Fix #422: [Model] TwoDimensionalConsecutiveSets

docs/paper/reductions.typ

@@ -136,10 +136,11 @@
   "SchedulingWithIndividualDeadlines": [Scheduling With Individual Deadlines],
   "StaffScheduling": [Staff Scheduling],
   "MinimumTardinessSequencing": [Minimum Tardiness Sequencing],
-  "SequencingToMinimizeWeightedTardiness": [Sequencing to Minimize Weighted Tardiness],
-  "SequencingToMinimizeMaximumCumulativeCost": [Sequencing to Minimize Maximum Cumulative Cost],
   "ConsecutiveOnesSubmatrix": [Consecutive Ones Submatrix],
+  "SequencingToMinimizeMaximumCumulativeCost": [Sequencing to Minimize Maximum Cumulative Cost],
+  "SequencingToMinimizeWeightedTardiness": [Sequencing to Minimize Weighted Tardiness],
   "SumOfSquaresPartition": [Sum of Squares Partition],
+  "TwoDimensionalConsecutiveSets": [2-Dimensional Consecutive Sets],
   "SequencingWithinIntervals": [Sequencing Within Intervals],
   "DirectedTwoCommodityIntegralFlow": [Directed Two-Commodity Integral Flow],
   "ConjunctiveBooleanQuery": [Conjunctive Boolean Query],
@@ -1906,6 +1907,52 @@ A classical NP-complete problem from Garey and Johnson @garey1979[Ch.~3, p.~76],
   ]
 }
 
+#{
+  let x = load-model-example("TwoDimensionalConsecutiveSets")
+  let n = x.instance.alphabet_size
+  let subs = x.instance.subsets
+  let m = subs.len()
+  let sol = (config: x.optimal_config, metric: x.optimal_value)
+  let config = sol.config
+  // Build groups from config: groups.at(g) = list of symbols in group g
+  let groups = range(n).map(g => range(n).filter(s => config.at(s) == g))
+  // Only non-empty groups
+  let nonempty = groups.enumerate().filter(((_, g)) => g.len() > 0)
+  let k = nonempty.len()
+  let fmt-set(s) = "${" + s.map(e => str(e)).join(", ") + "}$"
+  [
+    #problem-def("TwoDimensionalConsecutiveSets")[
+      Given finite alphabet $Sigma = {0, 1, dots, n - 1}$ and collection $cal(C) = {Sigma_1, dots, Sigma_m}$ of subsets of $Sigma$, determine whether $Sigma$ can be partitioned into disjoint sets $X_1, X_2, dots, X_k$ such that each $X_i$ has at most one element in common with each $Sigma_j$, and for each $Sigma_j in cal(C)$ there is an index $l(j)$ with $Sigma_j subset.eq X_(l(j)) union X_(l(j)+1) union dots.c union X_(l(j)+|Sigma_j|-1)$.
+    ][
+    This problem generalizes the Consecutive Sets problem (SR18) by requiring not just that each subset's elements appear consecutively in an ordering, but that they be spread across consecutive groups of a partition where each group contributes at most one element per subset. Shown NP-complete by Lipski @lipski1977fct via transformation from Graph 3-Colorability. The problem arises in information storage and retrieval where records must be organized in contiguous blocks. It remains NP-complete if all subsets have at most 5 elements, but is solvable in polynomial time if all subsets have at most 2 elements. The brute-force algorithm assigns each of $n$ symbols to one of up to $n$ groups, giving $O^*(n^n)$ time#footnote[No algorithm improving on brute-force enumeration is known for this problem.].
+
+    *Example.* Let $Sigma = {0, 1, dots, #(n - 1)}$ and $cal(C) = {#range(m).map(i => $Sigma_#(i + 1)$).join(", ")}$ with #subs.enumerate().map(((i, s)) => $Sigma_#(i + 1) = #fmt-set(s)$).join(", "). A valid partition uses $k = #k$ groups: #nonempty.map(((g, elems)) => $X_#(g + 1) = #fmt-set(elems)$).join(", "). Each group intersects every subset in at most one element, and each subset's elements span exactly $|Sigma_j|$ consecutive groups. For instance, $Sigma_1 = {0, 1, 2}$ maps to groups $X_1, X_2, X_3$ (consecutive), and $Sigma_5 = {0, 5}$ maps to groups $X_1, X_2$ (consecutive). Multiple valid partitions exist for this instance, differing only by unused or shifted group labels.
+
+    #figure(
+      canvas(length: 1cm, {
+        import draw: *
+        // Draw groups as labeled columns
+        let gw = 1.4
+        let gh = 0.45
+        for (col, (g, elems)) in nonempty.enumerate() {
+          let x0 = col * (gw + 0.3)
+          // Group header
+          content((x0 + gw / 2, 0.5), $X_#(g + 1)$, anchor: "south")
+          // Draw box for the group
+          rect((x0, -elems.len() * gh), (x0 + gw, 0),
+            stroke: 0.5pt + black, fill: rgb("#e8f0fe"))
+          // Elements inside
+          for (row, elem) in elems.enumerate() {
+            content((x0 + gw / 2, -row * gh - gh / 2), text(size: 9pt, str(elem)))
+          }
+        }
+      }),
+      caption: [2-Dimensional Consecutive Sets: partition of $Sigma = {0, dots, 5}$ into #k groups satisfying intersection and consecutiveness constraints for all #m subsets.],
+    ) <fig:two-dim-consecutive-sets>
+    ]
+  ]
+}
+
 == Optimization Problems
 
