From e8e22876d2d9ed2dd29dc677a11c07763a10980b Mon Sep 17 00:00:00 2001 From: CGH0S7 <776459475@qq.com> Date: Mon, 19 Jan 2026 01:17:31 +0800 Subject: [PATCH] backtracking lab finished --- backtracking/cost_approx_level.csv | 434 ++ backtracking/cost_approx_n.csv | 6 + backtracking/cost_ratio_level.png | Bin 0 -> 99300 bytes backtracking/cost_ratio_n.png | Bin 0 -> 117149 bytes backtracking/knapsack_bnb | Bin 0 -> 38552 bytes backtracking/labtemplate.typ | 166 + backtracking/main.pdf | 10273 +++++++++++++++++++++++++++ backtracking/main.typ | 257 + backtracking/mc_estimation.csv | 6 + backtracking/mc_estimation.png | Bin 0 -> 126722 bytes backtracking/mk_nodes.png | Bin 0 -> 124235 bytes backtracking/mk_time.png | Bin 0 -> 122643 bytes backtracking/multiple_knapsack.csv | 9 + backtracking/new_cost_analysis.csv | 11 + backtracking/new_cost_nodes.png | Bin 0 -> 127070 bytes backtracking/new_cost_time.png | Bin 0 -> 122431 bytes backtracking/plotter.py | 103 + backtracking/src/main.cpp | 491 ++ backtracking/task.txt | 13 + 19 files changed, 11769 insertions(+) create mode 100644 backtracking/cost_approx_level.csv create mode 100644 backtracking/cost_approx_n.csv create mode 100644 backtracking/cost_ratio_level.png create mode 100644 backtracking/cost_ratio_n.png create mode 100755 backtracking/knapsack_bnb create mode 100644 backtracking/labtemplate.typ create mode 100644 backtracking/main.pdf create mode 100644 backtracking/main.typ create mode 100644 backtracking/mc_estimation.csv create mode 100644 backtracking/mc_estimation.png create mode 100644 backtracking/mk_nodes.png create mode 100644 backtracking/mk_time.png create mode 100644 backtracking/multiple_knapsack.csv create mode 100644 backtracking/new_cost_analysis.csv create mode 100644 backtracking/new_cost_nodes.png create mode 100644 backtracking/new_cost_time.png create mode 100644 backtracking/plotter.py create mode 100644 backtracking/src/main.cpp create mode 100644 backtracking/task.txt diff --git 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+sns.set_theme(style="whitegrid") +plt.rcParams['font.sans-serif'] = ['SimHei'] # Use SimHei for Chinese characters if needed, or just English +plt.rcParams['axes.unicode_minus'] = False + +output_dir = "backtracking" +if not os.path.exists(output_dir): + os.makedirs(output_dir) + +def save_plot(filename): + plt.tight_layout() + plt.savefig(os.path.join(output_dir, filename), dpi=300) + plt.close() + +# 1. Monte Carlo Estimation +try: + df_mc = pd.read_csv(os.path.join(output_dir, "mc_estimation.csv")) + plt.figure(figsize=(8, 5)) + sns.lineplot(data=df_mc, x="n", y="estimated_nodes", marker="o") + plt.title("Monte Carlo Estimation of Search Tree Size") + plt.xlabel("Number of Item Types (n)") + plt.ylabel("Estimated Nodes (Log Scale)") + plt.yscale("log") + save_plot("mc_estimation.png") +except Exception as e: + print(f"Error plotting MC: {e}") + +# 2. Cost Approximation Analysis +try: + df_level = pd.read_csv(os.path.join(output_dir, "cost_approx_level.csv")) + # Filter for a specific n, e.g., n=15 or 20 + target_n = 20 + df_level_n = df_level[df_level["n"] == target_n] + + if not df_level_n.empty: + plt.figure(figsize=(10, 6)) + sns.boxplot(data=df_level_n, x="level", y="ratio") + plt.title(f"Cost Function Approximation Ratio vs Level (n={target_n})") + plt.xlabel("Search Tree Level") + plt.ylabel("Ratio (Bound / True Value)") + plt.axhline(1.0, color='r', linestyle='--') + save_plot("cost_ratio_level.png") + + df_n = pd.read_csv(os.path.join(output_dir, "cost_approx_n.csv")) + plt.figure(figsize=(8, 5)) + sns.lineplot(data=df_n, x="n", y="avg_ratio", marker="o") + plt.title("Average Cost Approximation Ratio vs Input Size") + plt.xlabel("Number of Item Types (n)") + plt.ylabel("Average Ratio") + save_plot("cost_ratio_n.png") +except Exception as e: + print(f"Error plotting Cost Analysis: {e}") + +# 3. New Cost Function Comparison +try: + df_new = pd.read_csv(os.path.join(output_dir, "new_cost_analysis.csv")) + + # Nodes + plt.figure(figsize=(8, 5)) + sns.lineplot(data=df_new, x="n", y="nodes", hue="method", marker="o") + plt.title("Nodes Visited: Fractional vs Simple Bound") + plt.xlabel("n") + plt.ylabel("Nodes Visited (Log Scale)") + plt.yscale("log") + save_plot("new_cost_nodes.png") + + # Time + plt.figure(figsize=(8, 5)) + sns.lineplot(data=df_new, x="n", y="time_us", hue="method", marker="o") + plt.title("Execution Time: Fractional vs Simple Bound") + plt.xlabel("n") + plt.ylabel("Time (microseconds)") + save_plot("new_cost_time.png") +except Exception as e: + print(f"Error plotting New Cost Analysis: {e}") + +# 