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Fix #330: Add MinMaxMulticenter (p-Center) model #630

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2cd5967
Add plan for #330: MinMaxMulticenter model
zazabap Mar 13, 2026
2bf3566
Merge remote-tracking branch 'origin/main' into issue-330-minmax-mult…
GiggleLiu Mar 16, 2026
466b81c
Add MinMaxMulticenter (vertex p-center) satisfaction problem model
GiggleLiu Mar 16, 2026
403e92e
Add MinMaxMulticenter problem-def to paper
GiggleLiu Mar 16, 2026
471f16f
Review fixes: add trait_consistency entry and non-unit edge length test
GiggleLiu Mar 16, 2026
cf36614
chore: remove plan file after implementation
GiggleLiu Mar 16, 2026
ab2f5d3
fix: address PR #630 review findings
GiggleLiu Mar 16, 2026
4d47043
fix: tighten PR #630 MinMaxMulticenter CLI review follow-ups
GiggleLiu Mar 16, 2026
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45 changes: 45 additions & 0 deletions docs/paper/reductions.typ
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Expand Up @@ -99,6 +99,7 @@
"MinimumFeedbackArcSet": [Minimum Feedback Arc Set],
"MinimumFeedbackVertexSet": [Minimum Feedback Vertex Set],
"ShortestCommonSupersequence": [Shortest Common Supersequence],
"MinMaxMulticenter": [Min-Max Multicenter],
"MinimumSumMulticenter": [Minimum Sum Multicenter],
"SteinerTree": [Steiner Tree],
"SubgraphIsomorphism": [Subgraph Isomorphism],
Expand Down Expand Up @@ -1003,6 +1004,50 @@ NP-completeness was established by Garey, Johnson, and Stockmeyer @gareyJohnsonS
]
}

#{
let x = load-model-example("MinMaxMulticenter")
let nv = graph-num-vertices(x.instance)
let edges = x.instance.graph.inner.edges.map(e => (e.at(0), e.at(1)))
let K = x.instance.k
let B = x.instance.bound
let desired-config = (0, 1, 0, 0, 1, 0)
let sol = x.optimal.find(o => o.config == desired-config)
let centers = sol.config.enumerate().filter(((i, v)) => v == 1).map(((i, _)) => i)
[
#problem-def("MinMaxMulticenter")[
Given a graph $G = (V, E)$ with vertex weights $w: V -> ZZ_(>= 0)$, edge lengths $l: E -> ZZ_(>= 0)$, a positive integer $K <= |V|$, and a rational bound $B > 0$, does there exist $S subset.eq V$ with $|S| = K$ such that $max_(v in V) w(v) dot d(v, S) <= B$, where $d(v, S) = min_(s in S) d(v, s)$ is the shortest weighted-path distance from $v$ to the nearest vertex in $S$?
][
Also known as the _vertex p-center problem_ (Garey & Johnson A2 ND50). The goal is to place $K$ facilities so that the worst-case weighted distance from any demand point to its nearest facility is at most $B$. NP-complete even with unit weights and unit edge lengths (Kariv and Hakimi, 1979).

Closely related to Dominating Set: on unweighted unit-length graphs, a $K$-center with radius $B = 1$ is exactly a dominating set of size $K$. The best known exact algorithm runs in $O^*(1.4969^n)$ via binary search over distance thresholds combined with dominating set computation @vanrooij2011. An optimal 2-approximation exists (Hochbaum and Shmoys, 1985); no $(2 - epsilon)$-approximation is possible unless $P = "NP"$ (Hsu and Nemhauser, 1979).

Variables: $n = |V|$ binary variables, one per vertex. $x_v = 1$ if vertex $v$ is selected as a center. A configuration is satisfying when exactly $K$ centers are selected and $max_(v in V) w(v) dot d(v, S) <= B$.

*Example.* Consider the graph $G$ on #nv vertices with unit weights $w(v) = 1$, unit edge lengths, edges ${#edges.map(((u, v)) => $(#u, #v)$).join(", ")}$, $K = #K$, and $B = #B$. Placing centers at $S = {#centers.map(i => $v_#i$).join(", ")}$ gives maximum distance $max_v d(v, S) = 1 <= B$, so this is a feasible solution. In fact, #x.optimal.len() distinct center placements satisfy the bound.

