Median problems with positive and negative weights: some new results - - PowerPoint PPT Presentation

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Median problems with positive and negative weights: some new results

10th Combinatorial Optimization Workshop Aussois

Rainer E. Burkard Johannes Hatzl

Technische Universität Graz hatzl@opt.math.tu-graz.ac.at

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Overview

• 1. Problem formulation (classical p-median problem) and some

results

• 2. Semi-obnoxious p-median problems
• Different objective functions (MWD, WMD)
• Some properties of optimal solutions
• 3. Obnoxious p-median problems on trees
• 4. Semi-obnoxious 2-median problem on cycles

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• 1. Problem formulation and some results

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Formal Problem Definition

Given: graph G = (V, E) vertex weights w : V → R+ edge lengths l : E → R+ number of facilities: p

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Formal Problem Definition

Given: graph G = (V, E) vertex weights w : V → R+ edge lengths l : E → R+ number of facilities: p Task: Find a set Xp = {x1, . . . , xp} ⊂ G of p points that minimizes F(Xp) :=

• v∈V
• wv min

1≤j≤p d(v, xj)

• .

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p-median Problem

1 5 5 6 2 5 5 7 8 6 2 1 7 2 3 3 8 3 3 3 2 5 4 9 7 8 1

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p-median Problem

1 5 5 6 2 5 5 7 8 6 2 1 7 2 3 3 8 3 3 3 2 5 4 9 7 8 1 1 5 5 6 2 5 5 7 8 6 2 1 3 3 3 4 7 1

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Classical Results

Theorem (vertex optimality property, Hakimi 1964). There exists an

• ptimal solution Xp = {x1, . . . , xp} ⊂ V .

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Classical Results

Theorem (vertex optimality property, Hakimi 1964). There exists an

• ptimal solution Xp = {x1, . . . , xp} ⊂ V .

Theorem (NP-hard, Hakimi and Kariv 1979). The p-median problem is

NP-hard, even if G is a planar graph of maximum degree 3.

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Classical Results

Theorem (vertex optimality property, Hakimi 1964). There exists an

• ptimal solution Xp = {x1, . . . , xp} ⊂ V .

Theorem (NP-hard, Hakimi and Kariv 1979). The p-median problem is

NP-hard, even if G is a planar graph of maximum degree 3.

Theorem (Hua 1961). The optimal solution of the 1-median problem on a tree is independent of the edge lengths and can be found in linear time.

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Classical Results

Theorem (vertex optimality property, Hakimi 1964). There exists an

• ptimal solution Xp = {x1, . . . , xp} ⊂ V .

Theorem (NP-hard, Hakimi and Kariv 1979). The p-median problem is

NP-hard, even if G is a planar graph of maximum degree 3.

Theorem (Hua 1961). The optimal solution of the 1-median problem on a tree is independent of the edge lengths and can be found in linear time. Theorem (Tamir 1996). The p-median problem on trees can be solved in

O(pn2) time.

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Hua 1961

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Hua 1961

When no cycles are among the roads, take each of the ends, let the minimum move to the next. When cycles are among the roads, get rid of one edge of each cycle, to reduce to the case of no cycles, and then, compute in the previous way. For all different ways of taking off cycles, compute the results one by one, by comparing all results, the optimum will be obtained.

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2.Semi-obnoxious p-median problems

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The semi-obnoxious case: wi ≷ 0

For semi-obnoxious p-median problems (p ≥ 2) there are two different models (Burkard, Çela, Dollani, 2000): FWMD(X) =

• v∈V
• wi min

1≤j≤p d(xj, v)

• (WMD)

The sum of the weighted minimum distances is minimized.

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The semi-obnoxious case: wi ≷ 0

For semi-obnoxious p-median problems (p ≥ 2) there are two different models (Burkard, Çela, Dollani, 2000): FWMD(X) =

• v∈V
• wi min

1≤j≤p d(xj, v)

• (WMD)

The sum of the weighted minimum distances is minimized. FMWD(X) =

• v∈V

min

1≤j≤p (wi d(xj, v))

(MWD)

The sum of minimum weighted distances is minimized. In this model the vertices with negative weights are assigned to the farthest facility!

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Why is MWD easier?

• If X ⊂ Y then

FMWD(X) ≥ FMWD(Y ).

• Let X = {x1, . . . xp} and V = Vk. Then

FMWD(X) ≤

p

• k=1
• v∈Vk

wv d(v, xk) and FMWD(X) = min

V1,...,Vk p

• k=1
• v∈Vk

wv d(v, xk).

