Fix #216: Add HamiltonianCircuit model

docs/paper/reductions.typ

@@ -47,6 +47,7 @@
   "MinimumVertexCover": [Minimum Vertex Cover],
   "MaxCut": [Max-Cut],
   "GraphPartitioning": [Graph Partitioning],
+  "HamiltonianCircuit": [Hamiltonian Circuit],
   "HamiltonianPath": [Hamiltonian Path],
   "IsomorphicSpanningTree": [Isomorphic Spanning Tree],
   "KColoring": [$k$-Coloring],
@@ -619,6 +620,21 @@ One of the most intensely studied NP-hard problems, with applications in logisti
 caption: [Complete graph $K_4$ with weighted edges. The optimal tour $v_0 -> v_1 -> v_2 -> v_3 -> v_0$ (blue edges) has cost 6.],
 ) <fig:k4-tsp>
 ]
+#problem-def("HamiltonianCircuit")[
+  *Instance:* An undirected graph $G = (V, E)$.
+
+  *Question:* Does $G$ contain a _Hamiltonian circuit_ --- a closed path that visits every vertex exactly once?
+][
+  The Hamiltonian Circuit problem is one of Karp's original 21 NP-complete problems @karp1972, and is listed as GT37 in Garey & Johnson @garey1979.
+  It is closely related to the Traveling Salesman Problem: while TSP seeks to minimize the total weight of a Hamiltonian cycle on a weighted complete graph, the Hamiltonian Circuit problem simply asks whether _any_ such cycle exists on a general (unweighted) graph.
+
+  A configuration is a permutation $pi$ of the vertices, interpreted as the order in which they are visited.
+  The circuit is valid when every consecutive pair $(pi(i), pi(i+1 mod n))$ is an edge in $G$.
+
+  *Algorithms.*
+  The classical Held--Karp dynamic programming algorithm @heldkarp1962 solves the problem in $O(n^2 dot 2^n)$ time and $O(n dot 2^n)$ space.
+  Björklund's randomized "Determinant Sums" algorithm achieves $O^*(1.657^n)$ time for general graphs and $O^*(sqrt(2)^n)$ for bipartite graphs @bjorklund2014.
+]
 #problem-def("OptimalLinearArrangement")[
   Given an undirected graph $G=(V,E)$ and a non-negative integer $K$, is there a bijection $f: V -> {0, 1, dots, |V|-1}$ such that $sum_({u,v} in E) |f(u) - f(v)| <= K$?
 ][

docs/paper/references.bib

@@ -237,6 +237,17 @@ @inproceedings{zamir2021
   doi       = {10.4230/LIPIcs.ICALP.2021.113}
 }
 
+@article{bjorklund2014,
+  author  = {Andreas Bj\"{o}rklund},
+  title   = {Determinant Sums for Undirected {H}amiltonicity},
+  journal = {SIAM Journal on Computing},
+  volume  = {43},
+  number  = {1},
+  pages   = {280--299},
+  year    = {2014},
+  doi     = {10.1137/110839229}
+}
+
 @article{bjorklund2009,
   author  = {Andreas Bj\"{o}rklund and Thore Husfeldt and Mikko Koivisto},
   title   = {Set Partitioning via Inclusion-Exclusion},

docs/src/reductions/problem_schemas.json

@@ -142,6 +142,17 @@
       }
     ]
   },
+  {
+    "name": "HamiltonianCircuit",
+    "description": "Does the graph contain a Hamiltonian circuit?",
+    "fields": [
+      {
+        "name": "graph",
+        "type_name": "G",
+        "description": "The undirected graph G=(V,E)"
+      }
+    ]
+  },
   {
     "name": "HamiltonianPath",
     "description": "Find a Hamiltonian path in a graph",