In this section we slightly change the problem formulation of the Minimum Cost Buffered Steiner Tree Problem. This allows a simplification of notation, a reduction of variables, and usage of a slightly more general delay model.
4.4.1 Shortening the Model: Eliminating Pin Properties
First, we may assume that the capacitance µ(t) is zero for each sink pin t ∈ N\{s}. This can be achieved by adding a new vertex t0 to G connected with t by an edge e = (t, t0).
Edge e will receive a µ-value equal to the original µ-value of t and cost c(e) = 0. The function F (e, .) can be set to be the constant zero function. Vertex t0 replaces t in N.
By a similar technique we can assume that the source delay function F (s, .) is constantly zero. As before, we add a new node s0 as well as an edge e = (s0, s) to G and replace s by
s0 in N. We set F (e, .) to the original function F (s, .) and c(e) = µ(e) = 0.
4.4.2 Shortening the Model: Representing Repeaters by Loops
Now we show how to avoid dealing with repeaters directly. Instead, we model the effect of inserting a repeater by an edge traversal as follows. For each repeater l ∈ L and each
v ∈ V (G) such that inserting repeater l at position v is allowed, we add a loop edge el= (v, v)to G. Traversal of el models adding a repeater of type l at position v. We set
c(el) := c(v) ·size(l) + cpower·power(l) and F (el, .) := F (l, .). To model the capacitance
change after repeater insertion we replace the former function µ by a function ∆ : E(G) × R≥0 → R≥0
such that ∆(el, .) is the constant function with value µ(l). For edges e ∈ E(G) that do not
model repeater insertion we define ∆(e, .) to be the function x 7→ x + µ(e) while ∆(◦, .) is the identity function R≥0→ R≥0.
The capacitance cap(ν) of a vertex ν ∈ V (A) is then recursively defined as cap(ν) = 0 for ν ∈ N\{s} and cap(ν) = X ζ=(ν,ω)∈δ+(ν) ∆(κ(ζ),cap(ω)) otherwise. 4.4.3 Problem Formulation
The following problem definition summarizes our new model and the resulting optimization problem. The Minimum Cost Buffered Steiner Tree Problem is a special case of this new problem.
Cost-Delay Minimum Steiner Tree Problem with Loops
Instance: A digraph G with edge costs c : E(G) → R≥0, possibly containing loops.
A net N ⊆ V (G) with source s and sink delay costs λ : N\{s} → R≥0.
A function F : E(G) × R≥0 → R≥0 such that F (e, .) is monotonically
non-decreasing for each e ∈ E(G).
A function ∆ : E(G) × R≥0 → R≥0 such that ∆(e, .)
• is a constant function if e is a loop edge and
• is monotonically non-decreasing with ∆(e, x) ≥ x for all x ∈ R≥0
otherwise.
We define c(◦) = 0, F (◦, .) to be the constant zero function, and ∆(◦, .) to be the identity function
Output: A Steiner tree (A, κ) for N in G minimizing
X ζ∈E(A) c(κ(ζ)) + X t∈N \{s} λ(t) · X ζ=(ν,ω)∈E(A[s,t]) F (κ(ζ),cap(ω)) ,
where for ω ∈ V (A), cap(ω) is recursively defined as
cap(ω) = X
ζ=(ω,ω0)∈δ+ A(ω)
∆(κ(ζ),cap(ω0)).
Note that the recursive definition of cap already implies that cap(t) = 0 for t ∈ N\{s}. We call the special case of the Cost-Delay Minimum Steiner Tree Problem with Loops where N = {s, t}the Cost-Delay Minimum Steiner Path Problem with Loops.
4.4.4 Necessity of Conservative Edge Costs
Recall that for the Standard Minimum Steiner Tree Problem and for the Shortest Path Problem it is essential to restrict the edge costs such that there are no negative cycles. An edge cost function for which no cycle with negative total cost exists is called conservative edge cost. Finding a shortest path in a graph with non-conservative edge costs is NP-hard as a straightforward reduction from the NP-hard Longest Path Problem shows ([GJ79] page 213).
In our case, the restriction that ∆(e, x) ≥ x for all non-loop edges e ∈ E(G) and all x ∈ R≥0 is necessary to ensure that traversal of cycles does not decrease the overall cost of
a Steiner tree. Note that this condition is fulfilled for the Elmore delay model.