 #{

docs/paper/references.bib

@@ -944,20 +944,25 @@ @article{papadimitriou1982
   doi     = {10.1145/322307.322309}
 }
 
-@book{papadimitriou-steiglitz1982,
-  author    = {Christos H. Papadimitriou and Kenneth Steiglitz},
-  title     = {Combinatorial Optimization: Algorithms and Complexity},
-  publisher = {Prentice-Hall},
-  address   = {Englewood Cliffs, NJ},
-  year      = {1982}
+@article{booth1976,
+  author  = {Booth, Kellogg S. and Lueker, George S.},
+  title   = {Testing for the consecutive ones property, interval graphs, and graph planarity using {PQ}-tree algorithms},
+  journal = {Journal of Computer and System Sciences},
+  volume  = {13},
+  number  = {3},
+  pages   = {335--379},
+  year    = {1976}
 }
 
-@techreport{Heidari2022,
-  author      = {Heidari, Shahrokh and Dinneen, Michael J. and Delmas, Patrice},
-  title       = {An Equivalent {QUBO} Model to the Minimum Multi-Way Cut Problem},
-  institution = {Centre for Discrete Mathematics and Theoretical Computer Science, University of Auckland},
-  year        = {2022},
-  number      = {CDMTCS-565}
+@article{boothlueker1976,
+  author  = {Kellogg S. Booth and George S. Lueker},
+  title   = {Testing for the Consecutive Ones Property, Interval Graphs, and Graph Planarity Using {PQ}-Tree Algorithms},
+  journal = {Journal of Computer and System Sciences},
+  volume  = {13},
+  number  = {3},
+  pages   = {335--379},
+  year    = {1976},
+  doi     = {10.1016/S0022-0000(76)80045-1}
 }
 
 @phdthesis{booth1975,
@@ -967,25 +972,25 @@ @phdthesis{booth1975
   year    = {1975}
 }
 
-@article{booth1976,
-  author  = {Booth, Kellogg S. and Lueker, George S.},
-  title   = {Testing for the consecutive ones property, interval graphs, and graph planarity using {PQ}-tree algorithms},
-  journal = {Journal of Computer and System Sciences},
-  volume  = {13},
-  number  = {3},
-  pages   = {335--379},
-  year    = {1976}
+@article{chopra1996,
+  author  = {Sunil Chopra and Jonathan H. Owen},
+  title   = {Extended formulations for the A-cut problem},
+  journal = {Mathematical Programming},
+  volume  = {73},
+  pages   = {7--30},
+  year    = {1996},
+  doi     = {10.1007/BF02592096}
 }
 
