This paper studies $k$-claw-free graphs, exploring the connection between an extremal combinatorics question and the power of a convex program in approximating the maximum-weight independent set in this graph class. For the extremal question, we consider the notion, that we call \textit{conditional $\chi$-boundedness} of a graph: Given a graph $G$ that is assumed to contain an independent set of a certain (constant) size, we are interested in upper bounding the chromatic number in terms of the clique number of $G$. This question, besides being interesting on its own, has algorithmic implications (which have been relatively neglected in the literature) on the performance of SDP relaxations in estimating the value of maximum-weight independent set. For $k=3$, Chudnovsky and Seymour (JCTB 2010) prove that any $3$-claw-free graph $G$ with an independent set of size three must satisfy $\chi(G) \leq 2 \omega(G)$. Their result implies a factor $2$-estimation algorithm for the maximum weight independent set via an SDP relaxation (providing the first non-trivial result for maximum-weight independent set in such graphs via a convex relaxation). An obvious open question is whether a similar conditional $\chi$-boundedness phenomenon holds for any $k$-claw-free graph. Our main result answers this question negatively. We further present some evidence that our construction could be useful in studying more broadly the power of convex relaxations in the context of approximating maximum weight independent set in $k$-claw free graphs. In particular, we prove a lower bound on families of convex programs that are stronger than known convex relaxations used algorithmically in this context.

翻译：本文研究 $k$‑无爪图，探讨一个极值组合学问题与凸规划在近似该类图中最大权独立集方面的能力之间的联系。针对极值问题，我们考虑一个称之为图的条件 $\chi$‑有界性的概念：给定一个假设包含某个（常数）大小的独立集的图 $G$，我们感兴趣的是用 $G$ 的团数来给出其色数的上界。这个问题除了本身具有趣味性外，还对 SDP 松弛在估计最大权独立集值时的性能具有算法上的意义（这在文献中相对被忽视）。对于 $k=3$，Chudnovsky 和 Seymour（JCTB 2010）证明，任何包含大小为三的独立集的 $3$‑无爪图 $G$ 必满足 $\chi(G) \leq 2 \omega(G)$。他们的结果通过 SDP 松弛给出了一个因子 $2$ 的最大权独立集估计算法（首次为此类图中的最大权独立集通过凸松弛提供了非平凡结果）。一个明显的开放问题是：对于任意 $k$‑无爪图，是否也存在类似的条件 $\chi$‑有界性现象？我们的主要结果对此问题给出了否定答案。我们进一步提供证据表明，我们的构造可能有助于更广泛地研究凸松弛在近似 $k$‑无爪图中最大权独立集方面的能力。特别地，我们证明了在某些比此背景下已知算法性凸松弛更强的凸规划族上的下界。