SCIP-Jack—a solver for STP and variants with parallelization extensions

Abstract

The Steiner tree problem in graphs is a classical problem that commonly arises in practical applications as one of many variants. While often a strong relationship between different Steiner tree problem variants can be observed, solution approaches employed so far have been prevalently problem-specific. In contrast, this paper introduces a general-purpose solver that can be used to solve both the classical Steiner tree problem and many of its variants without modification. This versatility is achieved by transforming various problem variants into a general form and solving them by using a state-of-the-art MIP-framework. The result is a high-performance solver that can be employed in massively parallel environments and is capable of solving previously unsolved instances.

Appendices

Appendix A: Proofs

This section provides proofs to the lemmata stated in the course of this paper. For the respective transformation corresponding to each of these lemmata a one-to-one correspondence between the solution sets of the original and the transformed problem is proven as well as the linear relation between the respective solutions values. This implies that all these problems can be solved on their transformed solution spaces.

1.1 A.1: Proof of Lemma 1 (NWSTP to SAP)

Proof

To see the one-to-one correspondence let $S = (V_S,E_S) \in \mathcal {S}$ and proceed as follows:

Surjective. Initially set $V'_{S'} := V_S$ and $A'_{S'} := \emptyset$. Traverse $(V_S,E_S)$, e.g. using breadth-first search, starting from $r'$ and add for each $w \in V_S$ visited from $v \in V_S$ the arc (v, w) to $A'_{S'}$. $S' := (V'_{S'}, A'_{S'})$ is a solution to $P'$ and by applying (9) and (10), S is obtained.

Injective. $S'$ is the only solution to $P'$ that is mapped by (9) and (10) to S: Each $\tilde{S}' \in \mathcal {S'}$, $\tilde{S}' \ne S'$ contains at least one arc (v, w) such that $(v,w) \notin A'_{S'}$ and $(w,v) \notin A'_{S'}$, since only substituting arcs in $A'_{S'}$ by there anti-parallel counterparts would not allow directed paths from the root to all vertices. Therefore, $\tilde{S}'$ is not mapped onto S.

To acknowledge (11) one readily observes that for each node of $S'$ except for the root there is exactly one incoming arc, so:

$$\begin{aligned} \sum _{(v,w) \in A'_{S'}} c'_{(v,w)} = \sum _{(v,w) \in A'_{S'}} \left( c_{\{ v,w\} } + p_w\right) = \sum _{\{ v,w\} \in E_S} c_{\{ v,w\} } + \sum _{w \in V_S} p_w - p_{r'}, \end{aligned}$$

which implies (11).

1.2 A.2: Proof of Lemma 2 (RPCSTP to SAP)

Proof

To acknowledge that (14) and (15) constitute a mapping $\mathcal {S'} \rightarrow \mathcal {S}$ it can be observed that first the root node is conserved and second the set of all arcs corresponding to edges in the original graph (V, E) forms a tree. To prove that a bijection is given, let $S = (V_S,E_S) \in \mathcal {S}$ and $T = \{ t_1,\ldots ,t_s\}$ as defined in Transformation 2.

Surjective Initially, set $V'_{S'} := V_S$ and $A'_{S'} := \emptyset$. Analogously to the proof of Lemma 1, add for each edge in $E_S$ an arc to $A'_{S'}$ in such a way that finally there is for each $v' \in V'_{S'}$ a directed path from $r'$ to $v'$. Next, for each $i \in \{ 1,\ldots s \}$ set $a_i:= (t_i, t_i' )$ if $t_i \in V_S$, otherwise $a_i:=(r', t_i' )$ and add $a_i$ to $A'_{S'}$. Thereupon, $S' := (V'_{S'}, A'_{S'})$ is a solution to $P'$ and by applying (14) and (15), we obtain S.

Injective Define the set of all arcs of $P'$ corresponding to the edges of P as $A := \{ (v,w) \in A' :~ \{v,w \} \in E \}$ and accordingly $A_{S'} := A'_{S'} \cap A$. Since (13) has been assumed, it holds that: $(t_i, t_i') \in A'_{S'} \Leftrightarrow t_i \in V_S'$ and $(r', t_i') \in A'_{S'} \Leftrightarrow t_i \notin V_S'$. This implies that $A'_{S'}$ is already determined by $A_{S'}$. Now let $\tilde{S}' =(\tilde{V}_{S}',\tilde{A}_{S}') \in \mathcal {S'}$, $\tilde{S}' \ne S'$. Consequently, there is at least one arc $(v,w) \in \tilde{A}_{S}'$ such that $(w,v) \notin A_{S'}$ and $(w,v) \notin A_{S'}$ and therefore is $\tilde{S}'$ not mapped to S.

Finally, using the above notation one observes that:

$$\begin{aligned} \sum _{a \in {A}_{S'}'} c_a' = \sum _{a \in A_{S'}} c_a'+ \sum _{a \in {A}_{S'}' {\setminus } A_{S'}} c_a' = \sum _{e \in E_S} c_e + \sum _{v \in V {\setminus } V_S } p_v, \end{aligned}$$

so the costs of $S'$ and S are equal.

1.3 A.3: Proof of Lemma 3 (PCSTP to rcSAP)

Proof

Likewise to the proof of Lemma 2 one observes that (18) and (19) constitute a mapping $\mathcal {S'} \rightarrow \mathcal {S}$. Let $S = (V_S,E_S) \in \mathcal {S}$ and $T = \{ t_1,\ldots ,t_s\}$ defined as in Transformation 3.