#figure({
let blue = graph-colors.at(0)
let gray = luma(200)
canvas(length: 1cm, {
import draw: *
let verts = ((-1.5, 0.8), (0, 1.5), (1.5, 0.8), (1.5, -0.8), (0, -1.5), (-1.5, -0.8))
for (u, v) in edges {
g-edge(verts.at(u), verts.at(v), stroke: 1pt + gray)
}
for (k, pos) in verts.enumerate() {
let is-center = centers.any(c => c == k)
g-node(pos, name: "v" + str(k),
fill: if is-center { blue } else { white },
label: if is-center { text(fill: white)[$v_#k$] } else { [$v_#k$] })
}
})
},
caption: [Min-Max Multicenter with $K = #K$, $B = #B$ on a #{nv}-vertex graph. Centers #centers.map(i => $v_#i$).join(" and ") (blue) ensure every vertex is within distance $B$ of some center.],
) <fig:min-max-multicenter>
]
]
}

== Set Problems

#{
Expand Down
4 changes: 3 additions & 1 deletion docs/src/cli.md
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Expand Up @@ -275,6 +275,7 @@ pred create QUBO --matrix "1,0.5;0.5,2" -o qubo.json
pred create KColoring --k 3 --graph 0-1,1-2,2-0 -o kcol.json
pred create SpinGlass --graph 0-1,1-2 -o sg.json
pred create MaxCut --graph 0-1,1-2,2-0 -o maxcut.json
pred create MinMaxMulticenter --graph 0-1,1-2,2-3 --weights 1,1,1,1 --edge-weights 1,1,1 --k 2 --bound 1 -o pcenter.json
pred create SteinerTree --graph 0-1,0-3,1-2,1-3,2-3,2-4,3-4 --edge-weights 2,5,2,1,5,6,1 --terminals 0,2,4 -o steiner.json
pred create Factoring --target 15 --bits-m 4 --bits-n 4 -o factoring.json
pred create Factoring --target 21 --bits-m 3 --bits-n 3 -o factoring2.json
Expand Down Expand Up @@ -393,6 +394,7 @@ Stdin is supported with `-`:
```bash
pred create MIS --graph 0-1,1-2,2-3 | pred solve -
pred create MIS --graph 0-1,1-2,2-3 | pred solve - --solver brute-force
pred create MinMaxMulticenter --graph 0-1,1-2,2-3 --weights 1,1,1,1 --edge-weights 1,1,1 --k 2 --bound 1 | pred solve - --solver brute-force
```

