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

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 - p. 1/19

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 - p. 2/19

1. Problem formulation and some results - p. 3/19

Formal Problem Definition Given: graph G = ( V, E ) vertex weights w : V → R + edge lengths l : E → R + number of facilities: p - p. 4/19

Formal Problem Definition Given: graph G = ( V, E ) vertex weights w : V → R + edge lengths l : E → R + number of facilities: p Find a set X p = { x 1 , . . . , x p } ⊂ G of p points that minimizes Task: � � � F ( X p ) := w v min 1 ≤ j ≤ p d ( v, x j ) . v ∈ V - p. 4/19

p -median Problem 1 2 1 7 5 3 5 3 8 2 6 3 2 5 5 3 3 2 5 4 7 7 9 8 6 1 8 - p. 5/19

p -median Problem 1 1 2 1 2 1 7 5 3 5 5 5 3 8 3 2 6 3 2 6 3 2 5 5 5 3 3 2 3 5 5 4 7 4 7 7 9 8 7 6 1 8 6 1 8 - p. 5/19

Classical Results Theorem (vertex optimality property, Hakimi 1964). There exists an optimal solution X p = { x 1 , . . . , x p } ⊂ V . - p. 6/19

Classical Results Theorem (vertex optimality property, Hakimi 1964). There exists an optimal solution X p = { x 1 , . . . , x p } ⊂ 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. - p. 6/19

Classical Results Theorem (vertex optimality property, Hakimi 1964). There exists an optimal solution X p = { x 1 , . . . , x p } ⊂ 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. - p. 6/19

Classical Results Theorem (vertex optimality property, Hakimi 1964). There exists an optimal solution X p = { x 1 , . . . , x p } ⊂ 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 ( pn 2 ) time. - p. 6/19

Hua 1961 - p. 7/19

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. - p. 7/19

2.Semi-obnoxious p -median problems - p. 8/19

The semi-obnoxious case: w i ≷ 0 For semi-obnoxious p -median problems ( p ≥ 2 ) there are two different models (Burkard, Çela, Dollani, 2000): � � � F WMD ( X ) = w i min 1 ≤ j ≤ p d ( x j , v ) (WMD) v ∈ V The sum of the weighted minimum distances is minimized. - p. 9/19

The semi-obnoxious case: w i ≷ 0 For semi-obnoxious p -median problems ( p ≥ 2 ) there are two different models (Burkard, Çela, Dollani, 2000): � � � F WMD ( X ) = w i min 1 ≤ j ≤ p d ( x j , v ) (WMD) v ∈ V The sum of the weighted minimum distances is minimized. � F MWD ( X ) = 1 ≤ j ≤ p ( w i d ( x j , v )) min (MWD) v ∈ V The sum of minimum weighted distances is minimized. In this model the vertices with negative weights are assigned to the farthest facility! - p. 9/19

Why is MWD easier? ● If X ⊂ Y then F MWD ( X ) ≥ F MWD ( Y ) . ● Let X = { x 1 , . . . x p } and V = � V k . Then p � � F MWD ( X ) ≤ w v d ( v, x k ) k =1 v ∈ V k and p � � F MWD ( X ) = min w v d ( v, x k ) . V 1 ,...,V k k =1 v ∈ V k - p. 10/19

The semi-obnoxious case: w i ≷ 0 The vertex optimality property does not hold any more. a 1 1 d c -2 1 1 1 1 b 1 - p. 11/19

The semi-obnoxious case: w i ≷ 0 The vertex optimality property does not hold any more. a 1 f ( a ) = − 2 1 1 d c 2 f ( b ) = − 2 m -2 1 1 f ( c ) = 0 1 f ( d ) = 5 1 2 f ( m ) = − 2 . 5 b 1 - p. 11/19

