[Model] SteinerTree

[zazabap]

Motivation

One of Karp's 21 NP-complete problems; foundational in network design. Telecommunications backbone routing, VLSI chip interconnect, pipeline network planning, and phylogenetic tree construction all reduce to Steiner Tree. Has known reductions to ILP and SAT. Opens a new network design category in the reduction graph.

Definition

Name: SteinerTree
Reference: Karp, 1972; Hwang, Richards & Winter, The Steiner Tree Problem, 1992

Given an undirected weighted graph $G = (V, E, w)$ where $w: E \to \mathbb{R}_{\geq 0}$, and a set of terminal vertices $T \subseteq V$, find a tree $S = (V_S, E_S)$ in $G$ such that $T \subseteq V_S$, minimizing the total edge weight:

$$\sum_{e \in E_S} w(e)$$

The vertices in $V_S \setminus T$ are called Steiner vertices — intermediate relay points that reduce total cost.

Variables

• Count: $m = |E|$ (one variable per edge)
• Per-variable domain: binary ${0, 1}$
• Meaning: $y_e = 1$ if edge $e$ is included in the Steiner tree

Schema (data type)

Type name: SteinerTree
Variants: graph Steiner tree, Euclidean Steiner tree, rectilinear Steiner tree

Field	Type	Description
graph	SimpleGraph	The weighted undirected graph $G = (V, E, w)$
terminals	Vec<usize>	The terminal vertex set $T \subseteq V$

Problem Size

Metric	Expression	Description
num_vertices	$n$	Number of vertices
num_edges	$m$	Number of edges
num_terminals	$\lvert T \rvert$	Number of terminal vertices

Complexity

• Decision complexity: NP-complete (Karp, 1972)
• Best known exact algorithm: $O(3^{|T|} \cdot n + 2^{|T|} \cdot n^2)$ via Dreyfus-Wagner dynamic programming over terminal subsets
• Approximation: 1.39-approximation (Byrka et al., 2013); the classic 2-approximation via minimum spanning tree of the terminal distance graph
• References: Karp 1972; Dreyfus & Wagner 1971; Byrka et al. 2013

Extra Remark

When $T = V$ (all vertices are terminals), the Steiner tree reduces to the minimum spanning tree (polynomial). The NP-hardness arises from choosing which Steiner vertices to include. Industry-standard tools include SteinLib (benchmark library) and GeoSteiner (exact solver for Euclidean instances). The rectilinear variant is critical in VLSI routing (Cadence, Synopsys).

How to solve

• It can be solved by (existing) bruteforce.
• It can be solved by reducing to integer programming, through a new SteinerTree → ILP rule issue (to be filed).

Bruteforce: enumerate all subsets of edges, check if terminals are connected and the selected edges form a tree, return minimum weight.

Example Instance

$n = 5$ vertices, $m = 7$ edges, terminals $T = {0, 2, 4}$:

    (2)     (2)
 0 ----- 1 ----- 2
          |       |
      (1) |       | (6)
          |       |
          3 ----- 4
             (1)

Also: $(0,3)$ weight 5, $(3,2)$ weight 5 (not shown for clarity).

Edge	Weight
$(0,1)$	2
$(1,2)$	2
$(1,3)$	1
$(3,4)$	1
$(0,3)$	5
$(3,2)$	5
$(2,4)$	6

Optimal Steiner tree: edges ${(0,1), (1,2), (1,3), (3,4)}$, cost $= 2+2+1+1 = 6$.

Steiner vertices used: ${1, 3}$. Vertex 1 acts as a hub connecting terminals 0 and 2; vertex 3 relays to terminal 4.

Compare: the only direct terminal-terminal edge is $(2,4)$ with weight 6 — equaling the entire Steiner tree cost by itself. This demonstrates why Steiner points are essential.

[isPANN]

Closing as duplicate — this problem is covered by the Garey & Johnson extraction (P31 SteinerTree, G&J Appendix A2, ND7). The extracted reference issue in references/Garey&Johnson/issues/models/P31_steiner_tree.md provides a comprehensive definition from the original source.