Surjective. Initially, define $V'_{S'} := V_S$, $A'_{S'} := \{ (r,t_{i_0}) \}$, with $i_0 := \min {\{ i ~|~ t_i \in V_{S'}' \} }$. Then extend $A'_{S'}$ analogously to the proof of Lemma 2. The so constructed $S' := (V'_{S'}, A'_{S'})$ is a solution to $P'$ and applying (18) and (19) S is obtained.

Injective Parallelly to the proof of Lemma 2 it can be shown that for a solution $\tilde{S}' \ne S'$ to $P'$ there must be at least one arc $(v,w) \in A_{\tilde{S}'}$ such that $(v,w) \notin A_{S'}$ and $(w,v) \notin A_{S'}$ with A defined as in the proof of Lemma 2. Therefore it follows that $\tilde{S}'$ is not mapped to S.

The equality of the solution values of S and $S'$ can be seen likewise.

1.4 A.4: Proof of Lemma 4 (MWCS to rcSAP)

Proof

The one-to-one correspondence between the sets of solutions to P and $P''$ can be seen analogously to the proof of Lemma 3.

To prove (22) let $S = (V_S,E_S)$ be a solution to P and $S'' = (V''_{S''},A''_{S''})$ the corresponding solution to $P''$, obtained by applying (20) and (21). Further, define $A := \{ (v,w) \in A'' :~ \{v,w \} \in E \}$ and $A_{S''} = A \cap A''_{S''}$. First, one observes that for each $v \in S$ such that $p_v \le 0$ there is exactly one incoming arc $a \in A_{S''}$, so:

$$\begin{aligned} \sum _{v \in V_S: p_v \le 0} p_v = - \sum _{a \in A_{S''}} c_a''. \end{aligned}$$

(23)

Second:

$$\begin{aligned} \sum _{v \in V_S: p_v> 0} p_v = \sum _{v \in V: p_v> 0} p_v - \sum _{v \in V {\setminus } V_S: p_v> 0} p_v = \sum _{v \in V: p_v > 0} p_v - \sum _{a \in {A''}_{S''} {\setminus } A_{S''}} c_a''. \end{aligned}$$

(24)

Finally, by adding (23) and (24) the equation:

$$\begin{aligned} \sum _{v \in V_S} p_v = \sum _{v \in V: p_v > 0} p_v - \sum _{a \in {A''}_{S''}} c_a'' \end{aligned}$$

(25)

is obtained, which coincides with (22).

Appendix B: Abbreviations of reduction methods

This section provides the names for all reduction methods used by SCIP-Jack, which are listed in Table 2 in abbreviated form. All methods are described in detail in [16]. Note that several methods can be used for several variants, see Table 2, but in this case almost always require (considerable) adaptations. These adaptations can likewise be found in [16].

Table 14 lists in the first column the respective abbreviation of each reduction method used by SCIP-Jack. The second column provides the full name of the method, while the third and last column states all Steiner problem variants that use this particular reduction method (in adapted form).

Table 14 Abbrevitations of reduction methods

Appendix C: Detailed computational results

This section presents detailed instance-wise results from the experiments performed in this paper for all test sets discussed in Sects. 2 and 3. The tables list the original and the presolved problem size, i.e., number of nodes | V |, arcs | A |, and terminals | T | as well as the preprocessing time (column t [s] in the Presolved columns). Moreover, the tables show the Dual and Primal bound upon termination and the corresponding Gap in percent. If an instance was solved to optimality, the optimal value is printed instead of primal and dual bound, and the gap is omitted. Similarly, “–” is printed if no primal bound was present at the time of termination. Additionally, the number of branch-and-bound nodes (N), and the total solving time in seconds (last column) is listed. The total solving time includes the preprocessing time. A timeout is marked by “>” before the termination time. In case of the DCSTP for which SCIP-Jack does not perform preprocessing, the statistics about the presolved model are omitted (Tables 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40).

Table 15 Detailed computational results for the STP, test set X

Table 16 Detailed computational results for the STP, test set E

Table 17 Detailed computational results for the STP, test set ALUE

Table 18 Detailed computational results for the STP, test set I640

Table 19 Detailed computational results for the STP, test set PUC

Table 20 Detailed computational results for the STP, test set vienna-i-simple

Table 21 Detailed computational results for the SAP, test set gene

Table 22 Detailed computational results for the SAP, test set gene2002

Table 23 Detailed computational results for the RSMTP, test set estein40

Table 24 Detailed computational results for the RSMTP, test set estein50

Table 25 Detailed computational results for the RSMTP, test set estein60

Table 26 Detailed computational results for the RSMTP, test set solids

Table 27 Detailed computational results for the RSMTP, test set cancer

Table 28 Detailed computational results for the PCSTP, test set JMP

Table 29 Detailed computational results for the PCSTP, test set CRR

Table 30 Detailed computational results for the PCSTP, test set PUCNU

Table 31 Detailed computational results for the RPCSTP, test set cologne1

Table 32 Detailed computational results for the RPCSTP, test set cologne2

Table 33 Detailed computational results for the MWCSP, test set ACTMOD

Table 34 Detailed computational results for the MWCSP, test set JMPALMK

Table 35 Detailed computational results for the STP, test set TreeFam

Table 36 Detailed computational results for the GSTP, test set GSTP1

Table 37 Detailed computational results for the GSTP, test set GSTP2

Table 38 Detailed computational results for the STP, test set gr12

Table 39 Detailed computational results for the STP, test set gr14

Table 40 Detailed computational results for the HCDSTP, test set gr16