[Rule] SteinerTree to ILP

[zazabap]

Source: SteinerTree
Target: ILP
Motivation: Enables solving Steiner Tree via ILP solvers; the flow-based formulation is the standard in network optimization and VLSI routing tools.
Reference: Wong, 1984; Koch & Martin, 1998

Reduction Algorithm

Notation:

• Source: undirected weighted graph $G = (V, E, w)$, terminals $T \subseteq V$, $n = |V|$, $m = |E|$, $|T| = k$
• Target: ILP with binary and continuous variables

Step 1 — Choose a root: Pick any terminal $r \in T$ as the root. Let $T' = T \setminus {r}$ ($k-1$ non-root terminals).

Step 2 — Variables:

• Edge selection: $y_e \in {0, 1}$ for each edge $e \in E$ (include in tree?)
• Flow: $f^t_{uv} \in [0, 1]$ for each directed arc $(u,v)$ (both directions of each edge) and each non-root terminal $t \in T'$. Represents one unit of flow from $r$ to $t$.

Step 3 — Constraints:

1. Flow conservation (for each $t \in T'$ and each vertex $v \in V$):

$$\sum_{u:(u,v) \in A} f^t_{uv} - \sum_{u:(v,u) \in A} f^t_{vu} = \begin{cases} -1 & \text{if } v = r \ +1 & \text{if } v = t \ 0 & \text{otherwise} \end{cases}$$

2. Capacity linking (for each directed arc $(u,v)$ and each $t \in T'$):

$$f^t_{uv} \leq y_e \quad \text{where } e = {u,v}$$

Objective: minimize $\sum_{e \in E} w_e \cdot y_e$

Solution extraction: Select edges with $y_e = 1$; these form the Steiner tree.

Size Overhead

Target metric (code name)	Polynomial (using symbols above)
num_vars	$m + 2m(k-1)$ (edge selection + flow variables)
num_constraints	$n(k-1) + 2m(k-1)$ (conservation + capacity)

Validation Method

Closed-loop testing: solve SteinerTree by brute-force (enumerate edge subsets), solve the reduced ILP, and verify both give the same optimal cost.

Example

Source: 5 vertices, 7 edges, terminals $T = {0, 2, 4}$.

Edge index	Edge	Weight
$e_0$	$(0,1)$	2
$e_1$	$(1,2)$	2
$e_2$	$(1,3)$	1
$e_3$	$(3,4)$	1
$e_4$	$(0,3)$	5
$e_5$	$(3,2)$	5
$e_6$	$(2,4)$	6

ILP construction: root $r = 0$, non-root terminals $T' = {2, 4}$ ($k - 1 = 2$ commodities).

Variables (35 total):

7 edge-selection variables:

$$y_0, y_1, y_2, y_3, y_4, y_5, y_6 \in {0, 1}$$

14 flow variables for commodity $t = 2$:

$$f^2_{01},; f^2_{10},; f^2_{12},; f^2_{21},; f^2_{13},; f^2_{31},; f^2_{34},; f^2_{43},; f^2_{03},; f^2_{30},; f^2_{32},; f^2_{23},; f^2_{24},; f^2_{42} \in [0,1]$$

14 flow variables for commodity $t = 4$:

$$f^4_{01},; f^4_{10},; f^4_{12},; f^4_{21},; f^4_{13},; f^4_{31},; f^4_{34},; f^4_{43},; f^4_{03},; f^4_{30},; f^4_{32},; f^4_{23},; f^4_{24},; f^4_{42} \in [0,1]$$

Constraints (38 total):

Flow conservation for $t = 2$ (5 constraints):

Vertex	RHS	Constraint
$v=0$ (root)	$-1$	$f^2_{10} + f^2_{30} - f^2_{01} - f^2_{03} = -1$
$v=1$	$0$	$f^2_{01} + f^2_{21} + f^2_{31} - f^2_{10} - f^2_{12} - f^2_{13} = 0$
$v=2$ (sink)	$+1$	$f^2_{12} + f^2_{32} + f^2_{42} - f^2_{21} - f^2_{23} - f^2_{24} = 1$
$v=3$	$0$	$f^2_{13} + f^2_{03} + f^2_{23} + f^2_{43} - f^2_{31} - f^2_{30} - f^2_{32} - f^2_{34} = 0$
$v=4$	$0$	$f^2_{34} + f^2_{24} - f^2_{43} - f^2_{42} = 0$

Flow conservation for $t = 4$ (5 constraints):

