From: Martin Mares Date: Sat, 13 Sep 2008 17:56:25 +0000 (+0200) Subject: Better (and more explicit) bounds on class density; Hadwiger numbers. X-Git-Url: http://mj.ucw.cz/gitweb/?a=commitdiff_plain;h=0338fe7d3ca1d6b7b7b3599cd4250673ddc71945;p=saga.git Better (and more explicit) bounds on class density; Hadwiger numbers. --- diff --git a/TODO b/TODO index b3291ca..8f8561e 100644 --- a/TODO +++ b/TODO @@ -18,15 +18,3 @@ Typography: Diaz: > 2: simplify, remove proof sketches -> 3.1.7, 3.1.10: skip proof -> 3.5.1: does the first alg give any insight? -> 5: mention graphs with moving vertices? - -Patrice: - -> Remark on 2.5.1: polynomial time could be replaced by sub-exponential time. -> In 3.1.12 and 3.1.16, you should make explicit the dependence of the - running time with respect, for instance, to the Hadwiger number of the - graph or to the maximal density nabla(G) of a minor of the graph, as - considering a minor closed class or another does not change the - algorithm but only the bound on its running time. diff --git a/adv.tex b/adv.tex index f0370bb..dace909 100644 --- a/adv.tex +++ b/adv.tex @@ -75,60 +75,62 @@ Let $G$ be a~graph and $\cal C$ be a class of graphs. We define the \df{edge den $\varrho(G)$ of~$G$ as the average number of edges per vertex, i.e., $m(G)/n(G)$. The edge density $\varrho(\cal C)$ of the class is then defined as the infimum of $\varrho(G)$ over all $G\in\cal C$. -\thmn{Mader \cite{mader:dens}}\id{maderthm}% +\obs +Let us consider a~non-trivial minor-closed class~${\cal C} = \Forb({\cal H})$ +and a~graph $X\in{\cal H}$ with the minimum number of vertices. +Obviously, $\Forb({\cal H}) \subseteq \Forb(X)$, because excluding additional +minors cannot make the class richer. Also, if we denote the number of vertices +of~$X$ by~$k$, we have $X\minorof K_k$ and hence $\Forb(X) \subseteq \Forb(K_k)$. +When we put these two inclusions together, we get ${\cal C} \subseteq \Forb(K_k)$ and +so $\varrho({\cal C}) \le \varrho(\Forb(K_k))$. It is therefore sufficient to +bound the density of classes that exclude a~single complete graph. +Moreover, our parameter~$k$ is equal to the well-known Hadwiger number: + +\defn\id{hadwiger}% +The \df{Hadwiger number} $H(G)$ is the smallest~$k$ such that the complete +graph~$K_k$ is not a~minor of~$G$. We can easily extend it to graph classes: +$H({\cal C})$ is the minimum of~$H(G)$ over all~$G\in{\cal C}$. + +\cor +$\varrho({\cal C}) \le \varrho(\Forb(K_{H({\cal C})}))$ +for any non-trivial minor-closed class~${\cal C}$. + +\thmn{Mader \cite{mader:dens}, see also Lemma 3.5.1 in Diestel \cite{diestel:gt}}\id{maderthm}% For every $k\in{\bb N}$ there exists $h(k)\in{\bb R}$ such that every graph of average degree at least~$h(k)$ contains a~subdivision of~$K_{k}$ as a~subgraph. -\proofsketch -(See Lemma 3.5.1 in \cite{diestel:gt} for a~complete proof in English.) - -Let us fix~$k$ and prove by induction on~$m$ that every graph of average -degree at least~$2^m$ contains a~subdivision of some graph with $k$~vertices -and $m$~edges (for $k\le m\le {k\choose 2}$). When we reach $m={k\choose 2}$, the theorem follows -as the only graph with~$k$ vertices and~$k\choose 2$ edges is~$K_k$. - -The base case $m=k$: Let us observe that when the average degree -is~$a$, removing any vertex of degree less than~$a/2$ does not decrease the -average degree. A~graph with $a\ge 2^k$ therefore has a~subgraph -with minimum degree $\delta\ge a/2=2^{k-1}$. Such subgraph contains -a~cycle on more than~$\delta$ vertices, in other