Let tw(G) denote the treewidth of graph G. Given a graph G and a positive integer k such that tw(G) <= k + 1, we are to decide if tw(G) <= k. We give a certifying algorithm RTW ("R" for recursive) for this task: it returns one or more tree-decompositions of G of width <= k if the answer is YES and a minimal contraction H of G such that tw(H) > k otherwise. RTW uses a heuristic variant of Tamaki's PID algorithm for treewidth (ESA2017), which we call HPID. RTW, given G and k, interleaves the execution of HPID with recursive calls on G /e for edges e of G, where G / e denotes the graph obtained from G by contracting edge e. If we find that tw(G / e) > k, then we have tw(G) > k with the same certificate. If we find that tw(G / e) <= k, we "uncontract" the bags of the certifying tree-decompositions of G / e into bags of G and feed them to HPID to help progress. If the question is not resolved after the recursive calls are made for all edges, we finish HPID in an exhaustive mode. If it turns out that tw(G) > k, then G is a certificate for tw(G') > k for every G' of which G is a contraction, because we have found tw(G / e) <= k for every edge e of G. This final round of HPID guarantees the correctness of the algorithm, while its practical efficiency derives from our methods of "uncontracting" bags of tree-decompositions of G / e to useful bags of G, as well as of exploiting those bags in HPID. Experiments show that our algorithm drastically extends the scope of practically solvable instances. In particular, when applied to the 100 instances in the PACE 2017 bonus set, the number of instances solved by our implementation on a typical laptop, with the timeout of 100, 1000, and 10000 seconds per instance, are 72, 92, and 98 respectively, while these numbers are 11, 38, and 68 for Tamaki's PID solver and 65, 82, and 85 for his new solver (SEA 2022).

翻译：设 tw(G) 表示图 G 的树宽度。给定一个图 G 和一个正整数 k，且满足 tw(G) ≤ k+1，我们需要判定是否 tw(G) ≤ k。我们为此任务提出一种可认证算法 RTW（“R”代表递归）：若答案为是，则返回一个或多个宽度≤k 的树分解；若答案为否，则返回 G 的一个最小收缩子图 H，使得 tw(H) > k。RTW 采用了 Tamaki 的树宽度 PID 算法（ESA2017）的启发式变体，记为 HPID。对于给定的 G 和 k，RTW 将 HPID 的执行与对 G/e（e 为 G 的边，G/e 表示通过收缩边 e 得到的图）的递归调用交替进行。若发现 tw(G/e) > k，则可直接判定 tw(G) > k 并给出相同证书。若发现 tw(G/e) ≤ k，则将 G/e 的可认证树分解的袋子“解收缩”为 G 的袋子，并输入 HPID 以促进算法进展。若对所有边完成递归调用后问题仍未解决，则以穷举模式完成 HPID。若最终判定 tw(G) > k，则由于已发现 G 的每条边 e 均满足 tw(G/e) ≤ k，G 本身可作为任何以 G 为收缩的图 G' 满足 tw(G') > k 的证书。HPID 的最终轮次保证了算法的正确性，而其实际效率得益于我们将 G/e 的树分解袋子“解收缩”为 G 的有用袋子，以及在 HPID 中利用这些袋子的方法。实验表明，我们的算法大幅扩展了实际可解实例的范围。具体而言，在 PACE 2017 奖励集的 100 个实例上，当单实例超时时间设为 100、1000 和 10000 秒时，我们的实现在典型笔记本电脑上分别解出 72、92 和 98 个实例；而 Tamaki 的 PID 求解器对应数值为 11、38 和 68，其新求解器（SEA 2022）为 65、82 和 85。