When the problem is not ILP, the solver automatically reduces it to ILP, solves, and maps the solution back. The auto-reduction is shown in the output:
Expand All @@ -418,7 +420,7 @@ Source evaluation: Valid(2)
```

> **Note:** The ILP solver requires a reduction path from the target problem to ILP.
> Some problems (e.g., QUBO, SpinGlass, MaxCut, CircuitSAT) do not have this path yet.
> Some problems (e.g., QUBO, SpinGlass, MaxCut, CircuitSAT, MinMaxMulticenter) do not have this path yet.
> Use `--solver brute-force` for these, or reduce to a problem that supports ILP first.

## Shell Completions
Expand Down
31 changes: 31 additions & 0 deletions docs/src/reductions/problem_schemas.json
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Expand Up @@ -366,6 +366,37 @@
}
]
},
{
"name": "MinMaxMulticenter",
"description": "Determine if K centers can be placed so max weighted distance is at most B (vertex p-center)",
"fields": [
{
"name": "graph",
"type_name": "G",
"description": "The underlying graph G=(V,E)"
},
{
"name": "vertex_weights",
"type_name": "Vec<W>",
"description": "Vertex weights w: V -> R"
},
{
"name": "edge_lengths",
"type_name": "Vec<W>",
"description": "Edge lengths l: E -> R"
},
{
"name": "k",
"type_name": "usize",
"description": "Number of centers to place"
},
{
"name": "bound",
"type_name": "W::Sum",
"description": "Upper bound B on maximum weighted distance"
}
]
},
{
"name": "MinimumDominatingSet",
"description": "Find minimum weight dominating set in a graph",
Expand Down
76 changes: 43 additions & 33 deletions docs/src/reductions/reduction_graph.json
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Expand Up @@ -349,6 +349,16 @@
"doc_path": "models/set/struct.MaximumSetPacking.html",
"complexity": "2^num_sets"
},
{
"name": "MinMaxMulticenter",
"variant": {
"graph": "SimpleGraph",
"weight": "i32"
},
"category": "graph",
"doc_path": "models/graph/struct.MinMaxMulticenter.html",
"complexity": "1.4969^num_vertices"
},
{
"name": "MinimumDominatingSet",
"variant": {
Expand Down Expand Up @@ -569,7 +579,7 @@
},
{
"source": 4,
"target": 54,
"target": 55,
"overhead": [
{
"field": "num_spins",
Expand Down Expand Up @@ -629,7 +639,7 @@
},
{
"source": 12,
"target": 49,
"target": 50,
"overhead": [
{
"field": "num_vars",
Expand Down Expand Up @@ -670,7 +680,7 @@
},
{
"source": 19,
"target": 49,
"target": 50,
"overhead": [
{
"field": "num_vars",
Expand All @@ -696,7 +706,7 @@
},
{
"source": 20,
"target": 49,
"target": 50,
"overhead": [
{
"field": "num_vars",
Expand All @@ -722,7 +732,7 @@
},
{
"source": 21,
"target": 49,
"target": 50,
"overhead": [
{
"field": "num_vars",
Expand All @@ -733,7 +743,7 @@
},
{
"source": 21,
"target": 58,
"target": 59,
"overhead": [
{
"field": "num_elements",
Expand All @@ -744,7 +754,7 @@
},
{
"source": 22,
"target": 51,
"target": 52,
"overhead": [
{
"field": "num_clauses",
Expand All @@ -763,7 +773,7 @@
},
{
"source": 23,
"target": 49,
"target": 50,
"overhead": [
{
"field": "num_vars",
Expand All @@ -789,7 +799,7 @@
},
{
"source": 25,
"target": 54,
"target": 55,
"overhead": [
{
"field": "num_spins",
Expand Down Expand Up @@ -969,7 +979,7 @@
},
{
"source": 31,
"target": 45,
"target": 46,
"overhead": [
{
"field": "num_vertices",
Expand Down Expand Up @@ -1104,7 +1114,7 @@
},
{
"source": 37,
"target": 49,
"target": 50,
"overhead": [
{
"field": "num_vars",
Expand Down Expand Up @@ -1159,7 +1169,7 @@
"doc_path": "rules/maximumsetpacking_casts/index.html"
},
{
"source": 39,
"source": 40,
"target": 12,
"overhead": [
{
Expand All @@ -1174,7 +1184,7 @@
"doc_path": "rules/minimumdominatingset_ilp/index.html"
},
{
"source": 42,
"source": 43,
"target": 12,
"overhead": [
{
Expand All @@ -1189,7 +1199,7 @@
"doc_path": "rules/minimumsetcovering_ilp/index.html"
},
{
"source": 45,
"source": 46,
"target": 31,
"overhead": [
{
Expand All @@ -1204,8 +1214,8 @@
"doc_path": "rules/minimumvertexcover_maximumindependentset/index.html"
},
{
"source": 45,
"target": 42,
"source": 46,
"target": 43,
"overhead": [
{
"field": "num_sets",
Expand All @@ -1219,7 +1229,7 @@
"doc_path": "rules/minimumvertexcover_minimumsetcovering/index.html"
},
{
"source": 49,
"source": 50,
"target": 12,
"overhead": [
{
Expand All @@ -1234,8 +1244,8 @@
"doc_path": "rules/qubo_ilp/index.html"
},
{
"source": 49,
"target": 53,
"source": 50,
"target": 54,
"overhead": [
{
"field": "num_spins",
Expand All @@ -1245,7 +1255,7 @@
"doc_path": "rules/spinglass_qubo/index.html"
},
{
"source": 51,
"source": 52,
"target": 4,
"overhead": [
{
Expand All @@ -1260,7 +1270,7 @@
"doc_path": "rules/sat_circuitsat/index.html"
},
{
"source": 51,
"source": 52,
"target": 16,
"overhead": [
{
Expand All @@ -1275,7 +1285,7 @@
"doc_path": "rules/sat_coloring/index.html"
},
{
"source": 51,
"source": 52,
"target": 21,
"overhead": [
{
Expand All @@ -1290,7 +1300,7 @@
"doc_path": "rules/sat_ksat/index.html"
},
{
"source": 51,
"source": 52,
"target": 30,
"overhead": [
{
Expand All @@ -1305,8 +1315,8 @@
"doc_path": "rules/sat_maximumindependentset/index.html"
},
{
"source": 51,
"target": 39,
"source": 52,
"target": 40,
"overhead": [
{
"field": "num_vertices",
Expand All @@ -1320,8 +1330,8 @@
"doc_path": "rules/sat_minimumdominatingset/index.html"
},
{
"source": 53,
"target": 49,
"source": 54,
"target": 50,
"overhead": [
{
"field": "num_vars",
Expand All @@ -1331,7 +1341,7 @@
"doc_path": "rules/spinglass_qubo/index.html"
},
{
"source": 54,
"source": 55,
"target": 25,
"overhead": [
{
Expand All @@ -1346,8 +1356,8 @@
"doc_path": "rules/spinglass_maxcut/index.html"
},
{
"source": 54,
"target": 53,
"source": 55,
"target": 54,
"overhead": [
{
"field": "num_spins",
Expand All @@ -1361,7 +1371,7 @@
"doc_path": "rules/spinglass_casts/index.html"
},
{
"source": 59,
"source": 60,
"target": 12,
"overhead": [
{
Expand All @@ -1376,8 +1386,8 @@
"doc_path": "rules/travelingsalesman_ilp/index.html"
},
{
"source": 59,
"target": 49,
"source": 60,
"target": 50,
"overhead": [
{
"field": "num_vars",
Expand Down
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