Add transversal example (#790)

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---------

Co-authored-by: ThomSerg <[email protected]>

Author: nandorsieben

examples/transversals.ipynb

@@ -0,0 +1,150 @@
+{
+ "cells": [
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "## Minimal transversals\n",
+    "\n",
+    "Let $\\mathcal{S}$ be a family of sets. A set $T$ is a *transversal* (or *hitting set*) of $\\mathcal{S}$ if $T$ intersects every $S\\in\\mathcal{S}$. The family of minimal transversals of $\\mathcal{S}$ is denoted by  $\\text{Tr}(\\mathcal{S})$.\n",
+    "\n",
+    "The minimality of $T$ can be characterized by the following condition:\n",
+    "If $x\\in T$, then $S\\cap T= \\{x\\}$ for some $S\\in\\mathcal{S}$. "
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "\"\"\"\n",
+    "Author: Nandor Sieben (https://github.com/nandorsieben)\n",
+    "\"\"\"\n",
+    "\n",
+    "import cpmpy as cp\n",
+    "from typing import List"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 7,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "def Tr(sets:List):\n",
+    "    \"\"\"\n",
+    "    Given a family of sets, returns all minimal transversals.\n",
+    "    \"\"\"\n",
+    "    X = {s for S in sets for s in S} # all elements across all sets, deduplicated\n",
+    "    \n",
+    "    # Decision Variables\n",
+    "    t = {x:cp.boolvar() for x in X} # whether to inlude an element into the transversal\n",
+    "\n",
+    "    # Model\n",
+    "    model = cp.Model()\n",
+    "\n",
+    "    # Constraints\n",
+    "    # 1) transversal\n",
+    "    for S in sets:\n",
+    "        model += cp.any([t[s] for s in S]) # must hit each set by intersecting with at least one element\n",
+    "    # 2) minimal transversal\n",
+    "    #       for each element x, if it is included in our transversal T, \n",
+    "    #       there must exist a set S which intersects with T in only that single element x\n",
+    "    for x in X: \n",
+    "        model += t[x].implies(\n",
+    "            cp.any(\n",
+    "              [cp.all([~t[s] for s in S if s!=x]) \n",
+    "                for S in sets if x in S]\n",
+    "              )\n",
+    "            )\n",
+    "    \n",
+    "    # Solve and collect results\n",
+    "    result = []\n",
+    "    model.solveAll(display=lambda: result.append({a for a in t if t[a].value()}))\n",
+    "    return result "
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 8,
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "[{2}, {1, 3}]"
+      ]
+     },
+     "execution_count": 8,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "Tr([{1,2},{2,3}])"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 9,
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "[{2, 3}, {1, 2}]"
+      ]
+     },
+     "execution_count": 9,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "Tr([{2},{1,3}])"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 10,
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "[{1, 4, 5}, {1, 3, 4}, {1, 5, 8}, {1, 5, 7}, {2, 7}, {2, 8}, {2, 4}]"
+      ]
+     },
+     "execution_count": 10,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "Tr([{1,2},{2,4,5},{2,3,5},{4,7,8}])"
+   ]
+  }
+ ],
+ "metadata": {
+  "kernelspec": {
+   "display_name": ".venv",
+   "language": "python",
+   "name": "python3"
+  },
+  "language_info": {
+   "codemirror_mode": {
+    "name": "ipython",
+    "version": 3
+   },
+   "file_extension": ".py",
+   "mimetype": "text/x-python",
+   "name": "python",
+   "nbconvert_exporter": "python",
+   "pygments_lexer": "ipython3",
+   "version": "3.12.8"
+  }
+ },
+ "nbformat": 4,
+ "nbformat_minor": 4
+}