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The semi-obnoxious case: wi ≷ 0

The vertex optimality property does not hold any more. 1 a

• 2

d 1 c 1 b 1 1 1 1

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The semi-obnoxious case: wi ≷ 0

The vertex optimality property does not hold any more. 1 a

• 2

d 1 c 1 b m 1 1 1

1 2 1 2

f(a) = −2 f(b) = −2 f(c) = f(d) = 5 f(m) =−2.5

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MWD p-median problem

Theorem (Bk. and H. 2005). There exists an optimal solution

X∗ = {x1, . . . , xp} of the MWD p-median problem such that one of the

following statements holds for each xi:

• 1. xi ∈ V .
• 2. xi is in the inner of edge (i, j) with the following property:

There exists a vertex v ∈ V with wv < 0 such that

d(v, i) + d(i, xi) = d(v, j) + d(j, xi).

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MWD p-median problem

Theorem (Bk. and H. 2005). There exists an optimal solution

X∗ = {x1, . . . , xp} of the MWD p-median problem such that one of the

following statements holds for each xi:

• 1. xi ∈ V .
• 2. xi is in the inner of edge (i, j) with the following property:

There exists a vertex v ∈ V with wv < 0 such that

d(v, i) + d(i, xi) = d(v, j) + d(j, xi). Property 2 implies that there exists a cycle in G!

• Corollary. There is an optimal solution of the MWD p-median problem on a

tree such that all points of the solution are vertices.

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• 3. Obnoxious p-median problems on

trees

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The obnoxious case: wi < 0

If all vertex weights are negative and p = 1 the objective function is min

x

• v∈V

wi d(v, x) = max

x

• v∈V

−wi d(v, x)

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The obnoxious case: wi < 0

If all vertex weights are negative and p = 1 the objective function is min

x

• v∈V

wi d(v, x) = max

x

• v∈V

−wi d(v, x)

Theorem (Zelinka 1968, Ting 1984). The optimal location of the 1-maxian problem in a tree is a leaf and can be found in O(n) time. Theorem (Tamir 1991). The 1-maxian problem on general graphs can be solved in O(mn) time.

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MWD obnoxious 2-median problem on trees

Lemma (Bk., Fathali and Kakhki 2005). Let P(vr, vk) be a longest path in T . Then {vr, vk} is an optimal solution for the 2-median problem.

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MWD obnoxious 2-median problem on trees

Sketch of the proof: vr vk vi vj a b a′ b′ mrk mij Tab T c

ab

Ta′b′ mrk ... midpoint of path P(vr, vk) Delete edge [a, b] which contains mrk: we get subtrees Tab and T c

ab

fT ′(v) =

• vi∈T ′

wid(v, vi)

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MWD obnoxious 2-median problem on trees

Sketch of the proof: vr vk vi vj a b a′ b′ mrk mij Tab T c

ab

Ta′b′ FMWD(vr, vk) = fTab(vk) + fT c

ab(vr)

One can show that FMWD(vr, vk) ≤ FMWD(vi, vj) holds.

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MWD obnoxious 2-median problem on trees

Theorem (Bk., Fathali and Kakhki 2005). Any set X with |X| = p which contains the two endpoints of a longest path in the tree is a p-median of G.

X can be found in linear time.

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• 4. Semi-obnoxious 2-median problem on

cycles

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MWD 2-median problem on cycles

Pab a b mba mab

We define for a vertex x and a path P with x ∈ P fP (x) =

• v∈P

max(0, wv)d(v, x) +

• v∈P c

min(0, wv)d(v, x).

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MWD 2-median problem on cycles

Then FMWD(a, b) = fPab(a) + fP c

ab(b)

We also have FMWD(a, b) ≤ fP (a) + fP c(b) ∀ paths P ⊂ C. Thus, it suffices to solve min

P

• min

a,b (fP (a) + fP c(b))

• =

min

P

• min

a∈P fP (a) + min b∈P c fP c(b)

• .

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MWD 2-median problem on cycles

We have to solve min

P

• min

a∈P fP (a) + min b∈P c fP c(b)

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MWD 2-median problem on cycles

We have to solve min

P

• min

a∈P fP (a) + min b∈P c fP c(b)

• In an optimal solution X = (a∗, b∗)

d(ma∗ b∗, mb∗ a∗) = L 2 , where L is the length of the cycle. ⇒ We only have to look at paths that are “almost” of some length. O(n) such paths.

• For a given path P the function fP (a) is convex.
• If the problem mina∈P fP (a) is solved for a path P, then we

do not need to start from the scratch for the problem mina∈P +v fP (a).

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MWD 2-median problem on cycles

We have to solve min

P

• min

a∈P fP (a) + min b∈P c fP c(b)

• Theorem (Bk. and H., 2005). The MWD 2-median problem on cycles can

be solved in linear time.