diff --git a/A4/CSC_373_EC3.ipynb b/A4/CSC_373_EC3.ipynb index 4bbc399..7896195 100644 --- a/A4/CSC_373_EC3.ipynb +++ b/A4/CSC_373_EC3.ipynb @@ -2,7 +2,7 @@ "cells": [ { "cell_type": "code", - "execution_count": 2, + "execution_count": 35, "id": "0744d366-ab31-4ed2-8845-e1d26ca90912", "metadata": { "cellView": "form", @@ -143,7 +143,7 @@ }, { "cell_type": "code", - "execution_count": 3, + "execution_count": 36, "id": "674a699c-22b5-42cf-859a-8ac38f68389e", "metadata": { "colab": { @@ -230,7 +230,7 @@ "\n" ], "text/plain": [ - "" + "" ] }, "metadata": {}, @@ -263,7 +263,7 @@ }, { "cell_type": "code", - "execution_count": 4, + "execution_count": 37, "id": "66347cb5-2752-4369-aa47-96f29c483688", "metadata": { "colab": { @@ -302,7 +302,7 @@ }, { "cell_type": "code", - "execution_count": 5, + "execution_count": 38, "id": "12758b83-6e78-4335-9fb3-e7b0f08485f2", "metadata": { "colab": { @@ -432,7 +432,7 @@ "\n" ], "text/plain": [ - "" + "" ] }, "metadata": {}, @@ -475,7 +475,7 @@ }, { "cell_type": "code", - "execution_count": 6, + "execution_count": 39, "id": "18438025-ce74-4abe-9ae7-c0b7cfefad8b", "metadata": { "colab": { @@ -486,35 +486,6 @@ "outputId": "21ab529a-163e-42ed-b400-25f3accad5a0" }, "outputs": [ - { - "data": { - "text/html": [ - " \n", - " " - ] - }, - "metadata": {}, - "output_type": "display_data" - }, { "data": { "application/vnd.plotly.v1+json": { @@ -3155,9 +3126,9 @@ } }, "text/html": [ - "
" + ] + }, + "metadata": {}, + "output_type": "display_data" } ], "source": [ @@ -29721,327 +54925,8 @@ "for cs_index, cs_student in enumerate(cs_students): # order is deterministic (list)\n", " for business_index, business_student in enumerate(business_students): # order is deterministic (list)\n", " pairing_index = cs_index * len(cs_students) + business_index\n", - " for availability in cs_student.availability: # O(1) since availability must be less than or equal to constant number of time slots\n", - " if availability in business_student.availability:\n", - " c[pairing_index] = c[pairing_index] + 1\n", + " c[pairing_index] = len(business_student.availability & cs_student.availability)\n", "\n", - "print(c)" - ] - }, - { - "cell_type": "code", - "execution_count": 145, - "id": "2fe82c44", - "metadata": {}, - "outputs": [ - { - "name": "stdout", - "output_type": "stream", - "text": [ - "[[1. 1. 1. 1. 1. 1. 1. 1. 1. 1. 1. 1. 1. 1. 1. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n", - " 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n", - 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"text": [ - "[1. 1. 1. 1. 1. 1. 1. 1. 1. 1. 1. 1. 1. 1. 1. 1. 1. 1. 1. 1. 1. 1. 1. 1.\n", - " 1. 1. 1. 1. 1. 1.]\n" - ] - } - ], - "source": [ "\n", "b = np.ones(cs_students_count + business_students_count)\n", - "print(b)\n" - ] - }, - { - "cell_type": "code", - "execution_count": 159, - "id": "55063d21", - "metadata": {}, - "outputs": [ - { - "name": "stdout", - "output_type": "stream", - "text": [ - " message: Optimization terminated successfully. (HiGHS Status 7: Optimal)\n", - " success: True\n", - " status: 0\n", - " fun: -20.0\n", - " x: [ 0.000e+00 0.000e+00 ... 0.000e+00 0.000e+00]\n", - " nit: 28\n", - " lower: residual: [ 0.000e+00 0.000e+00 ... 0.000e+00\n", - " 0.000e+00]\n", - " marginals: [ 2.000e+00 1.000e+00 ... 1.000e+00\n", - " 2.000e+00]\n", - " upper: residual: [ inf inf ... inf\n", - " inf]\n", - " marginals: [ 0.000e+00 0.000e+00 ... 0.000e+00\n", - " 0.000e+00]\n", - " eqlin: residual: []\n", - " marginals: []\n", - " ineqlin: residual: [ 0.000e+00 0.000e+00 ... 0.000e+00\n", - " 0.000e+00]\n", - " marginals: [-1.000e+00 -0.000e+00 ... -1.000e+00\n", - " -1.000e+00]\n", - " mip_node_count: 0\n", - " mip_dual_bound: 0.0\n", - " mip_gap: 0.0\n", - "[ 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. -0. 1. 0. 0. 0. 0.\n", - " 0. 0. 1. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n", - " 0. 1. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n", - " 1. 0. 0. 0. 0. 0. 0. 0. 1. 0. 0. 0. 0. 0. -0. 0. 0. 0.\n", - " 0. 0. 0. 0. 0. 0. 0. 1. 0. 0. 0. 0. -0. 0. 0. 0. 0. 0.\n", - " 0. 0. 0. 1. 0. 0. 0. 0. 0. 0. 0. 0. -0. 0. 0. 0. 0. 0.