-@article{lawler1972,
-  author  = {Eugene L. Lawler},
-  title   = {A Procedure for Computing the $K$ Best Solutions to Discrete Optimization Problems and Its Application to the Shortest Path Problem},
-  journal = {Management Science},
-  volume  = {18},
-  number  = {7},
-  pages   = {401--405},
+@article{coffman1972,
+  author  = {Edward G. Coffman and Ronald L. Graham},
+  title   = {Optimal Scheduling for Two-Processor Systems},
+  journal = {Acta Informatica},
+  volume  = {1},
+  number  = {3},
+  pages   = {200--213},
   year    = {1972},
-  doi     = {10.1287/mnsc.18.7.401}
+  doi     = {10.1007/BF00288685}
 }
 
 @article{eppstein1992,
@@ -999,14 +1004,23 @@ @article{eppstein1992
   doi     = {10.1007/BF01994880}
 }
 
-@article{chopra1996,
-  author  = {Sunil Chopra and Jonathan H. Owen},
-  title   = {Extended formulations for the A-cut problem},
-  journal = {Mathematical Programming},
-  volume  = {73},
-  pages   = {7--30},
-  year    = {1996},
-  doi     = {10.1007/BF02592096}
+@techreport{Heidari2022,
+  author      = {Heidari, Shahrokh and Dinneen, Michael J. and Delmas, Patrice},
+  title       = {An Equivalent {QUBO} Model to the Minimum Multi-Way Cut Problem},
+  institution = {Centre for Discrete Mathematics and Theoretical Computer Science, University of Auckland},
+  year        = {2022},
+  number      = {CDMTCS-565}
+}
+
+@article{hu1961,
+  author  = {Te Chiang Hu},
+  title   = {Parallel Sequencing and Assembly Line Problems},
+  journal = {Operations Research},
+  volume  = {9},
+  number  = {6},
+  pages   = {841--848},
+  year    = {1961},
+  doi     = {10.1287/opre.9.6.841}
 }
 
 @article{kou1977,
@@ -1020,26 +1034,15 @@ @article{kou1977
   doi     = {10.1137/0206005}
 }
 
-@article{boothlueker1976,
-  author  = {Kellogg S. Booth and George S. Lueker},
-  title   = {Testing for the Consecutive Ones Property, Interval Graphs, and Graph Planarity Using {PQ}-Tree Algorithms},
-  journal = {Journal of Computer and System Sciences},
-  volume  = {13},
-  number  = {3},
-  pages   = {335--379},
-  year    = {1976},
-  doi     = {10.1016/S0022-0000(76)80045-1}
-}
-
-@article{ullman1975,
-  author  = {Jeffrey D. Ullman},
-  title   = {NP-Complete Scheduling Problems},
-  journal = {Journal of Computer and System Sciences},
-  volume  = {10},
-  number  = {3},
-  pages   = {384--393},
-  year    = {1975},
-  doi     = {10.1016/S0022-0000(75)80008-0}
+@article{lawler1972,
+  author  = {Eugene L. Lawler},
+  title   = {A Procedure for Computing the $K$ Best Solutions to Discrete Optimization Problems and Its Application to the Shortest Path Problem},
+  journal = {Management Science},
+  volume  = {18},
+  number  = {7},
+  pages   = {401--405},
+  year    = {1972},
+  doi     = {10.1287/mnsc.18.7.401}
 }
 
 @article{lenstra1978,
@@ -1053,26 +1056,24 @@ @article{lenstra1978
   doi     = {10.1287/opre.26.1.22}
 }
 
-@article{coffman1972,
-  author  = {Edward G. Coffman and Ronald L. Graham},
-  title   = {Optimal Scheduling for Two-Processor Systems},
-  journal = {Acta Informatica},
-  volume  = {1},
-  number  = {3},
-  pages   = {200--213},
-  year    = {1972},
-  doi     = {10.1007/BF00288685}
+@inproceedings{lipski1977fct,
+  author    = {Witold Lipski Jr.},
+  title     = {Two {NP}-Complete Problems Related to Information Retrieval},
+  booktitle = {Fundamentals of Computation Theory (FCT 1977)},
+  series    = {Lecture Notes in Computer Science},
+  volume    = {56},
+  pages     = {452--458},
+  publisher = {Springer},
+  year      = {1977},
+  doi       = {10.1007/3-540-08442-8_115}
 }
 