Vertex	RHS	Constraint
$v=0$ (root)	$-1$	$f^4_{10} + f^4_{30} - f^4_{01} - f^4_{03} = -1$
$v=1$	$0$	$f^4_{01} + f^4_{21} + f^4_{31} - f^4_{10} - f^4_{12} - f^4_{13} = 0$
$v=2$	$0$	$f^4_{12} + f^4_{32} + f^4_{42} - f^4_{21} - f^4_{23} - f^4_{24} = 0$
$v=3$	$0$	$f^4_{13} + f^4_{03} + f^4_{23} + f^4_{43} - f^4_{31} - f^4_{30} - f^4_{32} - f^4_{34} = 0$
$v=4$ (sink)	$+1$	$f^4_{34} + f^4_{24} - f^4_{43} - f^4_{42} = 1$

Capacity linking (28 constraints, 4 per edge):

Edge	Constraints
$e_0 = (0,1)$	$f^2_{01} \leq y_0$, $f^2_{10} \leq y_0$, $f^4_{01} \leq y_0$, $f^4_{10} \leq y_0$
$e_1 = (1,2)$	$f^2_{12} \leq y_1$, $f^2_{21} \leq y_1$, $f^4_{12} \leq y_1$, $f^4_{21} \leq y_1$
$e_2 = (1,3)$	$f^2_{13} \leq y_2$, $f^2_{31} \leq y_2$, $f^4_{13} \leq y_2$, $f^4_{31} \leq y_2$
$e_3 = (3,4)$	$f^2_{34} \leq y_3$, $f^2_{43} \leq y_3$, $f^4_{34} \leq y_3$, $f^4_{43} \leq y_3$
$e_4 = (0,3)$	$f^2_{03} \leq y_4$, $f^2_{30} \leq y_4$, $f^4_{03} \leq y_4$, $f^4_{30} \leq y_4$
$e_5 = (3,2)$	$f^2_{32} \leq y_5$, $f^2_{23} \leq y_5$, $f^4_{32} \leq y_5$, $f^4_{23} \leq y_5$
$e_6 = (2,4)$	$f^2_{24} \leq y_6$, $f^2_{42} \leq y_6$, $f^4_{24} \leq y_6$, $f^4_{42} \leq y_6$

Objective: minimize $2y_0 + 2y_1 + y_2 + y_3 + 5y_4 + 5y_5 + 6y_6$

Optimal ILP solution: $y_0 = y_1 = y_2 = y_3 = 1$, $y_4 = y_5 = y_6 = 0$, objective $= 2 + 2 + 1 + 1 = 6$.

Flow for terminal 2 ($0 \to 1 \to 2$): $f^2_{01} = 1, f^2_{12} = 1$, all others $0$.
Flow for terminal 4 ($0 \to 1 \to 3 \to 4$): $f^4_{01} = 1, f^4_{13} = 1, f^4_{34} = 1$, all others $0$.

[zazabap]

Issue Quality Check — Rule

Check	Result	Details
Usefulness	✅ Pass	SteinerTree not yet implemented; novel reduction
Non-trivial	✅ Pass	Multi-commodity flow ILP with directed arc variables
Correctness	⚠️ Warn	Both DOIs are incorrect
Well-written	✅ Pass	Complete algorithm, consistent symbols, verified example

Overall: 3 passed, 0 failed, 1 warning

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Usefulness

SteinerTree does not exist in the codebase. This flow-based ILP formulation is the standard approach in network optimization. Requires a [Model] issue for SteinerTree first.

Non-trivial

Constructs multi-commodity flow formulation: m edge-selection variables + 2m(k−1) directed flow variables, with flow-conservation and capacity-linking constraints. O(mk) variables — far from a simple relabeling.

Correctness — WARN

Both references are real papers but have incorrect DOIs:

Reference	Issue DOI	Correct DOI
Wong (1984), "A dual ascent approach for Steiner tree problems," Math. Programming 28, 271–287	10.1002/net.3230140304 (Networks journal)	10.1007/BF02612335 (Springer, Math. Programming)
Koch & Martin (1998), "Solving Steiner tree problems in graphs to optimality," Networks 32(3), 207–232	SICI DOI references page 171	Should reference page 207: 10.1002/(SICI)1097-0037(199810)32:3<207::AID-NET5>3.0.CO;2-O

The multi-commodity flow formulation is mathematically correct and consistent with both papers.

Recommendation: Consider also citing Goemans (1993), "A catalog of Steiner tree formulations," Networks 23(1) for systematic comparison of Steiner tree ILP formulations.

Well-written

All sections complete. Algorithm is detailed and implementable (root selection, variable definitions, constraint types, solution extraction). Overhead metrics num_vars and num_constraints match ILP's size_fields.

Example verified: 5 vertices, 7 edges, 3 terminals → 35 vars (7 + 2·7·2), 38 constraints (5·2 + 2·7·2). Optimal Steiner tree: edges {(0,1),(1,2),(1,3),(3,4)}, cost = 2+2+1+1 = 6. Flow paths for both commodities correctly traced.

