[Rule] MinimumMultiwayCut to ILP

[hmyuuu]

Source: MinimumMultiwayCut
Target: ILP
Motivation: Enables exact solving of the minimum multiway cut on any backend ILP solver; serves as the canonical exact formulation for testing and validation.
Reference: Chopra & Owen (1996), Mathematical Programming 73, pp. 7–30; Călinescu, Karloff & Rabani (2000), J. Comput. Syst. Sci. 60(3); Illinois lecture notes §7

Reduction Algorithm

Notation:

• Source: graph $G = (V, E, w)$ with $n = |V|$ vertices, $m = |E|$ edges, edge weights $w_e \in \mathbb{R}{>0}$, and $k = |T|$ terminal vertices $T = {t_0, t_1, \ldots, t{k-1}} \subseteq V$.
• Target: ILP with $kn + m$ binary variables.

Variable mapping:

Introduce two families of binary ILP variables (flattened into a single vector):

• $y_{iv} \in {0,1}$ for each $i \in {0,\ldots,k-1}$ and $v \in V$, at index $i \cdot n + v$: equals $1$ if vertex $v$ is assigned to the component of terminal $t_i$.
• $x_e \in {0,1}$ for each edge $e \in E$, at index $kn + e_{\text{idx}}$: equals $1$ if edge $e$ is in the cut.

Bounds (fixing terminals):

• $y_{i,t_i} = 1$ for each $i$ (terminal $t_i$ is fixed to its own component); implemented as bounds $[1, 1]$.
• $y_{j,t_i} = 0$ for all $j \neq i$ (terminal $t_i$ cannot belong to another component); implemented as bounds $[0, 0]$.
• All other $y_{iv}$ and $x_e$: binary bounds $[0, 1]$.

Constraints:

1. Partition constraint (equality, $n$ constraints): each vertex belongs to exactly one component:
   $$\sum_{i=0}^{k-1} y_{iv} = 1 \quad \forall v \in V$$

2. Edge-cut lower bounds (inequality, $2km$ constraints): if the endpoints of edge $e = (u, v)$ are in different components, then $x_e = 1$:
   $$x_e \geq y_{iu} - y_{iv} \quad \text{and} \quad x_e \geq y_{iv} - y_{iu} \quad \forall e = (u,v) \in E,\ i \in {0,\ldots,k-1}$$

Objective: Minimize total cut weight:
$$\min \sum_{e \in E} w_e \cdot x_e$$

Solution extraction: A solution to the ILP yields $x_e^* \in {0,1}$; the multiway cut is ${e \in E \mid x_e^* = 1}$ with cost $\sum_e w_e x_e^*$.

Size Overhead

Target metric (code name)	Polynomial (using symbols above)
num_vars	$kn + m$
num_constraints	$n + 2km$

Validation Method

• Compare ILP optimal value against brute-force multiway cut on small instances (e.g., $n \leq 6$, $k = 3$).
• Cross-check against PAAL library implementation: https://paal.mimuw.edu.pl/docs/multiway_cut.html
• Verify that the partition induced by $y_{iv}^$ matches the cut edges $x_e^$: for every cut edge $e=(u,v)$ in the solution, $u$ and $v$ must be in different components.

Example

Source instance (same as in model issue #184):

• $n = 5$ vertices $V = {0,1,2,3,4}$, $k = 3$ terminals $T = {0, 2, 4}$, $m = 6$ edges:

Edge index	$(u, v)$	$w_e$
0	(0, 1)	2
1	(1, 2)	3
2	(2, 3)	1
3	(3, 4)	2
4	(0, 4)	4
5	(1, 3)	5

ILP instance: $kn + m = 3 \cdot 5 + 6 = 21$ binary variables, $n + 2km = 5 + 2 \cdot 3 \cdot 6 = 41$ constraints.