4. Multiple Knapsack Comparison +try: + df_mk = pd.read_csv(os.path.join(output_dir, "multiple_knapsack.csv")) + + # Nodes + plt.figure(figsize=(8, 5)) + sns.lineplot(data=df_mk, x="n", y="nodes", hue="method", marker="o") + plt.title("Multiple Knapsack: Nodes Visited Comparison") + plt.xlabel("n") + plt.ylabel("Nodes Visited") + plt.yscale("log") + save_plot("mk_nodes.png") + + # Time + plt.figure(figsize=(8, 5)) + sns.lineplot(data=df_mk, x="n", y="time_us", hue="method", marker="o") + plt.title("Multiple Knapsack: Execution Time Comparison") + plt.xlabel("n") + plt.ylabel("Time (microseconds)") + save_plot("mk_time.png") +except Exception as e: + print(f"Error plotting Multiple Knapsack: {e}") diff --git a/backtracking/src/main.cpp b/backtracking/src/main.cpp new file mode 100644 index 0000000..e461d12 --- /dev/null +++ b/backtracking/src/main.cpp @@ -0,0 +1,491 @@ +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include + +using namespace std; + +struct Item { + int id; + int weight; + int value; + double density; + int limit; // For multiple knapsack, -1 for complete + + Item(int id, int w, int v, int l = -1) : id(id), weight(w), value(v), limit(l) { + density = (double)v / w; + } +}; + +bool compareItems(const Item& a, const Item& b) { + return a.density > b.density; +} + +// Global random engine +mt19937 rng(42); + +vector generate_items(int n, int max_w, int max_v, bool multiple = false, int max_k = 5) { + vector items; + uniform_int_distribution w_dist(1, max_w); + uniform_int_distribution v_dist(1, max_v); + uniform_int_distribution k_dist(1, max_k); + + for (int i = 0; i < n; ++i) { + int w = w_dist(rng); + int v = v_dist(rng); + int l = multiple ? k_dist(rng) : -1; + items.emplace_back(i, w, v, l); + } + // Sort by density descending for BnB and DP consistency + sort(items.begin(), items.end(), compareItems); + return items; +} + +// --- DP for True Value Calculation (Complete Knapsack) --- +// dp[i][w] = max value using items i...n-1 with capacity w +vector> compute_suffix_dp_complete(const vector& items, int capacity) { + int n = items.size(); + vector> dp(n + 1, vector(capacity + 1, 0)); + + for (int i = n - 1; i >= 0; --i) { + for (int w = 0; w <= capacity; ++w) { + // Option 1: Don't take item i + int res = dp[i + 1][w]; + // Option 2: Take at least 1 item i (if possible) -> then we are still at state i with w - weight + if (w >= items[i].weight) { + res = max(res, dp[i][w - items[i].weight] + items[i].value); + } + dp[i][w] = res; + } + } + return dp; +} + +// --- Cost Functions --- + +// 1. Fractional Bound (Standard) +double bound_fractional(int level, int current_val, int rem_cap, const vector& items) { + double bound = current_val; + int w = rem_cap; + for (int i = level; i < items.size(); ++i) { + if (w >= items[i].weight) { + // For complete knapsack, in fractional relaxation, we can take as many as fit... + // Actually, complete knapsack fractional relaxation: + // Just take the item with MAX density until full. + // Since items are sorted, items[level] is the best available. + // So we just fill the rest with items[level]. + bound += (double)w * items[i].density; + return bound; + } else { + // Should not happen if we just take max density, + // but if we handle Multiple Knapsack, it's different. + // For Complete Knapsack, sorted by density, just fill with the current best. + bound += (double)w * items[i].density; + return bound; + } + } + return bound; +} + +// 2. Simple Bound (Global Max Density) - A looser bound +double bound_simple(int level, int current_val, int rem_cap, double max_global_density) { + return current_val + rem_cap * max_global_density; +} + +// --- Branch and Bound Node --- +struct Node { + int level; // Index of item being considered + int current_val; + int current_weight; + double bound; + + // For analysis + double true_opt; + + bool operator<(const Node& other) const { + return bound < other.bound; // Priority Queue is Max Heap by default? No, default is max. We want Max Bound first? Yes, exploring promising nodes. + } +}; + +// --- BnB Solver for Complete Knapsack --- +struct BnBStats { + int nodes_visited; + long long time_us; + int max_val; + // Analysis data: Level -> List of (Bound, TrueVal) + map>> analysis_data; +}; + +BnBStats solve_bnb_complete(const vector& items, int capacity, const vector>& dp_table, bool collect_analysis, int bound_mode = 0) { + auto start_time = chrono::high_resolution_clock::now(); + BnBStats stats; + stats.nodes_visited = 0; + stats.max_val = 0; + + double max_global_density = 0; + if (!items.empty()) max_global_density = items[0].density; // Since sorted + + priority_queue pq; + + // Root node + double init_bound; + if (bound_mode == 1) init_bound = bound_simple(0, 0, capacity, max_global_density); + else init_bound = bound_fractional(0, 0, capacity, items); + + pq.push({0, 0, 0, init_bound, (double)dp_table[0][capacity]}); + + while (!pq.empty()) { + Node u = pq.top(); + pq.pop(); + stats.nodes_visited++; + + if (u.bound <= stats.max_val) continue; // Prune + + if (collect_analysis) { + stats.analysis_data[u.level].push_back({u.bound, u.true_opt}); + } + + if (u.level == items.size()) { + if (u.current_val > stats.max_val) { + stats.max_val = u.current_val; + } + continue; + } + + // Branching for Complete Knapsack: + // Commonly we branch on "count of item[level]". + // But infinite branching is hard for heap. + // Alternative: Binary branching. + // Left: Take 1 unit of item[level] (if fits). State: (level, w+wi, v+vi) -> Re-consider same item level. + // Right: Don't take item[level] anymore. State: (level+1, w, v) + + // Let's use Binary Branching: Take 1 (Stay at level) vs Don't Take (Go to level + 1) + + // Child 1: Take item[level] + if (u.current_weight + items[u.level].weight <= capacity) { + int new_w = u.current_weight + items[u.level].weight; + int new_v = u.current_val + items[u.level].value; + int rem = capacity - new_w; + + double b; + if (bound_mode == 1) b = bound_simple(u.level, new_v, rem, max_global_density); + else b = bound_fractional(u.level, new_v, rem, items); + + // True val for child 1 (Still at u.level) + double true_v = collect_analysis ? (new_v + dp_table[u.level][rem]) : 0; + + if (b > stats.max_val) { + pq.push({u.level, new_v, new_w, b, true_v}); + } + } + + // Child 2: Don't take item[level] (Move to next item) + { + int rem = capacity - u.current_weight; + double b; + // Bound calculation starts from level + 1 + if (bound_mode == 1) b = bound_simple(u.level + 1, u.current_val, rem, max_global_density); + else b = bound_fractional(u.level + 1, u.current_val, rem, items); + + // True val for child 2 (at u.level + 1) + double true_v = collect_analysis ? (u.current_val + dp_table[u.level + 1][rem]) : 0; + + if (b > stats.max_val) { + pq.push({u.level + 1, u.current_val, u.current_weight, b, true_v}); + } + } + } + + auto end_time = chrono::high_resolution_clock::now(); + stats.time_us = chrono::duration_cast(end_time - start_time).count(); + return stats; +} + +// --- Monte Carlo Estimation --- +long long monte_carlo_estimate(const vector& items, int capacity, int samples = 1000) { + long long total_nodes = 0; + + for (int k = 0; k < samples; ++k) { + long long path_nodes = 1; // Root + int curr_w = 0; + int level = 0; + long long current_multiplier = 1; + + // Using the same binary branching structure: Take (stay) or Don't Take (next) + while (level < items.size()) { + vector moves; // 0 = don't take, 1 = take + + // Can always "Don't take" -> move to level + 1 + moves.push_back(0); + + // Can "Take" if fits + if (curr_w + items[level].weight <= capacity) { + moves.push_back(1); + } + + int branching_factor = moves.size(); + total_nodes += current_multiplier; // Add current level nodes count estimation + + if (branching_factor == 0) break; // Should not happen as 0 is always there + + // Pick random move + int move = moves[rng() % branching_factor]; + + current_multiplier *= branching_factor; + + if (move == 0) { + level++; // Move to next item + } else { + curr_w += items[level].weight; // Take item, stay at level + } + } + } + + return total_nodes / samples; +} + +// --- Multiple Knapsack BnB --- +// Branching: Iterate k from 0 to limit[i] +// Cost Functions: +// 1. Fractional (Relaxed integer & limit constraint) +// 2. Simple (Global Max) - Not very useful for Multiple, let's use "Fractional with limits" vs "Fractional without limits" (Relaxed Limit) +// Wait, "Fractional Knapsack" normally respects limits in standard definition. +// Let's compare: +// MK Bound 1: Relaxed Integrity (Continuous 0..limit). +// MK Bound 2: Relaxed Integrity AND Relaxed Limit (Infinite count). (Looser) + +double bound_mk_tight(int level, int current_val, int rem_cap, const vector& items) { + double bound = current_val; + int w = rem_cap; + for (int i = level; i < items.size(); ++i) { + int count = min(items[i].limit, w / items[i].weight); // Integer max count we could take? + // No, fractional allows taking fraction of item. + // Standard Fractional Knapsack with Limits: + // Take as much as possible up to limit. + if (items[i].weight == 0) continue; + + int can_take_weight = items[i].limit * items[i].weight; + + if (w >= can_take_weight) { + w -= can_take_weight; + bound += items[i].value * items[i].limit; + } else { + bound += (double)w * items[i].density; + return bound; // Filled + } + } + return bound; +} + +double bound_mk_loose(int level, int current_val, int rem_cap, const vector& items) { + // Like Complete Knapsack - ignore limits + double bound = current_val; + int w = rem_cap; + for (int i = level; i < items.size(); ++i) { + if (w >= items[i].weight) { + bound += (double)w * items[i].density; + return bound; + } + } + return bound; +} + +struct NodeMK { + int level; + int current_val; + int current_weight; + double bound; + + bool operator<(const NodeMK& other) const { + return bound < other.bound; + } +}; + +BnBStats solve_bnb_mk(const vector& items, int capacity, int bound_mode) { + auto start_time = chrono::high_resolution_clock::now(); + BnBStats stats; + stats.nodes_visited = 0; + stats.max_val = 0; + + priority_queue pq; + double init_bound = (bound_mode == 0) ? bound_mk_tight(0, 0, capacity, items) : bound_mk_loose(0, 0, capacity, items); + + pq.push({0, 0, 0, init_bound}); + + while (!pq.empty()) { + NodeMK u = pq.top(); + pq.pop(); + stats.nodes_visited++; + + if (u.bound <= stats.max_val) continue; + + if (u.level == items.size()) { + if (u.current_val > stats.max_val) { + stats.max_val = u.current_val; + } + continue; + } + + // Branching: 0 to limit + // To be safe and simple, let's branch on specific counts? + // Or Binary: Take 1 (decrement limit) vs Don't Take (next level). + // Let's use Binary with state modification (decrement limit). + // Wait, Node needs to store remaining limits? That's too heavy. + // Standard MK branching: Loop k = 0 to Limit? That's a high branching factor. + // Binary branching is better: + // Option 1: Take 1 of Item[level]. (Requires tracking 'used' count for current level). + // Option 2: Don't take any more of Item[level]. (Move to level+1). + + // We need to store 'count_taken_current_level' in Node? + // Or just simplify: "level" implies items[level]. + // But we need to know how many we already took. + // Let's assume standard MK backtracking logic where we just loop k. + // But for Priority Queue, creating k children is fine. + + int k_max = min(items[u.level].limit, (capacity - u.current_weight) / items[u.level].weight); + + // Generate children for k = 0 to k_max + // Heuristic: Try largest k first? Priority Queue handles order. + + for (int k = 0; k <= k_max; ++k) { + int new_w = u.current_weight + k * items[u.level].weight; + int new_v = u.current_val + k * items[u.level].value; + int rem = capacity - new_w; + + double b; + if (bound_mode == 0) b = bound_mk_tight(u.level + 1, new_v, rem, items); + else b = bound_mk_loose(u.level + 1, new_v, rem, items); + + if (b > stats.max_val) { + pq.push({u.level + 1, new_v, new_w, b}); + } + } + } + + auto end_time = chrono::high_resolution_clock::now(); + stats.time_us = chrono::duration_cast(end_time - start_time).count(); + return stats; +} + + +int main() { + int W = 100; + + // --- Experiment 1: Monte Carlo Estimation --- + ofstream mc_file("backtracking/mc_estimation.csv"); + mc_file << "n,estimated_nodes\n"; + cout << "Running Monte Carlo Estimation..." << endl; + for (int n = 5; n <= 25; n += 5) { + auto items = generate_items(n, 40, 100); + long long est = monte_carlo_estimate(items, W, 2000); + mc_file << n << "," << est << "\n"; + } + mc_file.close(); + + // --- Experiment 2 & 3: Cost Function Analysis (Fixed n & Varying n) --- + // We need: Level, Cost, TrueVal + // Also "Approximation Effect at different input sizes" -> Avg Error vs n + + ofstream cost_level_file("backtracking/cost_approx_level.csv"); + cost_level_file << "n,level,cost,true_val,ratio\n"; + + ofstream cost_n_file("backtracking/cost_approx_n.csv"); + cost_n_file << "n,avg_ratio\n"; + + cout << "Running Cost Function Analysis..." << endl; + for (int n = 5; n <= 25; n += 5) { + double total_ratio_sum = 0; + long long total_nodes_count = 0; + + // Run multiple samples + for (int s = 0; s < 5; ++s) { + auto items = generate_items(n, 40, 100); + auto dp = compute_suffix_dp_complete(items, W); + + // Collect analysis data using Standard Bound (Mode 0) + auto res = solve_bnb_complete(items, W, dp, true, 0); + + for (auto const& [lvl, data] : res.analysis_data) { + for (auto const& pair : data) { + double cost = pair.first; + double true_v = pair.second; + double ratio = (true_v > 0) ? (cost / true_v) : 1.0; + + if (s == 0) { // Only log details for one sample to avoid huge files + cost_level_file << n << "," << lvl << "," << cost << "," << true_v << "," << ratio << "\n"; + } + + total_ratio_sum += ratio; + total_nodes_count++; + } + } + } + if (total_nodes_count > 0) + cost_n_file << n << "," << (total_ratio_sum / total_nodes_count) << "\n"; + } + cost_level_file.close(); + cost_n_file.close(); + + // --- Experiment 4: New Cost Function Analysis --- + // Compare Mode 0 (Fractional - Tight) vs Mode 1 (Simple - Loose) + ofstream new_cost_file("backtracking/new_cost_analysis.csv"); + new_cost_file << "n,method,nodes,time_us\n"; + + cout << "Running New Cost Function Analysis..." << endl; + for (int n = 5; n <= 25; n += 5) { + long long nodes0 = 0, time0 = 0; + long long nodes1 = 0, time1 = 0; + int runs = 10; + + for (int r = 0; r < runs; ++r) { + auto items = generate_items(n, 40, 100); + auto dp = compute_suffix_dp_complete(items, W); // Just for empty passing + + auto res0 = solve_bnb_complete(items, W, dp, false, 0); // Fractional + nodes0 += res0.nodes_visited; + time0 += res0.time_us; + + auto res1 = solve_bnb_complete(items, W, dp, false, 1); // Simple + nodes1 += res1.nodes_visited; + time1 += res1.time_us; + } + + new_cost_file << n << ",Fractional," << nodes0/runs << "," << time0/runs << "\n"; + new_cost_file << n << ",Simple," << nodes1/runs << "," << time1/runs << "\n"; + } + new_cost_file.close(); + + // --- Experiment 5: Multiple Knapsack (Additional) --- + ofstream mk_file("backtracking/multiple_knapsack.csv"); + mk_file << "n,method,nodes,time_us\n"; + + cout << "Running Multiple Knapsack Analysis..." << endl; + for (int n = 5; n <= 20; n += 5) { // Smaller range as it might be slower + long long nodes0 = 0, time0 = 0; // Tight + long long nodes1 = 0, time1 = 0; // Loose + int runs = 10; + + for (int r = 0; r < runs; ++r) { + auto items = generate_items(n, 40, 100, true, 3); // Limit up to 3 + + auto res0 = solve_bnb_mk(items, W, 0); + nodes0 += res0.nodes_visited; + time0 += res0.time_us; + + auto res1 = solve_bnb_mk(items, W, 1); + nodes1 += res1.nodes_visited; + time1 += res1.time_us; + } + mk_file << n << ",TightBound," << nodes0/runs << "," << time0/runs << "\n"; + mk_file << n << ",LooseBound," << nodes1/runs << "," << time1/runs << "\n"; + } + mk_file.close(); + + return 0; +} diff --git a/backtracking/task.txt b/backtracking/task.txt new file mode 100644 index 0000000..e584c45 --- /dev/null +++ b/backtracking/task.txt @@ -0,0 +1,13 @@ +运用回溯与分支限界算法求解完全背包问题并进行分析。具体要求如下： + + 针对完全背包问题，实现回溯法和分支限界算法； + 以物品种类数n为输入规模，固定n，随机产生大量测试样本； + 用蒙特卡洛法对分支数量进行估计，根据统计结果给出应用蒙特卡洛估计整数背包问题分支数量的建议； + 在分支限界法计算代价函数时，使用其他算法（如动态规划法）计算出真实值，记录结点的代价函数与真实值，分析在同一输入规模下不同层代价函数的近似效果； + 改变n，分析在不同输入规模下同一层代价函数的近似效果； + 针对实验结果，设计新的代价函数，并分析其有效性（近似能力、剪枝效果、增加的开销等）。 + +附加：运用分支限界法求解多重背包问题并进行分析，具体要求如下： + + 每种物品的数量有限，第i种物品的数量上限为ki个； + 设计2种以上代价函数，实现分支限界法，并进行对比分析。

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R 527 0 R 529 0 R 531 0 R 533 0 R 535 0 R 537 0 R 539 0 R] +>> +endobj + +2 0 obj +<< + /Type /Outlines + /First 3 0 R + /Last 17 0 R + /Count 7 +>> +endobj + +3 0 obj +<< + /Parent 2 0 R + /Next 4 0 R + /Title + /Dest 477 0 R +>> +endobj + +4 0 obj +<< + /Parent 2 0 R + /Next 5 0 R + /Prev 3 0 R + /Title + /Dest 478 0 R +>> +endobj + +5 0 obj +<< + /Parent 2 0 R + /Next 6 0 R + /Prev 4 0 R + /Title + /Dest 479 0 R +>> +endobj + +6 0 obj +<< + /Parent 2 0 R + /Next 12 0 R + /Prev 5 0 R + /First 7 0 R + /Last 11 0 R + /Count -2 + /Title + /Dest 485 0 R +>> +endobj + +7 0 obj +<< + /Parent 6 0 R + /Next 11 0 R + /First 8 0 R + /Last 10 0 R + /Count -3 + /Title + /Dest 483 0 R +>> +endobj + +8 0 obj +<< + /Parent 7 0 R + /Next 9 0 R + /Title + /Dest 480 0 R +>> +endobj + +9 0 obj +<< + /Parent 7 0 R + /Next 10 0 R + /Prev 8 0 R + /Title + /Dest 481 0 R +>> +endobj + +10 0 obj +<< + /Parent 7 0 R + /Prev 9 0 R + /Title + /Dest 482 0 R +>> +endobj + +11 0 obj +<< + /Parent 6 0 R + /Prev 7 0 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6 371 0 R 7 367 0 R 8 363 0 R 9 359 0 R 10 354 0 R 11 349 0 R 12 345 0 R 13 341 0 R 14 337 0 R 15 332 0 R 16 328 0 R 17 320 0 R 18 316 0 R 19 312 0 R 20 308 0 R 21 304 0 R 22 300 0 R 23 296 0 R 24 22 0 R 25 23 0 R 26 24 0 R 27 25 0 R 28 26 0 R 29 27 0 R 30 28 0 R 31 29 0 R] + >> + /ParentTreeNextKey 32 +>> +endobj + +20 0 obj +[396 0 R 396 0 R 396 0 R 396 0 R 396 0 R 396 0 R 396 0 R] +endobj + +21 0 obj +[396 0 R 395 0 R 394 0 R 394 0 R 394 0 R 394 0 R 394 0 R 394 0 R 394 0 R 394 0 R 394 0 R 394 0 R 394 0 R 394 0 R 394 0 R 394 0 R 394 0 R 394 0 R 394 0 R 394 0 R 394 0 R 394 0 R 394 0 R 394 0 R 394 0 R 394 0 R 394 0 R 394 0 R 394 0 R 394 0 R 394 0 R 394 0 R 394 0 R 394 0 R 393 0 R 393 0 R 393 0 R 393 0 R 393 0 R 393 0 R 393 0 R 393 0 R 393 0 R 393 0 R 393 0 R 393 0 R 393 0 R 393 0 R 