关于宽度和填充类图参数之间的权衡
On tradeoffs between width- and fill-like graph parameters
摘要：
在这项工作中，我们考虑两个双准则优化问题：给定一个输入图，目标是找到同时最小化边数和团数的区间（或弦）超图。对于区间超图，该问题可以重述为同时最小化输入图$G$的路径宽度$pw(G)$和轮廓$p(G)$。证明了对任意图$G$和整数$t\in\{1,\ldots,pw(G)+1\}$，存在$G$的区间超图$G'$，使得它的团数保持$ω(G')\leq(1+\frac{2}{t})(pw(G)+1)$，且它的边数以$|E(G')|\leq(t+2)p(G)$为界.换句话说，图的路径宽度和轮廓可以分别在因子$1+\frac{2}{t}$（加上一个小常数）和因子$t+2$内同时最小化。注意，对于固定的$t$，两个上限都提供常数因子近似。在消极的一面，我们展示了一个例子，证明了，对于某些图，没有解决方案，其中两个参数都是最佳的。在寻找弦超图的情况下，反映其团大小和边数的两个相应的图参数是树宽和填充。我们得到的树的宽度和填充问题也是“正交”的意义上说，对于某些图形，最小化这些参数之一的解决方案不能最小化其他。作为一个激励的例子，我们回顾图搜索游戏，说明了需要同时最小化这些对图参数。
英文摘要：
In this work we consider two two-criteria optimization problems: given an input graph, the goal is to find its interval (or chordal) supergraph that minimizes the number of edges and its clique number simultaneously. For the interval supergraph, the problem can be restated as simultaneous minimization of the pathwidth $pw(G)$ and the profile $p(G)$ of the input graph $G$. We prove that for an arbitrary graph $G$ and an integer $t\in\{1,\ldots,pw(G)+1\}$, there exists an interval supergraph $G'$ of $G$ such that for its clique number it holds $ω(G')\leq(1+\frac{2}{t})(pw(G)+1)$ and the number of its edges is bounded by $|E(G')|\leq(t+2)p(G)$. In other words, the pathwidth and the profile of a graph can be simultaneously minimized within the factors of $1+\frac{2}{t}$ (plus a small constant) and $t+2$, respectively. Note that for a fixed $t$, both upper bounds provide constant factor approximations. On the negative side, we show an example that proves that, for some graphs, there is no solution in which both parameters are optimal. In case of finding a chordal supergraph, the two corresponding graph parameters that reflect its clique size and number of edges are the treewidth and fill-in. We obtain that the treewidth and the fill-in problems are also `orthogonal' in the sense that for some graphs, a solution that minimizes one of those parameters cannot minimize the other. As a motivating example, we recall graph searching games which illustrates a need of simultaneous minimization of these pairs of graph parameters.
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