-@article{hu1961,
-  author  = {Te Chiang Hu},
-  title   = {Parallel Sequencing and Assembly Line Problems},
-  journal = {Operations Research},
-  volume  = {9},
-  number  = {6},
-  pages   = {841--848},
-  year    = {1961},
-  doi     = {10.1287/opre.9.6.841}
+@book{papadimitriou-steiglitz1982,
+  author    = {Christos H. Papadimitriou and Kenneth Steiglitz},
+  title     = {Combinatorial Optimization: Algorithms and Complexity},
+  publisher = {Prentice-Hall},
+  address   = {Englewood Cliffs, NJ},
+  year      = {1982}
 }
 
 @inproceedings{papadimitriou1979,
@@ -1083,3 +1084,14 @@ @inproceedings{papadimitriou1979
   year      = {1979},
   doi       = {10.1145/800135.804393}
 }
+
+@article{ullman1975,
+  author  = {Jeffrey D. Ullman},
+  title   = {NP-Complete Scheduling Problems},
+  journal = {Journal of Computer and System Sciences},
+  volume  = {10},
+  number  = {3},
+  pages   = {384--393},
+  year    = {1975},
+  doi     = {10.1016/S0022-0000(75)80008-0}
+}

docs/src/cli.md

@@ -502,6 +502,7 @@ 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 TwoDimensionalConsecutiveSets --alphabet-size 6 --sets "0,1,2;3,4,5;1,3;2,4;0,5" | 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:

problemreductions-cli/src/cli.rs

@@ -243,8 +243,9 @@ Flags by problem type:
   ComparativeContainment          --universe, --r-sets, --s-sets [--r-weights] [--s-weights]
   X3C (ExactCoverBy3Sets)         --universe, --sets (3 elements each)
   SetBasis                        --universe, --sets, --k
-  PrimeAttributeName              --universe, --deps, --query
   MinimumCardinalityKey           --num-attributes, --dependencies, --k
+  PrimeAttributeName              --universe, --deps, --query
+  TwoDimensionalConsecutiveSets   --alphabet-size, --sets
   BicliqueCover                   --left, --right, --biedges, --k
   BalancedCompleteBipartiteSubgraph --left, --right, --biedges, --k
   BiconnectivityAugmentation      --graph, --potential-edges, --budget [--num-vertices]
@@ -307,8 +308,9 @@ Examples:
   pred create UndirectedTwoCommodityIntegralFlow --graph 0-2,1-2,2-3 --capacities 1,1,2 --source-1 0 --sink-1 3 --source-2 1 --sink-2 3 --requirement-1 1 --requirement-2 1
   pred create X3C --universe 9 --sets \"0,1,2;0,2,4;3,4,5;3,5,7;6,7,8;1,4,6;2,5,8\"
   pred create SetBasis --universe 4 --sets \"0,1;1,2;0,2;0,1,2\" --k 3
+  pred create MinimumCardinalityKey --num-attributes 6 --dependencies \"0,1>2;0,2>3;1,3>4;2,4>5\" --k 2
   pred create PrimeAttributeName --universe 6 --deps \"0,1>2,3,4,5;2,3>0,1,4,5\" --query 3
-  pred create MinimumCardinalityKey --num-attributes 6 --dependencies \"0,1>2;0,2>3;1,3>4;2,4>5\" --k 2")]
+  pred create TwoDimensionalConsecutiveSets --alphabet-size 6 --sets \"0,1,2;3,4,5;1,3;2,4;0,5\"")]
 pub struct CreateArgs {
     /// Problem type (e.g., MIS, QUBO, SAT). Omit when using --example.
     #[arg(value_parser = crate::problem_name::ProblemNameParser)]
@@ -547,7 +549,7 @@ pub struct CreateArgs {
     /// Number of available workers for StaffScheduling
     #[arg(long)]
     pub num_workers: Option<u64>,
-    /// Alphabet size for LCS, SCS, or StringToStringCorrection (optional; inferred from the input strings if omitted)
+    /// Alphabet size for LCS, SCS, StringToStringCorrection, or TwoDimensionalConsecutiveSets (optional; inferred from the input strings if omitted)
     #[arg(long)]
     pub alphabet_size: Option<usize>,
     /// Number of attributes for AdditionalKey or MinimumCardinalityKey
@@ -597,6 +599,7 @@ Examples:
   pred solve reduced.json -o solution.json       # save result to file
   pred create MIS --graph 0-1,1-2 | pred solve - # read from stdin when an ILP path exists
   pred create StringToStringCorrection --source-string \"0,1,2,3,1,0\" --target-string \"0,1,3,2,1\" --bound 2 | pred solve - --solver brute-force
+  pred create TwoDimensionalConsecutiveSets --alphabet-size 6 --sets \"0,1,2;3,4,5;1,3;2,4;0,5\" | pred solve - --solver brute-force
   pred solve problem.json --timeout 10           # abort after 10 seconds
 