\n", - " 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 1. 0. 0. 0. 0. 0. 0.\n", - " 1. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. -0. 0. 0. 0. 0.\n", - " 0. 0. 0. 1. 0. 0. 1. 0. 0. 0. 0. 0. 0. -0. 0. 0. 0. 0.\n", - " 0. 0. 0. 0. 0. 0. 0. 0. 0. -0. 0. 0. 0. 1. 0. 0. 0. 0.\n", - " 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n", - " 0. 0. 0. 0. 0. 1. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n", - " 0. 0. 0. 0. -0. 1. 0. 0. 0.]\n", - "[('28', '0'), ('20', '1'), ('22', '2'), ('24', '3'), ('17', '4'), ('19', '5'), ('18', '6'), ('29', '7'), ('21', '8'), ('27', '9'), ('15', '10'), ('25', '11'), ('23', '13'), ('26', '14')]\n", - "20.0\n" - ] - } - ], - "source": [ + "print(b)\n", + "\n", "# Optimize\n", "opt_results = linprog(c=-c, A_ub=A, b_ub=b, bounds=(0,None))\n", "print(opt_results)\n", @@ -30140,14 +54953,11 @@ " index = cs_student_index * business_students_count + business_index\n", " if opt_results.x[index] == 1:\n", " t2_matching.append((business_student.student_id, cs_student.student_id))\n", - "print(t2_matching)\n", - "\n", - "# Visualize\n", - "plot_bipartite(t1_graph, left=business_student_ids, matching=t2_matching, weights=True)\n", + "print(f\"Student ID Matches (Optimizing for Overlap):\\n\\t{\"\\n\\t\".join([str(pairing) for pairing in t2_matching])}\")\n", "\n", "# Calculate weight\n", "t2_total_weight = np.sum(c * opt_results.x)\n", - "print(t2_total_weight)\n", + "print(f\"Total weight of LP method: {t2_total_weight}\")\n", "\n", "# Calculate max-flow weight\n", "t1_total_weight = 0\n", @@ -30156,11 +54966,23143 @@ " t1_matching_dict[business_student_id] = cs_student_id\n", "for cs_index, cs_student in enumerate(cs_students): # order is deterministic (list)\n", " for business_index, business_student in enumerate(business_students): # order is deterministic (list)\n", - " pairing_index = cs_index * len(cs_students) + business_index\n", - " if business_student.student_id in t1_matching_dict and cs_student_id == t1_matching_dict[business_student.student_id]:\n", - " t1_total_weight = c[pairing_index] + t1_total_weight\n", - "print(t1_total_weight)" + " pairing_index = cs_index * cs_students_count + business_index\n", + " if business_student.student_id in t1_matching_dict and t1_matching_dict[business_student.student_id] == cs_student.student_id:\n", + " t1_total_weight = t1_total_weight + len(business_student.availability & cs_student.availability)\n", + "print(f\"Total weight of Max-flow: {t1_total_weight}\")\n", + "\n", + "# Visualize\n", + "t2_graph = nx.DiGraph()\n", + "\n", + "business_student_ids = []\n", + "\n", + "for business_student in business_students:\n", + " t2_graph.add_node(business_student.student_id)\n", + " business_student_ids.append(business_student.student_id)\n", + "\n", + "for cs_index, cs_student in enumerate(cs_students): # order is deterministic (list)\n", + " t2_graph.add_node(cs_student.student_id)\n", + " for business_index, business_student in enumerate(business_students): # order is deterministic (list)\n", + " if len(business_student.availability & cs_student.availability) > 0:\n", + " pairing_index = cs_index * cs_students_count + business_index\n", + " t2_graph.add_edge(business_student.student_id, cs_student.student_id, weight=c[pairing_index])\n", + "\n", + "plot_bipartite(t2_graph, left=business_student_ids, matching=t2_matching, weights=True)" ] + }, + { + "cell_type": "code", + "execution_count": 49, + "id": "2fe82c44", + "metadata": {}, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "[[1. 1. 1. ... 0. 0. 0.]\n", + " [0. 0. 0. ... 0. 0. 0.]\n", + " [0. 0. 0. ... 0. 0. 0.]\n", + " ...\n", + " [0. 0. 0. ... 1. 0. 0.]\n", + " [0. 0. 0. ... 0. 1. 0.]\n", + " [0. 0. 0. ... 0. 0. 1.]]\n" + ] + } + ], + "source": [] + }, + { + "cell_type": "code", + "execution_count": 50, + "id": "44be23f4", + "metadata": {}, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "[1. 1. 1. 1. 1. 1. 1. 1. 1. 1. 1. 1. 1. 1. 1. 1. 1. 1. 1. 1. 1. 1. 1. 1.\n", + " 1. 1. 1. 1. 1. 1.]\n" + ] + } + ], + "source": [] + }, + { + "cell_type": "code", + "execution_count": 51, + "id": "55063d21", + "metadata": {}, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + " message: Optimization terminated successfully. (HiGHS Status 7: Optimal)\n", + " success: True\n", + " status: 0\n", + " fun: -21.0\n", + " x: [ 0.000e+00 0.000e+00 ... 