Large independent sets in recursive Markov random graphs
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arXiv
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| Natura: | Preprint |
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2022
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| _version_ | 1866914871598120960 |
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| author | Gupte, Akshay Zhu, Yiran |
| author_facet | Gupte, Akshay Zhu, Yiran |
| contents | Computing the maximum size of an independent set in a graph is a famously hard combinatorial problem that has been well-studied for various classes of graphs. When it comes to random graphs, only the classical Erdős-Rényi-Gilbert random graph $G_{n,p}$ has been analysed and shown to have largest independent sets of size $Θ(\log{n})$ w.h.p. This classical model does not capture any dependency structure between edges that can appear in real-world networks. We initiate study in this direction by defining random graphs $G^{r}_{n,p}$ whose existence of edges is determined by a Markov process that is also governed by a decay parameter $r\in(0,1]$. We prove that w.h.p. $G^{r}_{n,p}$ has independent sets of size $(\frac{1-r}{2+ε}) \frac{n}{\log{n}}$ for arbitrary $ε> 0$, which implies an asymptotic lower bound of $Ω(π(n))$ where $π(n)$ is the prime-counting function. This is derived using bounds on the terms of a harmonic series, Turán bound on stability number, and a concentration analysis for a certain sequence of dependent Bernoulli variables that may also be of independent interest. Since $G^{r}_{n,p}$ collapses to $G_{n,p}$ when there is no decay, it follows that having even the slightest bit of dependency (any $r < 1$) in the random graph construction leads to the presence of large independent sets and thus our random model has a phase transition at its boundary value of $r=1$. For the maximal independent set output by a greedy algorithm, we deduce that it has a performance ratio of at most $1 + \frac{\log{n}}{(1-r)}$ w.h.p. when the lowest degree vertex is picked at each iteration, and also show that under any other permutation of vertices the algorithm outputs a set of size $Ω(n^{1/1+τ})$, where $τ=1/(1-r)$, and hence has a performance ratio of $O(n^{\frac{1}{2-r}})$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2207_04514 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Large independent sets in recursive Markov random graphs Gupte, Akshay Zhu, Yiran Combinatorics Discrete Mathematics Optimization and Control Probability 90C27, 60J10, 05C80, 05C69 Computing the maximum size of an independent set in a graph is a famously hard combinatorial problem that has been well-studied for various classes of graphs. When it comes to random graphs, only the classical Erdős-Rényi-Gilbert random graph $G_{n,p}$ has been analysed and shown to have largest independent sets of size $Θ(\log{n})$ w.h.p. This classical model does not capture any dependency structure between edges that can appear in real-world networks. We initiate study in this direction by defining random graphs $G^{r}_{n,p}$ whose existence of edges is determined by a Markov process that is also governed by a decay parameter $r\in(0,1]$. We prove that w.h.p. $G^{r}_{n,p}$ has independent sets of size $(\frac{1-r}{2+ε}) \frac{n}{\log{n}}$ for arbitrary $ε> 0$, which implies an asymptotic lower bound of $Ω(π(n))$ where $π(n)$ is the prime-counting function. This is derived using bounds on the terms of a harmonic series, Turán bound on stability number, and a concentration analysis for a certain sequence of dependent Bernoulli variables that may also be of independent interest. Since $G^{r}_{n,p}$ collapses to $G_{n,p}$ when there is no decay, it follows that having even the slightest bit of dependency (any $r < 1$) in the random graph construction leads to the presence of large independent sets and thus our random model has a phase transition at its boundary value of $r=1$. For the maximal independent set output by a greedy algorithm, we deduce that it has a performance ratio of at most $1 + \frac{\log{n}}{(1-r)}$ w.h.p. when the lowest degree vertex is picked at each iteration, and also show that under any other permutation of vertices the algorithm outputs a set of size $Ω(n^{1/1+τ})$, where $τ=1/(1-r)$, and hence has a performance ratio of $O(n^{\frac{1}{2-r}})$. |
| title | Large independent sets in recursive Markov random graphs |
| topic | Combinatorics Discrete Mathematics Optimization and Control Probability 90C27, 60J10, 05C80, 05C69 |
| url | https://arxiv.org/abs/2207.04514 |