It is easy to see that allowing e. g. a function ∆(e, .) = (x 7→ α · x) for a constant α < 1 may lead to undesired situations: If we do not enforce that the κ function of a Steiner tree (A, κ) is injective, no finite optimum solution might exist. If we require injectivity, we obtain an optimization problem for which no approximation algorithm exists unless P = NP. The last fact is even true for the Cost-Delay Minimum Steiner Path Problem with Loops and can be proved by a straightforward reduction from the Hamiltonian s-t Path Problem:
Lemma 4.1 Let p be a polynomial. Unless P = NP there is no 2p(|V (G)|)-approximation algorithm for the modification of the Cost-Delay Minimum Steiner Path Problem with Loops in which we omit the condition ∆(e, x) ≥ x for non-loop edges e ∈ E(G) and require that the returned buffered Steiner tree ((A, κ), b) has the property |κ−1(e)| ≤ 1 for all e ∈ E(G).
Proof Let N = {s, t} and let G be a directed graph with n := |V (G)| vertices. We construct a graph G0 from G by adding vertices s0, t0 and edges (s0, s), (t, t0) with
F ((s0, s), x) = x, F ((t, t0), x) = 0, ∆((t, t0), x) = x + 1 for all x ∈ R≥0. We set
∆(e, x) = 2p(n+2)+1x and F (e, x) = 0 for all edges e ∈ E(G). Edge costs c(e) are zero
for all edges e ∈ E(G0). Then, the total cost of an s0-t0 path containing all vertices in G
is more than a factor 2p(n+2) smaller than the cost of any shorter s0-t0 path. Hence, no
2p(|V (G0)|)-approximation algorithm can exist unless P = NP.
While for the Shortest Steiner Tree Problems in Graphs with non-negative (and hence conservative) edge costs, loops do not play any role, they are essential in our case as they model repeater insertion. With introduction of loop edges we have relaxed the condition ∆(e, x) ≥ x that we require for non-loop edges e ∈ E(G) and x ∈ R≥0. Consequently,
optimum solutions can contain non-trivial cycles. Such a situation is depicted in Figure 4.2. Despite these situations we can still bound the length of an optimum solution as the next theorem states. Having such a bound is necessary to obtain a polynomial time algorithm for (special cases of) the Cost-Delay Minimum Steiner Tree Problem with Loops. Lemma 4.2 Let G, N, λ, F, c, ∆ be an instance of the Cost-Delay Minimum Steiner Tree Problem with Loops such that N = {s, t}. Let h be the number of loop edges in G. There exists an optimum solution ((A, κ), b) with |V (A)| ≤ (1 + h) · |V (G)|.
Proof Without loss of generality we may assume that λ(t) = 1. Let ((A, κ), b) be an optimum solution with |V (A)| minimum. Let v ∈ κ(A) such that κ−1(v) = {x
1, x2, . . . , xk}
and xi is reachable from xj in A if and only if i ≥ j (i. e. the xi appear in that order in
s v1 v2 t v3 v4 F (., x) = x ∆(., x) = x F (., x) = 0 ∆(., x) = x + 100 F (., x) = 0 ∆(., x) = 1
For black edges: F (., x) = 0 ∆(., x) = x + 1
(a) Graph G with functions ∆ and F . Here, N = {s, t}, c(e) = 0 for e ∈ E(G) and λ(t) = 1.
κ = s κ = v1 κ = v2 κ = t κ = s κ = v1 κ = v2 κ = t κ = v3 κ = v3 κ = v4 κ = v1 κ = v2
(b) Two Steiner trees. The above tree has cost 101, the tree below has cost 3 and uses a loop edge and a non-trivial cycle.
Figure 4.2: Instance of the Cost-Delay Minimum Steiner Tree Problem with Loops for which the optimum solution uses a non-trivial cycle in G.
If there is i ∈ {1, . . . , k − 1} such that cap(xi) ≥ cap(xi+1), removing A[xi,xi+1]− xi
from A but connecting xi with the successor of xi+1 in A (respectively identifying xi with
t if xi+1= t) would not increase the cost of A but would decrease |V (A)|. We conclude
that A[xi,xi+1] contains a loop edge for each i ∈ {1, . . . , k − 1}.