Required fixes

1. Fix Wong (1984) DOI to 10.1007/BF02612335
2. Fix Koch & Martin (1998) DOI page reference (171 → 207)

[zazabap]

Issue Quality Check — Rule

Check	Result	Details
Usefulness	⚠️ Warn	SteinerTree model does not yet exist in the codebase — reduction cannot be implemented until the model is added
Non-trivial	✅ Pass	Multicommodity flow formulation with auxiliary flow variables and capacity-linking constraints
Correctness	❌ Fail	Both DOIs are incorrect (point to wrong papers); see details below
Well-written	✅ Pass	Algorithm is complete, notation is consistent, example is fully worked

Overall: 2 passed, 1 failed, 1 warning

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Usefulness

SteinerTree is not currently implemented as a model in the codebase (pred show SteinerTree fails). There is no existing path from SteinerTree to ILP since the source problem does not exist yet. This means the reduction is novel, but it also means a [Model] issue for SteinerTree must be implemented first before this rule can be added.

The motivation is reasonable — solving Steiner Tree via ILP is the standard approach in network optimization and VLSI routing. However, the motivation section could be stronger by explaining why this specific formulation (multicommodity flow) is preferred over alternatives (e.g., cut-based formulations).

Non-trivial

The reduction constructs a multicommodity flow ILP with:

• Binary edge-selection variables plus continuous flow variables for each terminal-arc pair
• Flow conservation constraints ensuring unit flow from root to each non-root terminal
• Capacity-linking constraints tying flow to edge selection

This is a genuine structural transformation involving auxiliary variables (flow), gadget-like constraints (capacity linking), and a non-obvious variable encoding. Pass.

Correctness

Reference 1: Wong, 1984

• The DOI 10.1002/net.3230140304 is incorrect. That DOI resolves to "Euclidean shortest paths in the presence of rectilinear barriers" by Lee (1984) in Networks — an unrelated paper.
• The actual Wong 1984 paper is: R.T. Wong, "A dual ascent approach for Steiner tree problems on a directed graph," Mathematical Programming 28:271–287, 1984. DOI: 10.1007/BF02612335.
• The paper itself does exist and does describe dual ascent / directed formulations for Steiner tree problems. The mathematical content cited in the issue is consistent with Wong's work.

Reference 2: Koch & Martin, 1998

• The DOI 10.1002/(SICI)1097-0037(199810)32:3<171::AID-NET3>3.0.CO;2-K is incorrect. The correct DOI is 10.1002/(SICI)1097-0037(199810)32:3<207::AID-NET5>3.0.CO;2-O (pages 207–232, not starting at 171; article code NET5, not NET3).
• The paper does exist: T. Koch and A. Martin, "Solving Steiner tree problems in graphs to optimality," Networks 32(3):207–232, 1998.
• The flow-based ILP formulation described in the issue is consistent with what Koch & Martin present.

Verdict: Both papers are real and the mathematical claims are correct, but both DOIs are wrong. This is a Fail because the references as written do not resolve to the claimed papers.

Well-written

Information Completeness: All required sections are present and substantive — Source, Target, Motivation, Reference, Reduction Algorithm, Size Overhead, Validation Method, and Example.

Algorithm Completeness: The reduction algorithm is a complete, step-by-step procedure:

• All notation defined upfront (n, m, k, T, T', r, A)
• Variables clearly specified with domains
• Constraints enumerated with mathematical formulas
• Objective function stated
• Solution extraction described

Symbol and Notation Consistency: Symbols are consistent across sections. The overhead table uses the same symbols (m, k, n) defined in the Notation block. The code metric names (num_vars, num_constraints) match the ILP schema's size_fields.

Example Quality: The example has 5 vertices, 7 edges, 3 terminals — non-trivial enough to exercise the reduction. It shows the source instance, the ILP dimensions (35 variables, 38 constraints), the optimal solution, and the flow routing for both commodities. The numbers check out: 7 + 272 = 35 variables, 52 + 27*2 = 38 constraints.

Recommendations

• Fix the Wong 1984 DOI to 10.1007/BF02612335 and update the journal to Mathematical Programming (not Networks)
• Fix the Koch & Martin 1998 DOI to 10.1002/(SICI)1097-0037(199810)32:3<207::AID-NET5>3.0.CO;2-O
• Consider also citing Goemans & Myung (1993) "A catalog of Steiner tree formulations" (Networks 23:19–28), which surveys and compares multiple ILP formulations for Steiner tree including the multicommodity flow formulation
• A [Model] issue for SteinerTree should be created/implemented before this rule can proceed

[isPANN]

Issue Quality Check — Rule

Check	Result	Details
Usefulness	✅ Pass	Novel reduction — SteinerTree is currently an orphan node with 0 reductions
Non-trivial	✅ Pass	Multicommodity flow ILP with O(mk) auxiliary variables and capacity-linking constraints
Correctness	❌ Fail	Both DOIs resolve to wrong papers; mathematical content is correct
Well-written	❌ Fail	Example only summarizes ILP dimensions instead of enumerating constructed objects

Overall: 2 passed, 2 failed, 0 warnings

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Usefulness

No reduction path exists from SteinerTree to ILP (pred path fails). SteinerTree currently has zero incoming and zero outgoing reductions — it is an isolated node in the reduction graph. This reduction would connect it to the ILP hub, making it solvable via all downstream solvers. Pass.

