definitions in terms of paths
• Def Two nodes s, t are called k-edge (k-node)
  connected if there exist k edge-disjoint
  (node-disjoint) simple paths between s and t.
• Def A graph is called k-edge (k-node) connected
  if all pairs of distinct nodes of G are k-edge
  (k-node) connected.
• Def For graph G, the largest integer k such
  that G is k-edge connected (k-node connected) is
  called the edge connectivity (node connectivity)
  of G and denoted ?(G) ( ?(G) ).

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• Mengers Thm states that both definitions for
  edge-connectivity (node-connectivity) are
  equivalent.
• Thm (Menger) G, with V ? k1, is k-edge
  connected (k-node connected) if and only if any
  two distinct nodes of G are connected by at least
  k-edge disjoint simple paths (k-node disjoint
  simple paths).
• Pf) consider later after studying max flow min
  cut Theorem.
• Refer Bondy and Murty Chap 3 and p.203 - 205.

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• IP models for connectivity see handout
• Survivability requirements
• rst there must exist ? rst edge disjoint
  (s, t) paths in N (V, F)
• kst, dst removal of at most kst nodes
  (except s, t) must leave
• at least dst edge disjoint (s, t)
  paths.
• Formulation
• min ? ij?E cij xij
• s.t (i) ? i?W ? j?V\W xij ? rst , ? s, t ?
  V, s ? t and
• ? W ? V such that s ? W, t ? W
• (ii) ? i?W ? j?V \ (Z?W) xij ? dst , ? s, t
  ? V, s ? t and
• ? Z ? V\s, t with Z kst and
• ? W ? V\Z with s ? W, t ? W
• (iii) 0 ? xij ? 1, ? ij ? E
• (iv) xij integral, ? ij ? E

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• Special cases
• MST drop (ii), set rst 1 , ? s, t ? V
  (assuming cij ? 0 )
• Steiner tree problem drop (ii), set rst 1, ?
  s, t ? S
• 0, otherwise
• Min cost k-edge connected network drop (ii)
• rst k, ? s, t ? V
• Min cost k-node connected network drop (i)
• kst k-1, dst 1, ? s, t ? V
• Practical models
• define node type rs ? Z for s ? V, rst ?
  min rs, rt
• N (V, F) satisfies node survivability (edge
  survivability) conditions if for all s, t ? V,
  there exist rst node disjoint (edge disjoint) (s,
  t) paths.
• ( conduit level network design)
• There can be different IP formulations for the
  respective problems.

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• The structure of the survivable network may
  depend on the facilities used (e.g. ADM (Add-Drop
  Multiplexor) needs to configure ring networks )