[zazabap]

Issue Quality Check — Model

Check	Result	Details
Usefulness	✅ Pass	Novel problem not yet in codebase; opens network design category
Non-trivial	✅ Pass	Distinct feasibility constraints (terminal connectivity + tree structure) from all existing problems
Correctness	✅ Pass	All references verified; complexity claims accurate
Well-written	✅ Pass	All sections present, definition precise, example illustrative

Overall: 4 passed, 0 failed, 0 warnings

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Usefulness

SteinerTree does not exist in the codebase. It is Karp's 21 NP-complete problem #16 (ND5 in Garey-Johnson), and opens a new network design category in the reduction graph. The motivation is concrete: telecommunications routing, VLSI interconnect, pipeline planning, phylogenetic trees. The issue mentions planned reductions to ILP and SAT, which would connect it to existing problems. The "How to solve" section identifies bruteforce as viable, ensuring reduction rules can be verified.

Non-trivial

SteinerTree is genuinely distinct from all existing problems in the codebase. Its feasibility constraint — finding a subtree that spans a designated terminal set T while optionally including Steiner vertices — is fundamentally different from:

• MinimumVertexCover / MaximumIndependentSet — vertex selection problems without tree/connectivity constraints
• MinimumSetCovering — set system, no graph connectivity
• TravelingSalesman — requires visiting all vertices in a cycle, not building a tree over a subset
• MaxCut — edge partition, no connectivity requirement
• MinimumDominatingSet — domination, no tree structure

The problem's essence — choosing intermediate relay vertices to minimize total tree cost — has no equivalent in the current codebase.

Correctness

References verified:

1. Karp, 1972 — DOI 10.1007/978-1-4684-2001-2_9. Already in project bibliography (references.bib). Steiner Tree is indeed one of Karp's 21 NP-complete problems. Confirmed in references.md as problem Reduction-automaton, scalable way towards unifying computational hard problems #16. ✅

2. Hwang, Richards & Winter, 1992 — DOI 10.1016/S0167-5060(08)70656-2. Verified: published by North-Holland as Volume 53 of Annals of Discrete Mathematics. 339 pages. ISBN 9780444890986. Authoritative monograph on Steiner tree variants. ✅

3. Dreyfus & Wagner, 1971 — DOI 10.1002/net.3230010302. Verified: published in Networks, Vol. 1, pp. 195-207. The claimed complexity O(3^|T| * n + 2^|T| * n^2) matches the known Dreyfus-Wagner DP bound (confirmed via multiple sources including Codeforces, Springer, and the original paper). ✅

4. Byrka et al., 2013 — DOI 10.1145/2432622.2432628. Verified: published in Journal of the ACM, 2013. The approximation ratio is ln(4) + epsilon ≈ 1.386 + epsilon, which the issue rounds to "1.39-approximation". This is accurate. ✅

Complexity claims:

• NP-complete (Karp, 1972): Correct. ✅
• Dreyfus-Wagner DP bound: O(3^|T| * n + 2^|T| * n^2). Correct. ✅
• 1.39-approximation (Byrka et al.): Correct (ln(4) + epsilon). ✅
• 2-approximation via MST of terminal distance graph: Classic result, correct. ✅

Well-written

4a — Information completeness: All required sections present and substantive: Motivation, Name, Reference, Definition, Variables, Schema, Problem Size, Complexity, How to solve, Example Instance, Extra Remark. ✅

4b — Definition completeness: The formal definition is precise and implementable. Input structure (weighted undirected graph + terminal set), feasibility constraints (tree spanning all terminals), and objective (minimize total edge weight) are all clearly specified. ✅

4c — Symbol and notation consistency: Symbols are consistent across sections. G = (V, E, w) used consistently, T for terminals, V_S and E_S for the Steiner tree. The variables section correctly identifies m = |E| binary variables for edge inclusion. The Problem Size metrics (num_vertices, num_edges, num_terminals) are well-defined. ✅

4d — Example quality: The 5-vertex, 7-edge instance with T = {0, 2, 4} is non-trivial and illustrative. The optimal solution (cost 6, using Steiner vertices {1, 3}) is correct and hand-verifiable. The comparison with the direct terminal edge (2,4) at cost 6 effectively demonstrates why Steiner points matter. ✅

Recommendations

• Naming convention: The issue uses SteinerTree without a Minimum prefix. While this follows the pattern of TravelingSalesman (also a minimization problem without prefix), the codebase also uses MinimumVertexCover, MinimumDominatingSet, and MinimumSetCovering for other minimization problems. Consider whether MinimumSteinerTree would be more consistent with the Garey-Johnson and Compendium naming ("Minimum Steiner Tree"). Either choice is defensible.