The motivation (ILP solvers for network optimization / VLSI routing) is reasonable but could be strengthened by explaining why the multicommodity flow formulation specifically is preferred over alternatives (e.g., cut-based formulations).

Non-trivial

The reduction constructs a multicommodity flow ILP with:

• Binary edge-selection variables (m) plus continuous flow variables for each terminal-arc pair (2m(k−1))
• Flow conservation constraints ensuring unit flow from root to each non-root terminal
• Capacity-linking constraints tying flow to edge selection

This is a genuine structural transformation involving auxiliary variables (flow), gadget-like constraints (capacity linking), and a non-obvious variable encoding. Pass.

Correctness

Reference verification:

Reference	Issue DOI	Status	Correct DOI
Wong (1984), "A dual ascent approach for Steiner tree problems"	10.1002/net.3230140304	❌ Resolves to Lee & Preparata, "Euclidean shortest paths…" in Networks	10.1007/BF02612335 (Mathematical Programming 28:271–287)
Koch & Martin (1998), "Solving Steiner tree problems in graphs to optimality"	…32:3<171::AID-NET3>3.0.CO;2-K	❌ Wrong page (171→207) and article code (NET3→NET5)	…32:3<207::AID-NET5>3.0.CO;2-O (Networks 32(3):207–232)

Both papers are real and mathematically relevant — Wong (1984) introduced the directed flow formulation, Koch & Martin (1998) used it computationally — but the DOIs as written do not resolve to the claimed papers. Fail.

Neither reference is currently in docs/paper/references.bib.

Overhead verification: Verified against brute-force Python enumeration:

• num_vars = m + 2m(k−1): 7 + 2·7·2 = 35 ✓
• num_constraints = n(k−1) + 2m(k−1): 5·2 + 2·7·2 = 38 ✓
• Overhead expression variables map to SteinerTree's getters: m→num_edges(), n→num_vertices(), k→num_terminals() ✓
• Code metric names (num_vars, num_constraints) match ILP's size_fields ✓

Source/target definition match: SteinerTree schema (graph, edge_weights, terminals) matches the issue's description. ILP schema matches the constructed formulation. ✓

Example verification (brute-force): Optimal Steiner tree cost = 6 using edges {(0,1),(1,2),(1,3),(3,4)} ✓. ILP optimal matches ✓. Round-trip verified ✓. 11 suboptimal feasible Steiner trees found (well above the required 2) ✓.

Recommendation: Consider also citing Goemans & Myung (1993), "A catalog of Steiner tree formulations," Networks 23(1):19–28, for systematic comparison of Steiner tree ILP formulations.

Well-written

Sections: All required sections present and substantive (Source, Target, Motivation, Reference, Reduction Algorithm, Size Overhead, Validation Method, Example). ✓

Algorithm completeness: Every step numbered and unambiguous — root selection, variable definitions with domains, constraint types with full mathematical formulas, objective function, solution extraction. ✓

Symbol and notation consistency: All symbols defined before use (n=|V|, m=|E|, k=|T|). Consistent across sections. Overhead metric names match target's size_fields. Expression variables map to source's size_fields. ✓

Example: The example provides the source instance data (5 vertices, 7 edges, 3 terminals with weights) and the optimal solution with flow paths. However, it only summarizes the ILP dimensions ("7 edge variables + 28 flow variables = 35 variables", "10 conservation + 28 capacity = 38 constraints") without explicitly listing the constructed variables by name or writing out the constraints. Per the review standard, merely summarizing dimensions is not fully worked — the actual variables and constraints must appear. Fail.

Required fixes

1. Fix Wong (1984) DOI to 10.1007/BF02612335 and note journal is Mathematical Programming, not Networks
2. Fix Koch & Martin (1998) DOI to 10.1002/(SICI)1097-0037(199810)32:3<207::AID-NET5>3.0.CO;2-O
3. Expand the example to explicitly enumerate all constructed ILP variables (edge-selection y_e for each edge, flow f^t_{uv} for each arc-terminal pair) and write out representative constraints (at minimum: full conservation constraints for one commodity and capacity constraints for one arc)