393 0 R 393 0 R 393 0 R 393 0 R 393 0 R 393 0 R 393 0 R 393 0 R 393 0 R 393 0 R 393 0 R 393 0 R 393 0 R 393 0 R 393 0 R 393 0 R 393 0 R 393 0 R 393 0 R 393 0 R 393 0 R 393 0 R 393 0 R 393 0 R 393 0 R 393 0 R 393 0 R 393 0 R 392 0 R 392 0 R 392 0 R 392 0 R 392 0 R] +endobj + +22 0 obj +[391 0 R 388 0 R 387 0 R 387 0 R 387 0 R 384 0 R 383 0 R 383 0 R 383 0 R 380 0 R 379 0 R 379 0 R 379 0 R 376 0 R 375 0 R 375 0 R 375 0 R 372 0 R 371 0 R 371 0 R 371 0 R 368 0 R 367 0 R 367 0 R 367 0 R 364 0 R 363 0 R 363 0 R 363 0 R 360 0 R 359 0 R 359 0 R 359 0 R 355 0 R 354 0 R 354 0 R 354 0 R 350 0 R 349 0 R 349 0 R 349 0 R 346 0 R 345 0 R 345 0 R 345 0 R 342 0 R 341 0 R 341 0 R 341 0 R 338 0 R 337 0 R 337 0 R 337 0 R 333 0 R 332 0 R 332 0 R 332 0 R 329 0 R 328 0 R 328 0 R 328 0 R 324 0 R 321 0 R 320 0 R 320 0 R 320 0 R 317 0 R 316 0 R 316 0 R 316 0 R 316 0 R 313 0 R 312 0 R 312 0 R 312 0 R 312 0 R 312 0 R 312 0 R 309 0 R 308 0 R 308 0 R 308 0 R 305 0 R 304 0 R 304 0 R 304 0 R 301 0 R 300 0 R 300 0 R 300 0 R 297 0 R 296 0 R 296 0 R 296 0 R] +endobj + +23 0 obj +[292 0 R 292 0 R 289 0 R 289 0 R 289 0 R 289 0 R 289 0 R 289 0 R 289 0 R 289 0 R 289 0 R 289 0 R 289 0 R 289 0 R 289 0 R 289 0 R 289 0 R 288 0 R 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235 0 R 235 0 R 235 0 R 235 0 R 235 0 R 235 0 R 235 0 R 235 0 R 235 0 R 235 0 R 234 0 R 234 0 R 233 0 R 233 0 R] +endobj + +24 0 obj +[232 0 R 232 0 R 232 0 R 232 0 R 232 0 R 232 0 R 232 0 R 232 0 R 232 0 R 232 0 R 232 0 R 232 0 R 232 0 R 232 0 R 232 0 R 232 0 R 232 0 R 232 0 R 232 0 R 231 0 R 231 0 R 231 0 R 231 0 R 231 0 R 231 0 R 231 0 R 231 0 R 231 0 R 231 0 R 231 0 R 230 0 R 225 0 R 225 0 R 228 0 R 228 0 R 225 0 R 225 0 R 227 0 R 227 0 R 225 0 R 225 0 R 225 0 R 226 0 R 225 0 R 225 0 R 224 0 R 223 0 R 221 0 R 221 0 R 222 0 R 222 0 R 222 0 R 222 0 R 222 0 R 222 0 R 222 0 R 222 0 R 222 0 R 222 0 R 222 0 R 222 0 R 222 0 R 222 0 R 222 0 R 221 0 R 219 0 R 218 0 R 216 0 R 216 0 R 217 0 R 217 0 R 217 0 R 217 0 R 217 0 R 217 0 R 217 0 R 217 0 R 217 0 R 217 0 R 217 0 R 216 0 R 212 0 R 212 0 R 212 0 R 212 0 R 212 0 R 213 0 R 213 0 R 213 0 R 212 0 R 212 0 R 212 0 R 211 0 R 210 0 R 210 0 R 210 0 R 209 0 R 208 0 R 208 0 R 208 0 R 208 0 R 208 0 R 208 0 R 208 0 R 208 0 R 208 0 R 208 0 R 208 0 R 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171 0 R 171 0 R 170 0 R 170 0 R 170 0 R 169 0 R 169 0 R] +endobj + +25 0 obj +[168 0 R 168 0 R 168 0 R 168 0 R 168 0 R 168 0 R 167 0 R 165 0 R 165 0 R 164 0 R 164 0 R 160 0 R 160 0 R 161 0 R 161 0 R 160 0 R 160 0 R 160 0 R 160 0 R 159 0 R 158 0 R 158 0 R 158 0 R 157 0 R 155 0 R 154 0 R 154 0 R 154 0 R 152 0 R 152 0 R 152 0 R 153 0 R 153 0 R 152 0 R 152 0 R 149 0 R 149 0 R 149 0 R 149 0 R 149 0 R 149 0 R 149 0 R 149 0 R 149 0 R 149 0 R 149 0 R 149 0 R 149 0 R 149 0 R 149 0 R 149 0 R 149 0 R 149 0 R 149 0 R 149 0 R 149 0 R 149 0 R 149 0 R 149 0 R 149 0 R 149 0 R 149 0 R 149 0 R 149 0 R 149 0 R 149 0 R 148 0 R 148 0 R 148 0 R 148 0 R 148 0 R 148 0 R 147 0 R 147 0 R 147 0 R 147 0 R 147 0 R 147 0 R 146 0 R 146 0 R 146 0 R 146 0 R 146 0 R 146 0 R 146 0 R 146 0 R 146 0 R 146 0 R 146 0 R 146 0 R 146 0 R 146 0 R 146 0 R 146 0 R 146 0 R 146 0 R 146 0 R 146 0 R 146 0 R 146 0 R 146 0 R 145 0 R 145 0 R 145 0 R 145 0 R 145 0 R 145 0 R 145 0 R 145 0 R 145 0 R 145 0 R 145 0 R 145 0 R 145 0 R 145 0 R 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104 0 R 102 0 R 102 0 R 102 0 R 102 0 R 103 0 R 102 0 R 102 0 R 102 0 R 101 0 R 101 0 R 98 0 R 98 0 R 98 0 R 98 0 R 95 0 R 97 0 R 97 0 R 97 0 R 97 0 R] +endobj + +27 0 obj +[90 0 R 92 0 R 92 0 R 92 0 R 92 0 R 92 0 R 86 0 R 86 0 R 86 0 R 86 0 R 86 0 R 86 0 R 86 0 R 86 0 R 86 0 R 86 0 R 86 0 R 86 0 R 86 0 R 86 0 R 88 0 R 86 0 R 86 0 R 86 0 R 86 0 R 87 0 R 86 0 R 86 0 R 85 0 R 85 0 R 82 0 R 82 0 R 82 0 R 82 0 R 82 0 R 82 0 R 82 0 R 82 0 R 82 0 R 82 0 R 82 0 R 79 0 R 81 0 R 81 0 R] +endobj + +28 0 obj +[74 0 R 76 0 R 76 0 R 72 0 R 71 0 R 70 0 R 69 0 R 69 0 R 69 0 R 69 0 R 69 0 R 69 0 R 69 0 R 69 0 R 69 0 R 69 0 R 69 0 R 67 0 R 66 0 R 65 0 R 65 0 R 65 0 R 65 0 R 62 0 R 61 0 R 61 0 R 58 0 R 58 0 R 58 0 R 58 0 R 58 0 R 58 0 R 58 0 R 58 0 R 58 0 R 58 0 R 58 0 R 58 0 R 55 0 R 57 0 R 57 0 R] +endobj + +29 0 obj +[50 0 R 52 0 R 52 0 R 48 0 R 48 0 R 48 0 R 48 0 R 48 0 R 48 0 R 47 0 R 47 0 R 44 0 R 43 0 R 42 0 R 41 0 R 41 0 R 41 0 R 39 0 R 38 0 R 37 0 R 37 0 R 35 0 R 34 0 R 33 0 R 33 0 R 33 0 R 33 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qՇ~;!!!,ܔ ߿t=oY㮆jyq_tq_h.g(g f(g f(g f;6 7`􋤧'е8kvpRR8ҽ.g(g f(g f(g f;6 p츫!