Many real discrete optimization problems (DOPs) are $NP$-hard and contain a huge number of variables and/or constraints that make the models intractable for currently available solvers. Large DOPs can be solved due to their special tructure using decomposition approaches. An important example of decomposition approaches is tree decomposition with local decomposition algorithms using the special block matrix structure of constraints which can exploit sparsity in the interaction graph of a discrete optimization problem. In this paper, discrete optimization problems with a tree structural graph are solved by local decomposition algorithms. Local decomposition algorithms generate a family of related DO problems which have the same structure but differ in the right-hand sides. Due to this fact, postoptimality techniques in DO are applied.
Tree decompositions whose trees are subgraphs: An application of Simon's factorization
LaBRI, Université de Bordeaux（波尔多大学）；Département d’Informatique, Université libre de Bruxelles（布鲁塞尔自由大学）；Department of Theoretical Computer Science, Jagiellonian University（雅盖隆大学）；Faculty of Mathematics, Natural Sciences and Information Technologies and Andrej Marušič Institute, University of Primorska（普里莫斯卡大学）；Institute of Informatics, Faculty of Mathematics, Informatics, and Mechanics, University of Warsaw（华沙大学）
From Modular Decomposition Trees to Rooted Median Graphs
对称映射$δ\colon X\times X \to Υ$（或者等价地，一组对称二元关系、2-结构或边着色无向图）的模分解是在标记树中捕获$δ$的关键特征的自然构造。如果标记$δ(x,y)$与$T$中的$x$和$y$的最后共同祖先的标记一致，则通过顶点标记的有根树$(T,t)$来解释映射$δ$，即，如果是$δ(x,y)=t(\mathrm{lca}(x,y))$。仅映射其模分解不包含素节点，即，符号超度量可以用这种方式来解释。在这里，我们考虑根中值图作为（模分解）树的推广来解释对称映射。我们首先证明了每一个对称映射都可以用“扩展的”超立方体和半网格来解释。然后，我们推导出一个线性时间算法，逐步解决素顶点的模块化分解树，以获得一个有根和标记的中值图，解释了一个给定的对称地图$δ$。我们认为，由此产生的“树状”的中位数图可能是使用在遗传学作为一个模型的进化关系。
Optimal tree-decompositions with bags of bounded pathwidth
School of Mathematics, Monash University（莫纳什大学数学学院）；Département d’Informatique, Université libre de Bruxelles（布鲁塞尔自由大学计算机系）；Theoretical Computer Science Department, Faculty of Mathematics and Computer Science and Doctoral School of Exact and Natural Sciences, Jagiellonian University（雅盖隆大学数学与计算机科学学院理论计算机科学系及精确与自然科学博士学院）
An efficient tree decomposition method for permanents and mixed discriminants
我们提出了一个有效的算法来计算结构矩阵和多维数组（张量）的永久式，混合判别式和超行列式。我们描述的稀疏结构的数组中的一个图形，我们假设它的树宽，表示为$ω$，是小的。我们的算法需要$O(n 2^ω)$算术运算来计算持久式，而$O(n^2 + n 3^ω)$用于混合判别式和超行列式。我们最后表明，混合体积计算仍然是困难的有界树宽假设。
