求可生成近似指定权重随机环的图算法

Absolutely, there are several graph algorithms and practical approaches to solve this problem—let’s split them into two main categories based on your goal: finding the closest possible cycle to your target weight W*, or generating a random cycle within an acceptable error margin of W*.

Key Notes First #

First, a quick clarification: we’ll distinguish between two types of cycles here:

• Simple cycles: Paths that start and end at N, visit every other node at most once (no repeated nodes except the start/end).
• General closed paths: Paths that start and end at N, allowing repeated nodes and edges (this is often easier to handle, especially for larger graphs).

All approaches below leverage the fact that your graph has positive edge weights—this is critical because it ensures path weights increase monotonically with path length, avoiding infinite loops with zero net weight gain.

1. Finding the Cycle Closest to W* #

For Simple Cycles #

This is a variant of the Traveling Salesman Problem (TSP) and is NP-hard, meaning there’s no known efficient exact algorithm for large graphs. However, you have solid options:

• Dynamic Programming (Exact, Small Graphs): Use a DP state dp[mask][u] where mask represents the set of visited nodes, and u is the current node. Track all possible path weights for each state, then look for paths that return to N with weights closest to W*. Complexity is O(n²2ⁿ), so only feasible for graphs with ~15 nodes or fewer.
• A Heuristic Search*: Adapt A* to prioritize paths where the current weight is as close as possible to W*. The heuristic function can estimate the minimum/maximum possible weight needed to return to N from the current node (using shortest paths), allowing you to prune paths that can’t possibly get close to W*.
• Genetic/Simulated Annealing Algorithms: For larger graphs, use heuristic optimization. Generate initial random simple cycles, then iteratively mutate/crossover them, keeping track of cycles whose weights are closest to W*. These methods trade off exactness for speed on bigger graphs.

For General Closed Paths (Allowing Repeats) #

This is more flexible, and you can use combinatorial or shortest-path-based approaches:

• Shortest Path Combination: Precompute the shortest path weights from N to every other node (using Dijkstra’s algorithm, since weights are positive). You can then combine these paths (e.g., go N→u→N, repeat k times) and add other detours to adjust the total weight to get as close as possible to W*. This works because repeating a fixed path adds a consistent weight each time, making it easy to tune the total.
• Subset Sum Adaptation: Treat small closed paths (from N to N) as "coins" with their respective weights, then solve a subset sum-like problem to find a combination of these paths whose total weight is closest to W*. You can generate these small paths via random walks or BFS from N.

2. Generating Random Cycles Within an Error Margin of W* #

If you don’t need the absolute closest cycle, just a random one that’s within your error range, these approaches work well:

• Random Walk with Adjustment: Start at N and perform a random walk, keeping track of the total weight accumulated. When the current weight is within a certain range below W*, use Dijkstra’s algorithm to find the shortest path back to N—if the total weight (walk + return path) falls within your error margin, you’re done. If you overshoot, you can backtrack or restart the walk with adjusted parameters.
• Random Path Combination: Pre-generate a pool of small closed paths (different weights) starting/ending at N. Then randomly select and repeat these paths (like adding "blocks" of weight) until the total weight falls within W* ± your error. This is efficient if you have a diverse pool of path weights to choose from.
• Monte Carlo Filtering: Generate a large number of random closed paths (either simple or general) by randomly traversing edges from N, then filter out all paths whose total weight is outside your error margin. Randomly pick one of the remaining valid paths. Note: This can be slow for large graphs, so it’s best for small to medium-sized graphs or when you have time to spare.

内容的提问来源于stack exchange，提问作者JNevens