words a~subdivision of -the cycle~$C_k$. - -Induction step: Let~$G$ be a~graph with average degree at least~$2^m$ and -assume that the theorem already holds for $m-1$. Without loss of generality, -$G$~is connected. Consider a~maximal set $U\subseteq V$ such that the subgraph $G[U]$ -induced by~$U$ is connected and the graph $G\sgc U$ ($G$~with $U$~contracted to -a~single vertex) has average degree at least~$2^m$ (such~$U$ exists, because -$G=G\sgc U$ whenever $\vert U\vert=1$). Now consider the subgraph~$H$ induced -in~$G$ by the neighbors of~$U$. Every $v\in V(H)$ must have $\deg_H(v) \ge 2^{m-1}$, -as otherwise we can add this vertex to~$U$, contradicting its -maximality. By the induction hypothesis, $H$ contains a~subdivision of some -graph~$R$ with $k$~vertices and $m-1$ edges. Any two non-adjacent vertices -of~$R$ can be connected in the subdivision by a~path lying entirely in~$G[U]$, -which reveals a~subdivision of a~graph with $m$~edges. \qed - \thmn{Density of minor-closed classes, Mader~\cite{mader:dens}} -Every non-trivial minor-closed class of graphs has finite edge density. +A~non-trivial minor-closed class ${\cal C}$ has density $\varrho({\cal C}) \le 2h(k)$, +where~$h$ is the function from the previous theorem and $k=H({\cal C})$ is the +Hadwiger number of the class. \proof -Let~$\cal C$ be any such class, $X$~its excluded minor with the smallest number -of vertices~$x$. -As $X\minorof K_x$, the class $\cal C$ is entirely contained in ${\cal C}'=\Forb(K_x)$, so -$\varrho({\cal C}) \le \varrho({\cal C}')$ and therefore it suffices to prove the -theorem for classes excluding a~single complete graph~$K_x$. - -We will show that $\varrho({\cal C})\le 2h(x)$, where $h$~is the function -from the previous theorem. If any $G\in{\cal C}$ had more than $2h(x)\cdot n(G)$ -edges, its average degree would be at least~$h(x)$, so by the previous theorem -$G$~would contain a~subdivision of~$K_x$ and hence $K_x$ as a~minor. +We already know that it is sufficient to prove the theorem for the case when +${\cal C}$ excludes on the complete graph~$K_k$. + +We will prove the contrapositive. If $\varrho({\cal C}) > 2h(k)$, then there is some graph +$G\in{\cal C}$ such that $\varrho(G) > 2h(k)$. This implies that the average degree +of~$G$ is greater than~$h(k)$, so by the previous theorem $G$~contains a~subdivison +of~$K_k$ and hence also~$K_k$ as a~minor. \qed +\para +The Mader's original proof of Theorem \ref{maderthm} yields $h(k) \approx 2^{n^2}$, which is +very coarse. It was however vastly improved later: Kostochka +\cite{kostochka:lbh} and independently Thomason \cite{thomason:efc} have proven +that an~average degree $\Omega(k\sqrt{\log k})$ is sufficient to enforce~$K_k$ +as a~minor and that this is the best what we can get. Their result implies: + +\cor +$\varrho({\cal C}) = \O(k\sqrt{\log k})$ whenever ${\cal C}$ is a~minor-closed +class of graphs and~$k=H({\cal C})$ is its Hadwiger number. + Let us return to the analysis of our algorithm. \thmn{MST on minor-closed classes, Tarjan \cite{tarjan:dsna}}\id{mstmcc}% -For any fixed non-trivial minor-closed class~$\cal C$ of graphs, the Contractive Bor\o{u}vka's +For any non-trivial minor-closed class~$\cal C$ of graphs, the Contractive Bor\o{u}vka's algorithm (\ref{contbor}) finds the MST of any graph of this class in time -$\O(n)$. (The constant hidden in the~$\O$ depends on the class.) +$\O(n \cdot \varrho({\cal C}))$. \proof Following the proof for planar graphs (\ref{planarbor}), we denote the graph considered @@ -139,8 +141,10 @@ and we have $n_i \le n/2^i$, so it remains to show a linear bound for the $m_i$' Since each $G_i$ is produced from~$G_{i-1}$ by a sequence of edge contractions, all $G_i$'s are minors of the input graph.