Ƒ1|;fp9C =C1C =C1C =C1۱Y):uf] riԛeqB/t33d33d33d/~зCqW׮7sz8҆&_ .g(g f(g f(g f;6 w_WJۡkq츫_ M1m4]U}'=ÿ!!!,n-ED g͎>f8%[g33d33d33duEHzzB5;jQ ze^qZɦwg33d33d33d_ׯ7_{yMt Κw5|' +O_36nA %  %  % lf>xmfsO>}?3 +6m7vqW_a_6k_ghrz bz bz bc +/+azz8׿Qg͎>8ܥ{s2J2J2J2َ͢+yn: --W^{wqW/Yq6u % %  %  % lfҺLݟ}Ǐ{[Dq츫9l}&ٔ)Sf̘]x:f] _(x8X˱3d33d33d33db~f]WZq|tG=C1C =C1C =C1C =C1۱Y)7 Κw5ݕ5v  %  %  % lfᦌ=^ YF2.3q7}w3d33d33d33dΚw5֞C,_z bz bz bz bcYFr:p qh=EϐAPBϐAPBϐAPBϐAvl;8kvHBGN_ /HΚw57{q̞mcs]_!!!!,\e˖>oc=Pl?~_=zhG {8kvPK 4-qp65Fz bz bz bz bcpҥKcg}fLMMM{qڵ?ǾαcǺő5;jx% +6q[5wk3d33d33d33dk.\)--#ˋ}UVu[#a8kvD(켵xq|zښk  %  %  % lf?0]g͛g_}˟CDqW}3VW]nWg f(g f(g f(g f;6 vbg ̞={Yzug o!" q츫2q WJT_g f(g f(g f(g f;6 |go…7~n僷~Z-))]H Κw5\xՏqp蔒 k3d33d33d33dknS7>?f_/팋/Fp츫%OϐAPBϐAPBϐAPBϐAvl9qDtKKKȸ&L o;Μ9sb+**Fp츫'+oAϐAPBϐAPBϐAPBϐAvl ý{酅z…={~u˖-7:?cݻ]yw$g͎(q8WPU =C1C =C1C =C1C =C1۱Yiɱ!իp$4hP~Fѽ/]J̏5;j`g3좭%3d33d33d33d<ؽ{0z?kR??>:f] ?= 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ǜ8q"ڻk***xsF=Pk&#ƌ#1kfj fjJr[}e>z bzz bz b6cg`08|>eN:5`3;pҥDI]3cVC1'h壱9㚎uR2JJ2J2ٌ^~K^?O8qGv'++c>~pٲeI ?29/-?rзpbۯ>z bzz bz b6cgᤊ!CDw|^n۷o9˒sfee-Λ7G>Sz_}U2w͌Y Ĝ<ͧ䌋,ZpgonP=C1C =C =C1C =C1pئM/ݻo9vءCvܙ3a„z뭭aРA7pw͌Y Ĝ eK~:P53d333d33d; -_<%[n)))i׬Yy:uLkfj 愳]+얃f:Z>,z bzz bz b6cgኼ_%cҘ\oN29B->^2K!!!Y=W#FY'29%k[}ٲnQ=C1C =C =C1C =C1pWqq%KƎŃ#F6m޽{I7q̘ՐẢ⯸Tq'}@ϐAPBPBϐAPBϐAf,RDWWW~~c8'|קNp\H29!;[4)rзhRo'>`g f(g(g f(g f3v3kfj k>y g\`ъG>`g f(g(g f(g f3v3kfj pݗ>pjϘY2JJ2J2ٌ 29nV0P2`濅:Z>cơg f(g(g f(g f3v3kfj :Z7Ͷ[VZn'z bzz bz b6cgj~}}w H029ƪ[Ųn1C3d333d33d; ם?~_F9h 4hȑO=>&fƬ bE+|h  % %  % lEĉ AO,ѯ>_|GFkfj >ioѤAߢ/t!!!YyւXz +O25W3.rh#zbzz bz b6cgἊQFEwv_zFr@q̘ՐAWòu_k%k 4U}F=C1C =C =C1C =C1pXssZEM74yiӦ ӦM{o馘3fLKKۿ +fƬ bp%v:>#z bzz bz b6cg_|׾o6l(//wyc"/:~v w͌Y ljo){`Vb)!!!Y8=7֭ WP(aÆFw]3cVC1*K|O~ȑAE_B /$H<2Ngٯ˧^kD  % %  % l1ewny8dƍ ~$]3cVC1v‰1_"6z bzz bz b6cgᘟ9[#GFgϞMI53f5d嚎-x}l`G>[  % %  % l1,^믿zKD53f5dsO;g%j>#L2JJ2J2ٌclq?ۣϳe˖I]3cVC1G-˷ u}F\=C1C =C =C1C =C1pڵk{<틾1777gDq̘ՐA沼햃{޴BO&3d333d33d; Ǽ y֭[ϳm۶I]3cVC1B}E[O-߇y  % %  % l1oΜ9q?ϫ !!!Y8… nȓx .$H +22<N-~'𞡄!!Y8}tr8aժU%$]3cVCF&ç,z1UDerCPBPBϐAPBϐAf,+D{ٟ|I}޽ ٳvR$w͌Y |K=lsGf I % %  % lI_dǗhѢ+~cgg%Kz y!F:ᮙ1!#c;-+lsS U % %  % las΍/1bĒ%KΜ9s?@ p̙ŋ1"__{5~ĉfƬL9T%`'P3"~3333d33d; uvv?0wyzy=_l}0awWAp̘ՐQ1wՕre}FKF m % %  % ly cƌ}1c%g53f5ddN̝{98|˙>  sz #CzF& f(g(g f(g f3v.:yӳ;AǓ3n㮙1!C>f˲Sؘ7.rH9JJ2J2ٌ몪vꫯ7n6lXVVְanϝ;wn]3cVCvV8T{`lsH$힑QJJ2J2ٌ 2c;-+=owH0ទiJJ2J2ٌ 2Tc5m|n9X[V(x=#333d33d;/{gAp̘Ր!sW]Y'햃 ,3")${Ff"f(g(g f(g f3viu,2b,9_Z~Dz=#c333d33d;sw{^ifƬ [9[p_`g> Kgd2bzz bz b6cgv)2db,.-zsjWMgD 333d33d;~0Mq̘Ր_g,]?3Rbzz bz b6cgv)2bwuVnϱ[Vlw}F8Dg !!YifƬt9P΋vXgsҽg !!YifƬxՓvYᨴ艘!!YifƬ􍹣\c/Oh ҷgbzz bz b6cgv)24sG +L,\`GnHӞ333d33d;~0Mq̘Րv1[v)Zꉮפ]πbzz bz b6cgv)2+f+ٻn9X`kgҫg!!YifƬ49Qv9Qπ1C =C =C1C =C1H;w͌Y st v}kp3}3pE % %  % l"LS53f5dE]5%oN[6eYgDJHAPBPBϐAPBϐAf,4]3cVCFQsɱ &"~U"f(g(g f(g f3vi`⮙1!#cnb. >Q|"x#f(g(g f(g f3vi`⮙1!#ec,v)ZɮRψ=}EPBPBϐAPBϐAf,ifƬԌ +j>[c,{`kgD*J͞8333d33d;$t]3cVCF +wTlg`~3"E`@|JJ2J2ٌEE6w~p̘Րj1[Kϴ[_gC,zFPBPBϐAPBϐAf, b⮙1!#b..zsr-rxH})3 % %  % l"F9}]3cVCFQxַdr͠o}u`H#3O % %  % lpw͌Y )sٯ O,\PgyWN"=GPBPBϐAPBϐAf,gp̘Րz̖e5aū+sHH_ $ +1C =C =C1C =C1]3cVC1[`ͧ햃e_ +59y;3d333d33d; 53f5ds^얃;~ g f(g(g f(g f3v3kfjp+`s]햃6X3'3d333d33d; 53f5dsWuQ,6~eYzΐAPBPBϐAPBϐAf,gp̘Ր|Ӿ%c.} k=,03d333d33d; 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"程景愉", + id1: "202302723005", + advisor: "罗磊", + jobtitle: "教授", + lab: "306-707", + date: "2026.1.19", + header_str: "回溯与分支限界算法分析实验报告", + minimal_cover: true, +) + +#set page(header: [ + #set par(spacing: 6pt) + #align(center)[#text(size: 11pt)[《算法设计与分析》实验报告]] + #v(-0.3em) + #line(length: 100%, stroke: (thickness: 1pt)) +],) + +#show heading: it => box(width: 100%)[ + #v(0.50em) + #set text(font: hei) + #it.body +] + +#outline(title: "目录",depth: 3, indent: 1em) +// #pagebreak() +#outline( + title: [图目录], + target: figure.where(kind: image), +) + +#show heading: it => box(width: 100%)[ + #v(0.50em) + #set text(font: hei) + #counter(heading).display() + #it.body +] +#set enum(indent: 0.5em,body-indent: 0.5em,) +#pagebreak() + += 实验介绍 +#para[ +回溯法（Backtracking）和分支限界法（Branch and Bound）是求解组合优化问题的两种重要算法。回溯法通过深度优先搜索状态空间树，利用剪枝函数避免无效搜索；分支限界法则常采用广度优先或最佳优先策略，利用代价函数（Bound）计算结点的上界（或下界），以剪除不可能产生最优解的分支，从而加速搜索。本实验旨在通过完全背包问题和多重背包问题，深入理解这两种算法的原理，特别是代价函数的设计对算法性能的影响，并掌握蒙特卡洛方法在估算搜索树规模中的应用。 +] + += 实验内容 +#para[ +本实验主要包含以下内容： +] + + 针对完全背包问题，实现回溯法与分支限界算法。 + + 利用蒙特卡洛方法对搜索树的分支数量进行估计。 + + 分析分支限界法中代价函数的准确性，通过与真实值（由动态规划求得）对比，分析不同层级和不同输入规模下的近似效果。 + + 设计并对比两种不同的代价函数（朴素界与分数背包界），分析其剪枝效果与计算开销。 + + （附加）针对多重背包问题，实现分支限界算法，并对比不同代价函数的性能。 + += 实验要求 +#para[ +具体要求如下： +] + + 以物品种类数 $n$ 为输入规模，随机生成测试样本。 + + 统计不同算法的运行时间、访问结点数。 + + 使用 Python 绘制数据图表，展示蒙特卡洛估计结果、代价函数近似比、以及不同算法的性能对比。 + + 分析实验结果，验证理论分析。 + += 实验步骤 + +== 算法设计 + +=== 完全背包问题的分支限界法 +#para[ +完全背包问题允许每种物品选择无限次。在分支限界法中，我们构建状态空间树。为了便于剪枝，我们将物品按价值密度（$v_i/w_i$）降序排列。 +] +```cpp +struct Item { + int id; int weight; int value; + double density; int limit; + Item(int id, int w, int v, int l = -1) : id(id), weight(w), value(v), limit(l) { + density = (double)v / w; + } +}; +bool compareItems(const Item& a, const Item& b) { + return a.density > b.density; +} +``` +#para[ +每个结点包含当前价值 $V_"cur"$、当前重量 $W_"cur"$ 和当前考虑的物品层级 $"level"$。我们使用二叉分支策略： +] +1. *左孩子*：选择当前物品一件，状态更新为 $("level", W_"cur"+w_i, V_"cur"+v_i)$，前提是未超重。 +2. *右孩子*：不再选择当前物品，转而考虑下一件物品，状态更新为 $("level"+1, W_"cur", V_"cur")$。 + +#para[ +为了进行剪枝，我们需要计算当前结点的价值上界（Upper Bound, UB）。如果 $"UB" <= "current_best"$，则剪枝。 +我们实现了两种代价函数： +] +1. *朴素界 (Simple Bound)*：假设剩余容量全部以全局最大单位价值填充。 + $ "UB" = V_"cur" + (W - W_"cur") times max(v_i/w_i) $ + 该界计算简单，但较为松弛。 + +2. *分数背包界 (Fractional Bound)*：即标准的分支限界法上界。将剩余空间用分数背包问题的贪心解填充（即优先装入密度大的物品，最后一件可分割）。由于物品已排序，该界能提供更紧密的上界。 +```cpp +double bound_fractional(int level, int current_val, int rem_cap, const vector& items) { + double bound = current_val; + int w = rem_cap; + for (int i = level; i < items.size(); ++i) { + if (w >= items[i].weight) { + // Take as many as possible (for complete knapsack fractional) + bound += (double)w * items[i].density; + return bound; + } + } + return bound; +} +``` + +=== 蒙特卡洛方法估算搜索树规模 +#para[ +对于大规模问题，直接遍历搜索树是不现实的。蒙特卡洛方法通过随机采样路径来估算树的结点总数。 +设路径上第 $i$ 层结点的度数为 $m_i$，则该路径代表的树规模估计值为： +] +$ N = 1 + m_0 + m_0 m_1 + m_0 m_1 m_2 + dots $ +```cpp +long long monte_carlo_estimate(const vector& items, int capacity, int samples = 1000) { + long long total_nodes = 0; + for (int k = 0; k < samples; ++k) { + long long current_multiplier = 1; + // ... (traversal logic) ... + int branching_factor = moves.size(); + total_nodes += current_multiplier; + current_multiplier *= branching_factor; + // ... + } + return total_nodes / samples; +} +``` +#para[ +通过多次采样取平均值，可得到搜索树规模的无偏估计。在完全背包问题中，由于分支因子变化较大（取决于剩余容量），该方法能有效预估问题难度。 +] + +=== 多重背包问题的分支限界法 +#para[ +多重背包问题中，每种物品的数量有限制 $k_i$。算法结构与完全背包类似，但在分支时需考虑物品数量限制。 +此处同样对比了两种代价函数： +] +1. *松弛界 (Loose Bound)*：忽略数量限制，视为完全背包求分数界。 +2. *紧致界 (Tight Bound)*：考虑数量限制求解分数背包问题。即在贪心填充时，不仅受容量限制，也受物品数量 $k_i$ 限制。 +```cpp +double bound_mk_tight(int level, int current_val, int rem_cap, const vector& items) { + double bound = current_val; + int w = rem_cap; + for (int i = level; i < items.size(); ++i) { + if (items[i].weight == 0) continue; + int can_take_weight = items[i].limit * items[i].weight; + if (w >= can_take_weight) { + w -= can_take_weight; + bound += items[i].value * items[i].limit; + } else { + bound += (double)w * items[i].density; + return bound; + } + } + return bound; +} +``` + +== 实验环境 +- 操作系统：Linux +- 编程语言：C++ (G++) +- 数据分析：Python (Pandas, Seaborn, Matplotlib) +- 硬件环境：标准 PC + += 实验结果与分析 + +== 蒙特卡洛搜索树规模估计 +#para[ +图 1 展示了随物品种类数 $n$ 增加，完全背包问题搜索树结点数的蒙特卡洛估计值（对数坐标）。 +] + +#figure( + image("mc_estimation.png", width: 80%), + caption: [搜索树规模的蒙特卡洛估计], +) + +#para[ +结果表明，搜索树规模随 $n$ 呈指数级增长。蒙特卡洛方法能够快速给出问题规模的数量级估计，对于判断是否能在有限时间内求出精确解具有指导意义。对于整数背包问题，当 $n$ 较大时，建议先使用蒙特卡洛方法预估，若规模过大则应考虑近似算法或启发式搜索。 +] + +== 代价函数准确性分析 +#para[ +为了评估代价函数（上界）的质量，我们记录了搜索过程中各结点的上界值与该状态下的真实最优值（通过动态规划预先计算得到）的比值。比值越接近 1，说明上界越紧致。 +] + +#figure( + image("cost_ratio_level.png", width: 80%), + caption: [不同层级下代价函数的近似比 (n=20)], +) + +#figure( + image("cost_ratio_n.png", width: 80%), + caption: [平均近似比随输入规模 n 的变化], +) + +#para[ +从图 2 可以看出，随着搜索深度的增加（Level 增大），剩余问题规模变小，代价函数的近似比逐渐趋向于 1，说明上界越来越精确。这是符合预期的，因为随着物品确定的越多，不确定性越小。 +图 3 展示了输入规模 $n$ 对平均近似比的影响。通常情况下，平均近似比相对稳定，不会随 $n$ 剧烈波动，这表明分数背包界具有良好的鲁棒性。 +] + +== 不同代价函数的性能对比 +#para[ +我们对比了“分数背包界 (Fractional)”与“朴素界 (Simple)”在完全背包问题上的性能。 +] + +#figure( + image("new_cost_nodes.png", width: 80%), + caption: [不同代价函数下的访问结点数对比], +) + +#figure( + image("new_cost_time.png", width: 80%), + caption: [不同代价函数下的运行时间对比], +) + +#para[ +实验结果显著： +] +1. *剪枝效果*：分数背包界（Fractional）的访问结点数远少于朴素界（Simple），常常相差数个数量级（注意图 4 为对数坐标）。这是因为分数背包界提供了更紧的上界，能更早地剪除无效分支。 +2. *运行时间*：尽管分数背包界的计算复杂度略高于朴素界（需要遍历剩余物品，而朴素界仅需常数/一次乘法），但由于其极强的剪枝能力，总运行时间反而大幅降低。 +#para[ +这说明在分支限界法中，设计一个计算稍复杂但更紧致的代价函数通常是值得的。 +] + += 附加：多重背包问题分析 +#para[ +在多重背包问题中，我们对比了考虑物品数量限制的“紧致界 (TightBound)”与忽略数量限制的“松弛界 (LooseBound)”。 +] + +#figure( + image("mk_nodes.png", width: 80%), + caption: [多重背包：不同代价函数的结点数对比], +) + +#figure( + image("mk_time.png", width: 80%), + caption: [多重背包：不同代价函数的运行时间对比], +) + +#para[ +结果显示，紧致界（TightBound）在性能上优于松弛界。因为忽略数量限制会导致上界过大，无法有效剪除那些虽然总重量满足但单种物品数量超标的分支。通过在代价函数中精确建模约束条件，可以显著提高算法效率。 +] + += 实验总结 +#para[ +本实验通过实现和分析完全背包及多重背包问题的分支限界算法，得出以下结论： +] +1. *代价函数的重要性*：代价函数的紧致程度直接决定了分支限界法的剪枝效率。更紧的界（如分数背包界）虽然单次计算开销稍大，但能指数级减少搜索空间，从而获得更好的总性能。 +2. *蒙特卡洛方法的实用性*：该方法能有效评估大规模组合优化问题的解空间大小，为算法选择提供依据。 +3. *真实值对比分析*：通过与 DP 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