• Improved exact algorithms: The Dreyfus-Wagner bound is the classic result, but Bjorklund et al. (2007, "Fourier meets Möbius") improved this to O*(2^|T|) using subset convolution over the Möbius algebra. Consider mentioning this as the best known exact algorithm parameterized by |T|.

[zazabap]

Issue Quality Check — Model

Check	Result	Details
Usefulness	✅ Pass	Novel problem not yet in the reduction graph; strong motivation
Non-trivial	✅ Pass	Distinct feasibility constraints (terminal connectivity + tree + weight minimization)
Correctness	✅ Pass	All references verified; complexity claims accurate
Well-written	✅ Pass	Complete, well-structured, implementable

Overall: 4 passed, 0 failed, 0 warnings

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Usefulness

SteinerTree does not exist in the current problem list (24 problems). The motivation is strong: it is one of Karp's original 21 NP-complete problems (problem #16), foundational in network design, with applications in VLSI routing, telecommunications, and phylogenetics. The issue explicitly mentions planned reductions to ILP and SAT, so it will not be an orphan node. It opens a new network design category alongside TravelingSalesman.

Non-trivial

SteinerTree is genuinely distinct from all existing problems. Its feasibility constraint requires selecting a subset of edges that (1) forms a tree and (2) spans a designated terminal set $T \subseteq V$, while minimizing total edge weight. No existing problem has this combination — the closest is TravelingSalesman (Hamiltonian cycle, not tree; visits all vertices, not a terminal subset). The terminal-set structure and Steiner vertex selection make it a fundamentally different optimization landscape.

Correctness

References verified:

Reference	Status	Details
Karp, 1972 (DOI: 10.1007/978-1-4684-2001-2_9)	✅ Verified	Steiner Tree is indeed #16 of Karp's 21 NP-complete problems. Already in project bibliography as karp1972.
Hwang, Richards & Winter, 1992 (DOI: 10.1016/S0167-5060(08)70656-2)	✅ Verified	Published as Volume 53 of Annals of Discrete Mathematics by Elsevier. Comprehensive monograph on the Steiner Tree Problem.
Dreyfus & Wagner, 1971 (DOI: 10.1002/net.3230010302)	✅ Verified	Classic DP algorithm. Complexity $O(3^{
Byrka et al., 2013 (DOI: 10.1145/2432622.2432628)	✅ Verified	Published in JACM Vol. 60, Issue 1. Achieves $\ln(4) + \varepsilon < 1.39$ approximation via iterative randomized rounding.

Complexity claims:

• NP-completeness via Karp (1972): ✅ correct
• Dreyfus-Wagner DP: $O(3^{|T|} \cdot n + 2^{|T|} \cdot n^2)$: ✅ confirmed
• 1.39-approximation (Byrka et al.): ✅ confirmed
• 2-approximation via MST of metric closure: ✅ well-known classical result

Well-written

4a — Information Completeness: All required sections present and substantive:

• ✅ Motivation: concrete use cases (VLSI, telecom, phylogenetics) and graph connectivity plan
• ✅ Name: SteinerTree — consistent with project convention (cf. TravelingSalesman for minimization without prefix)
• ✅ Reference: four literature citations with DOIs
• ✅ Definition: formal with input $(G, T, w)$, feasibility (tree spanning $T$), objective (minimize weight)
• ✅ Variables: $m = |E|$ binary edge variables with clear semantics
• ✅ Schema: type name, variants listed, field table with types
• ✅ Complexity: decision + exact + approximation with citations
• ✅ How to solve: brute-force checked with description
• ✅ Example instance: concrete 5-vertex graph with worked solution

4b — Definition Completeness: The formal definition clearly specifies input structure (weighted undirected graph + terminal set), feasibility (tree $S$ with $T \subseteq V_S$), and objective (minimize $\sum w(e)$). All quantifiers explicit.

4c — Symbol Consistency: Symbols ($G$, $V$, $E$, $w$, $T$, $n$, $m$) are defined and used consistently across sections. Size metric names (num_vertices, num_edges, num_terminals) follow project conventions.

4d — Example Quality: The 5-vertex, 7-edge instance with 3 terminals is non-trivial (exercises Steiner vertex selection), brute-force verifiable, and fully worked with step-by-step solution showing why Steiner points matter.