\foot{Technically, these are multigraph contractions, but followed by flattening, so they are equivalent to contractions on simple graphs.} -So they also belong to~$\cal C$ and by the Density theorem $m_i\le \varrho({\cal C})\cdot n_i$. -The time complexity is therefore $\sum_i \O(m_i) = \sum_i \O(n_i) = \O(\sum_i n/2^i) = \O(n)$. +So they also belong to~$\cal C$ and thus $m_i\le \varrho n_i$, where $\varrho=\varrho({\cal C})$ +is the density of the class~${\cal C}$. +The time complexity of the algorithm is therefore +$\sum_i \O(m_i) = \sum_i \O(\varrho n_i) = \O(\varrho\cdot\sum_i n/2^i) = \O(n\varrho)$. \qed \paran{Local contractions}\id{nobatch}% @@ -185,10 +189,9 @@ $\deg(v)\le 4\varrho$. \endalgo \thm -When $\cal C$ is a minor-closed class of graphs with density~$\varrho$, the +When $\cal C$ is a~minor-closed class of graphs with density~$\varrho$, the Local Bor\o{u}vka's Algorithm with the parameter~$t$ set to~$4\varrho$ -finds the MST of any graph from this class in time $\O(n)$. (The constant -in the~$\O$ depends on~the class.) +finds the MST of any graph from this class in time $\O(n\varrho)$. \proof Let us denote by $G_i$, $n_i$ and $m_i$ the graph considered by the @@ -214,11 +217,11 @@ It remains to analyse the time complexity of the algorithm. Since $G_i\in{\cal C $m_i\le \varrho n_i \le \varrho n/2^i$. We will show that the $i$-th iteration is carried out in time $\O(m_i)$. Steps 5 and~6 run in time $\O(\deg(v))=\O(t)$ for each~$v$, so summed -over all $v$'s they take $\O(tn_i)$, which is $\O(n_i)$ for a fixed class~$\cal C$. +over all $v$'s they take $\O(tn_i)$. Flattening takes $\O(m_i)$ as already noted in the analysis of the Contracting Bor\o{u}vka's Algorithm (see \ref{contiter}). -The whole algorithm therefore runs in time $\O(\sum_i m_i) = \O(\sum_i n/2^i) = \O(n)$. +The whole algorithm therefore runs in time $\O(\sum_i m_i) = \O(\varrho\cdot\sum_i n/2^i) = \O(n\varrho)$. \qed \paran{Back to planar graphs}% @@ -255,12 +258,6 @@ has degree~9. \figure{hexangle.eps}{\epsfxsize}{The construction from Remark~\ref{hexa}} -\rem -The bound on the average degree needed to enforce a~$K_k$ minor, which we get from Theorem \ref{maderthm}, -is very coarse. Kostochka \cite{kostochka:lbh} and independently Thomason \cite{thomason:efc} -have proven that an~average degree $\Omega(k\sqrt{\log k})$ is sufficient and that this -is the best what we can get. - \rem Minor-closed classes share many other interesting properties, for example bounded chromatic numbers of various kinds, as shown by Theorem 6.1 of \cite{nesetril:minors}. We can expect diff --git a/notation.tex b/notation.tex index 4a26a3e..ad8beb6 100644 --- a/notation.tex +++ b/notation.tex @@ -21,6 +21,8 @@ \n{$E(G)$}{set of edges of a graph~$G$} \n{$E$}{$E(G)$ when the graph~$G$ is clear from context} \n{${\E}X$}{expected value of a~random variable~$X$} +\n{$H(G)$}{Hadwiger number of a~graph~$G$: $H(G) := \min \{ k \mid K_k\not\minorof G \}$ \[hadwiger]} +\n{$H({\cal C})$}{Hadw.~number of a~graph class: $H({\cal C}) := \min \{ H(G) \mid G\in{\cal C} \}$ \[hadwiger]} \n{$K_k$}{complete graph on~$k$ vertices} \n{$\log n$}{binary logarithm of the number~$n$} \n{$\log^* n$}{iterated logarithm: $\log^*n := \min\{i \mid \log^{(i)}n \le 1\}$; the inverse of~$2\tower n$}