1.000e+00 0.000e+00]\n", + " nit: 38\n", + " lower: residual: [ 0.000e+00 0.000e+00 ... 1.000e+00\n", + " 0.000e+00]\n", + " marginals: [ 0.000e+00 2.000e+00 ... 0.000e+00\n", + " 3.000e+00]\n", + " upper: residual: [ inf inf ... inf\n", + " inf]\n", + " marginals: [ 0.000e+00 0.000e+00 ... 0.000e+00\n", + " 0.000e+00]\n", + " eqlin: residual: []\n", + " marginals: []\n", + " ineqlin: residual: [ 0.000e+00 0.000e+00 ... 0.000e+00\n", + " 0.000e+00]\n", + " marginals: [-1.000e+00 -1.000e+00 ... -0.000e+00\n", + " -2.000e+00]\n", + " mip_node_count: 0\n", + " mip_dual_bound: 0.0\n", + " mip_gap: 0.0\n", + "[ 0. 0. 0. 0. 0. 0. 0. 0. 0. 1. 0. 0. 0. 0. 0. 0. 0. 0.\n", + " 0. 0. 1. 0. -0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 1. 0. 0.\n", + " 0. 0. 0. 0. -0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 1. -0.\n", + " 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. -0. 0. 0. 0. 0. -0. 0. 0.\n", + " 0. 0. 1. 0. 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" + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], + "source": [] } ], "metadata": { diff --git a/A4/main.pdf b/A4/main.pdf index e9603e2..7ca8970 100644 Binary files a/A4/main.pdf and b/A4/main.pdf differ diff --git a/A4/main.tex b/A4/main.tex index bd1f21b..669649a 100644 --- a/A4/main.tex +++ b/A4/main.tex @@ -70,6 +70,34 @@ Here is the {\it Set-Cover} problem. You are given a set $E = \{ e_1, ..., e_n \ \qpart{c}{5} Let ${\sf OPT}$ be value of the optimal solution of the set-cover. Prove that the rounding algorithm in (b) gives an $f$-approximation. +\solution + +\solpart{b} We wish to show that the collection of subsets $S_j$ where $j \in I$ chosen by the rounding algorithm is a set cover. In order to do this, we must show that each element $e_i \in E$ is included by at least one subset in $S_j$. \\ + +Let $f_i$ be the number of subsets in which $e_i$ appears. + +To begin, note that for all $x^*_i$, at least one $x^*_i$ must be greater than $\frac{1}{f_i}$ due to constraints on the linear program (\textbf{claim 1}): + +\begin{align} + \forall i \in n, \sum_{j:e_i \in S_j} x_j &\geq 1 &(\text{LP constraints}) \\ + \forall i \in n, \sum_{j:e_i \in S_j} x^*_j &\geq 1 &(\text{Optimal solution substitution}) +\end{align} + +Then (\textbf{claim 2}), + +\begin{align} + f_i &\leq f \\ + \frac{1}{f_i} &\geq \frac{1}{f} +\end{align} + +since, for each element, there is at least one $x^*_i \geq \frac{1}{f_i}$ (claim 1), and $\frac{1}{f_i} \geq \frac{1}{f}$ (claim 2), then, (\textbf{claim 3}) $x^*_i \geq \frac{1}{f}$ ($x^*_i \geq \frac{1}{f_i} \geq \frac{1}{f}$). + +Claim 3 thus allows us to state that for every element, there exists at least one $x^*_i \geq \frac{1}{f}$. Since the rounding algorithm selects all subsets $S_i$ such that $x^*_i \geq \frac{1}{f}$ (as described), then, we can state that each element will appear at least once in at least one subset selected. + +Since our selection selection criteria is such that \(x^*_j \geq \frac{1}{f}\), Then, according to our first constraint in the relaxed LP, for any element $e_i$, they must exist we have that at least one subset $S_j$ exists such that $x_j* \geq \frac{1}{f}$ We know that it must be at least $1$. This means that the sum of all $x_j*$ must evaluate to 1 as well, meaning that the collection of subsets, which indeed span all elements in $E$ form a set cover. + +\qed + \newpage \question{15}{Traveling Salesman} diff --git a/A4/task1_code.png b/A4/task1_code.png new file mode 100644 index 0000000..c222df7 Binary files /dev/null and b/A4/task1_code.png differ diff --git a/A4/task1_plot.png b/A4/task1_plot.png new file mode 100644 index 0000000..10ac9f2 Binary files /dev/null and b/A4/task1_plot.png differ diff --git a/A4/task2.png b/A4/task2.png new file mode 100644 index 0000000..85c3f6f Binary files /dev/null and b/A4/task2.png differ diff --git a/A4/task2_plot.png b/A4/task2_plot.png new file mode 100644 index 0000000..42a3b4a Binary files /dev/null and b/A4/task2_plot.png differ