MWD p -median problem Theorem (Bk. and H. 2005). There exists an optimal solution X ∗ = { x 1 , . . . , x p } of the MWD p -median problem such that one of the following statements holds for each x i : 1. x i ∈ V . 2. x i is in the inner of edge ( i, j ) with the following property: There exists a vertex v ∈ V with w v < 0 such that d ( v, i ) + d ( i, x i ) = d ( v, j ) + d ( j, x i ) . - p. 12/19

MWD p -median problem Theorem (Bk. and H. 2005). There exists an optimal solution X ∗ = { x 1 , . . . , x p } of the MWD p -median problem such that one of the following statements holds for each x i : 1. x i ∈ V . 2. x i is in the inner of edge ( i, j ) with the following property: There exists a vertex v ∈ V with w v < 0 such that d ( v, i ) + d ( i, x i ) = d ( v, j ) + d ( j, x i ) . 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. - p. 12/19

3. Obnoxious p -median problems on trees - p. 13/19

The obnoxious case: w i < 0 If all vertex weights are negative and p = 1 the objective function is � � min w i d ( v, x ) = max − w i d ( v, x ) x x v ∈ V v ∈ V - p. 14/19

The obnoxious case: w i < 0 If all vertex weights are negative and p = 1 the objective function is � � min w i d ( v, x ) = max − w i d ( v, x ) x x v ∈ V v ∈ V 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. - p. 14/19

MWD obnoxious 2 -median problem on trees Lemma (Bk., Fathali and Kakhki 2005). Let P ( v r , v k ) be a longest path in T . Then { v r , v k } is an optimal solution for the 2-median problem. - p. 15/19

MWD obnoxious 2 -median problem on trees Sketch of the proof: T c T ab ab v j v k v r m rk b ′ a b m ij a ′ v i T a ′ b ′ m rk ... midpoint of path P ( v r , v k ) Delete edge [ a, b ] which contains m rk : we get subtrees T ab and T c ab � f T ′ ( v ) = w i d ( v, v i ) v i ∈ T ′ - p. 15/19

MWD obnoxious 2 -median problem on trees Sketch of the proof: T c T ab ab v j v k v r m rk b ′ a b m ij a ′ v i T a ′ b ′ F MWD ( v r , v k ) = f T ab ( v k ) + f T c ab ( v r ) One can show that F MWD ( v r , v k ) ≤ F MWD ( v i , v j ) holds. - p. 15/19

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. - p. 15/19

4. Semi-obnoxious 2-median problem on cycles - p. 16/19

MWD 2 -median problem on cycles P ab a m ba m ab b We define for a vertex x and a path P with x ∈ P � � f P ( x ) = max(0 , w v ) d ( v, x ) + min(0 , w v ) d ( v, x ) . v ∈ P c v ∈ P - p. 17/19

MWD 2 -median problem on cycles Then F MWD ( a, b ) = f P ab ( a ) + f P c ab ( b ) We also have F MWD ( a, b ) ≤ f P ( a ) + f P c ( b ) ∀ paths P ⊂ C. Thus, it suffices to solve � � min min a,b ( f P ( a ) + f P c ( b )) = P � � min min a ∈ P f P ( a ) + min b ∈ P c f P c ( b ) . P - p. 18/19

MWD 2 -median problem on cycles We have to solve � � min min a ∈ P f P ( a ) + min b ∈ P c f P c ( b ) P - p. 19/19

MWD 2 -median problem on cycles We have to solve � � min min a ∈ P f P ( a ) + min b ∈ P c f P c ( b ) P ● In an optimal solution X = ( a ∗ , b ∗ ) d ( m a ∗ b ∗ , m b ∗ 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 f P ( a ) is convex. ● If the problem min a ∈ P f P ( a ) is solved for a path P , then we do not need to start from the scratch for the problem min a ∈ P + v f P ( a ) . - p. 19/19

MWD 2 -median problem on cycles We have to solve � � min min a ∈ P f P ( a ) + min b ∈ P c f P c ( b ) P Theorem (Bk. and H., 2005). The MWD 2 -median problem on cycles can be solved in linear time. - p. 19/19