Now, note that each loop edge e is used by A at most once as otherwise, removing the part between the first and the last traversal of e (including the first traversal but excluding the last one) decreases |V (A)| without increasing the cost.
This implies that k ≤ h + 1 and the lemma follows.
The bound of Lemma 4.2 is tight up to a constant factor:
Lemma 4.3 For each n, h ∈ N there is an instance G, N, λ, F, c, ∆ of the Cost-Delay Minimum Steiner Tree Problem with Loops with |N| = 2, |V (G)| = 2 + h + n, and h loop edges in which each optimum solution (A, κ) has at least |V (A)| = (h + 1)(n + 2) vertices. Proof Let G be the directed graph with vertices
V (G) = {s, t, v1, . . . , vn, w1, . . . , wh}.
We add edges (s, v1), (vn, t), and (vi, vi+1) for i = 1, . . . , n − 1. For each j ∈ {1, . . . , h} we
insert edges (vn, wj), (wj, v1), and attach a loop edge ej at wj.
We define N = {s, t} with λ(t) = 1, c(e) = 0 for all e ∈ E(G), and for x ∈ R≥0
F ((s, v1), x) = x,
F ((wj, v1), x) =
(
0 if x ≤ j
1 otherwise, for all j = 1, . . . , h
F (e, x) = 0 for all edges e ∈ E(G)\δ−(v1),
∆((vn, t), x) = x + h,
∆(ej, x) = j − 1 for all j = 1, . . . , h,
∆(e, x) = x for all non-loop edges e ∈ E(G)\{(vn, t)}.
f e s v1 v2 v3 v4 v5= vn t w1 w2 w3= wh e1 e2 e3 f1 f2 f3 F (e, x) = 0 ∆(e, x) = x + 3 F (ej, x) = 0 ∆(ej, x) = j − 1 F (f, x) = x ∆(f, x) = x F (fi, x) = ( 0 if x ≤ i 1 otherwise ∆(fi, x) = x F ((v5, wj), x) = 0 F ((vi, vi+1), x) = 0 ∆((v5, wj), x) = x ∆((vi, vi+1), x) = x
(a) Graph G and timing functions. (b) A Steiner path with cost 0 and mini- mum number of vertices.
Figure 4.3: Instance for the Cost-Delay Minimum Steiner Tree Problem with Loops constructed in the proof of Lemma 4.3 with n = 5, h = 3. The number of edges in each optimum s-t Steiner path is large.
First observe that the Steiner path traversing edges
(s, v1), (v1, v2), . . . , (vn−1, vn), (vn, w1), e1, (w1, v1),
(v1, v2), . . . , (vn−1, vn), (vn, w2), e2, (w2, v1),
. . .
(v1, v2), . . . , (vn−1, vn), (vn, wh), eh, (wh, v1),
(v1, v2), . . . , (vn−1, vn), (vn, t)
in that order has cost 0 (see Figure 4.3(b)).
To complete the proof of the lemma it remains to show that this path has the least number of vertices among all Steiner paths with cost 0. Let A be such an s-t Steiner path with cost 0. By definition of F ((s, v1), .)and the ∆-functions, A must traverse e1 which
requires to traverse (w1, v1) afterwards.
Using the same argument we observe that each Steiner path with cost 0 that tra- verses edge (wj, v1) for j ∈ {1, . . . , h − 1} must also traverse ej+1 and thus (wj+1, v1).
Consequently, A traverses all edges entering v1.
As each non-trivial cycle in G containing v1 and a loop edge has n + 2 vertices and since
the part of A between the first and last occurrence (say including the first but excluding the last occurrence) must contain at least n + 2 vertices, we conclude |V (A)| ≥ (h + 1)(n + 2).
In instances arising in practice it is often the case that either no repeater or all repeaters can be inserted at a node. In this case, the bound of Lemma 4.2 improves to (|L|+1)·|V (G)| where L is the repeater library. This statement has essentially the same proof as Lemma 4.2 and tightness can easily be proved by attaching |L| loop edges at each inner node of an s-t path of length n + 2. This graph defines G and each of the |L| loop edges at each inner node represents a repeater of a certain type. By defining cost functions similar to the ones used in the proof of Lemma 4.3 we can make sure that an optimum Steiner path actually uses each of these loops.