 Typical workflow:

problemreductions-cli/src/commands/create.rs

@@ -445,15 +445,18 @@ fn example_for(canonical: &str, graph_type: Option<&str>) -> &'static str {
             "--universe 4 --r-sets \"0,1,2,3;0,1\" --s-sets \"0,1,2,3;2,3\" --r-weights 2,5 --s-weights 3,6"
         }
         "SetBasis" => "--universe 4 --sets \"0,1;1,2;0,2;0,1,2\" --k 3",
-        "PrimeAttributeName" => {
-            "--universe 6 --deps \"0,1>2,3,4,5;2,3>0,1,4,5\" --query 3"
-        }
         "LongestCommonSubsequence" => {
             "--strings \"010110;100101;001011\" --bound 3 --alphabet-size 2"
         }
         "MinimumCardinalityKey" => {
             "--num-attributes 6 --dependencies \"0,1>2;0,2>3;1,3>4;2,4>5\" --k 2"
         }
+        "PrimeAttributeName" => {
+            "--universe 6 --deps \"0,1>2,3,4,5;2,3>0,1,4,5\" --query 3"
+        }
+        "TwoDimensionalConsecutiveSets" => {
+            "--alphabet-size 6 --sets \"0,1,2;3,4,5;1,3;2,4;0,5\""
+        }
         "ShortestCommonSupersequence" => "--strings \"0,1,2;1,2,0\" --bound 4",
         "SequencingToMinimizeMaximumCumulativeCost" => {
             "--costs 2,-1,3,-2,1,-3 --precedence-pairs \"0>2,1>2,1>3,2>4,3>5,4>5\" --bound 4"
@@ -1794,6 +1797,27 @@ pub fn create(args: &CreateArgs, out: &OutputConfig) -> Result<()> {
             )
         }
 
+        // TwoDimensionalConsecutiveSets
+        "TwoDimensionalConsecutiveSets" => {
+            let alphabet_size = args.alphabet_size.or(args.universe).ok_or_else(|| {
+                anyhow::anyhow!(
+                    "TwoDimensionalConsecutiveSets requires --alphabet-size (or --universe) and --sets\n\n\
+                     Usage: pred create TwoDimensionalConsecutiveSets --alphabet-size 6 --sets \"0,1,2;3,4,5;1,3;2,4;0,5\""
+                )
+            })?;
+            let sets = parse_sets(args)?;
+            (
+                ser(
+                    problemreductions::models::set::TwoDimensionalConsecutiveSets::try_new(
+                        alphabet_size,
+                        sets,
+                    )
+                    .map_err(anyhow::Error::msg)?,
+                )?,
+                resolved_variant.clone(),
+            )
+        }
+
         // BicliqueCover
         "BicliqueCover" => {
             let usage = "pred create BicliqueCover --left 2 --right 2 --biedges 0-0,0-1,1-1 --k 2";