Variable layout:

• $y_{0,v}$: variables 0–4 (component 0, one per vertex); fixed: $y_{0,0}=1$, $y_{0,2}=0$, $y_{0,4}=0$
• $y_{1,v}$: variables 5–9 (component 1); fixed: $y_{1,0}=0$, $y_{1,2}=1$, $y_{1,4}=0$
• $y_{2,v}$: variables 10–14 (component 2); fixed: $y_{2,0}=0$, $y_{2,2}=0$, $y_{2,4}=1$
• $x_e$: variables 15–20 (edge cut indicators)

Objective: minimize $2x_0 + 3x_1 + x_2 + 2x_3 + 4x_4 + 5x_5$.

Optimal ILP solution: $x_0 = x_3 = x_4 = 1$ (all others 0), yielding objective $2 + 2 + 4 = \mathbf{8}$.

Corresponding partition: $V_0 = {0}$, $V_1 = {1, 2, 3}$, $V_2 = {4}$. Each terminal is in its own component. Verified to match brute-force optimal from issue #184.

[zazabap]

Issue Quality Check — Rule

Check	Result	Details
Usefulness	✅ Pass	MinimumMultiwayCut is a new model (not yet in the codebase); no existing path to ILP
Non-trivial	✅ Pass	Constructs auxiliary edge-cut indicator variables with linking constraints — genuine structural transformation
Correctness	✅ Pass	All three references verified; ILP formulation matches standard literature
Well-written	✅ Pass	All sections present, complete, and consistent; example fully worked

Overall: 4 passed, 0 failed, 0 warnings

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Usefulness

MinimumMultiwayCut does not exist in the codebase yet (pred show MinimumMultiwayCut returns "Unknown problem"). Since the source problem is entirely new, there is no existing path from MinimumMultiwayCut to ILP. This reduction is novel and enables exact solving of multiway cut instances via any ILP backend. Pass.

Non-trivial

The reduction introduces two families of binary variables (vertex-assignment variables y_{iv} and edge-cut indicators x_e), partition equality constraints, and 2km linking inequalities that enforce consistency between vertex assignments and edge cuts. Terminal vertices are fixed via variable bounds. This is a genuine structural transformation involving auxiliary variables and non-trivial constraint construction — not a simple relabeling or subtype coercion. Pass.

Correctness

Reference 1: Chopra & Owen (1996)

• Verified: "Extended formulations for the A-cut problem," Mathematical Programming 73, pp. 7–30, 1996. The A-cut problem is the multiway cut problem. The paper introduces extended formulations with both node and edge variables for multiway cut, which is exactly what this issue's ILP formulation is based on. Citation is accurate.

Reference 2: Călinescu, Karloff & Rabani (2000)

• Verified: "An Improved Approximation Algorithm for Multiway Cut," Journal of Computer and System Sciences 60(3), 2000. This is a well-known paper on multiway cut approximation (achieving ratio 1.5 − 1/k). The LP relaxation in that paper uses a similar variable structure. Citation is accurate.

Reference 3: Illinois lecture notes §7

• URL verified and accessible (cs598csc/sp2009/lectures/lecture_7.pdf). This is a standard reference for multiway cut LP/ILP formulations in a graduate algorithms course.

Formulation correctness: The ILP formulation uses the standard approach from the literature — vertex-assignment variables with partition constraints, edge-cut indicators with linking inequalities, and a weighted objective over cut edges. The constraint count (n + 2km) and variable count (kn + m) are correct. Pass.

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 fully specified with numbered steps: variable mapping with explicit index formulas, bounds for fixing terminals, partition constraints, linking inequalities, objective function, and solution extraction. A programmer could implement this directly.

Symbol consistency: All symbols are defined before use (n, m, k, T, w_e, y_{iv}, x_e, e_idx). The overhead table uses the correct ILP code metric names (num_vars, num_constraints) matching the size_fields from pred show ILP.

Example quality: The example uses n=5, k=3, m=6 — non-trivial enough to exercise the reduction meaningfully. The full variable layout (21 variables), constraint count (41), objective coefficients, and optimal solution (cost 8, partition V_0={0}, V_1={1,2,3}, V_2={4}) are all explicitly worked through. The solution is hand-verifiable. Pass.

Recommendations

• Note that Chopra & Owen's paper title is "Extended formulations for the A-cut problem" — the issue links correctly but does not mention the actual title. Adding the title would improve traceability.