arXiv subjects
Publications and source records attributed to Junichiro Fukuyama.
We demonstrate the truth of the sunflower conjecture by showing that a family $\mathcal{F}$ of sets each of cardinality at most $m$ includes a $k$-sunflower, if $|\mathcal{F}| > ( c k^4 )^{m}$ for a constant $c>0$ independent of $m$ and $k$, where $k$-sunflower means a family of $k$ different sets with a common pairwise intersection.
We show that a family $\mathcal{F}$ of sets each of cardinality $m \in \mathbb{Z}_{>2}$ includes a $k$-sunflower if $ |\mathcal{F}| \ge \left( \frac{c k^2 \ln m}{\ln \ln m} \right)^m$ for some constant $c>0$, where $k$-sunflower means a family of $k$ different sets with a common pairwise intersection. The base of the exponential lower bound is sub-logarithmic for each $k$ updating the current best-known result.
This paper explores the structure of the combinatorial domain $2^X$ in relation to sunflowers. The previous study found some intrinsic properties of the $l$-extension \[ Ext \left( \mathcal{F}, l \right) = \left\{ V ~:~ V \in {X \choose l},~ \exists U \in \mathcal{F}~ U \subset V \right\} \] of a family $\mathcal{F}$ of $m$-cardinality sets. Subsequently, it lead to the proof that such an $\mathcal{F}$ includes three mutually disjoint sets if it satisfies the $Γ(b)$-condition, that is, \[ \left| \mathcal{F}[S] \right| < b^{-|S|} |\mathcal{F}| \quad \textrm{for every nonempty set}~ S, \qquad \textrm{where} \quad \mathcal{F} [S] := \left\{ U : U \in \mathcal{F},~ S \subset U \right\}, \] for $b= m^{\frac{1}{2}+ ε}$ with an $m$ sufficiently larger than a given constant $1/ε$. It is stronger than the statement that $\mathcal{F}$ includes a 3-sunflower if $|\mathcal{F}| > b^m$, where $k$-sunflower refers to a family of $k$ different sets with a common pair-wise intersection. Further refining the theory, we show that an $\mathcal{F}$ includes $k$ mutually disjoint sets if it satisfies the $Γ\left( 8^{\sqrt{\log_2 m}} \sqrt m ~k \log_2 k \right)$-condition with an $m$ sufficiently larger than $k$.
A sunflower with $k$ petals, or $k$-sunflower, is a family of $k$ sets every two of which have a common intersection. Known since 1960, the sunflower conjecture states that a family ${\mathcal F}$ of sets each of cardinality $m$ includes a $k$-sunflower if $|{\mathcal F}| \ge c_k^m$ for some $c_k \in {\mathbb R}_{>0}$ depending only on $k$. The case $k=3$ of the conjecture was especially emphasized by Erdös, for which Kostochka's bound $c m! \left( \frac{\log \log \log m}{\log \log m} \right)^m$ on $|{\mathcal F}|$ without a 3-sunflower had been the best-known since 1997 until the recent development to update it to $c \log m$. This paper proves with an entirely different combinatorial approach that ${\mathcal F}$ includes three mutually disjoint sets if it satisfies the $Γ\left( c m^{\frac{1}{2}+ δ} \right)$-condition for any given $δ\in (0, 1/2)$. Here $c$ is a constant depending only on $δ$, and the $Γ$-condition refers to \[ | \left\{ U~:~ U \in {\mathcal F} \textrm{~and~} S \subset U \right\}| < \left( c m^{\frac{1}{2}+ δ} \right)^{-|S|} |{\mathcal F}|, \] for every nonempty set $S$. This poses an alternative proof of the 3-sunflower bound $\left( c m^{\frac{1}{2}+ δ} \right)^m$.
We consider the selection problem on a completely connected network of $n$ processors with no shared memory. Each processor initially holds a given numeric item of $b$ bits allowed to send a message of $\max ( b, \lg n )$ bits to another processor at a time. On such a communication network ${\cal G}$, we show that the $k$th smallest of the $n$ inputs can be detected in $O ( \log n )$ time with $O ( n \log \log n )$ messages. The possibility of such a parallel algorithm for this distributed $k$-selection problem has been unknown despite the intensive investigation on many variations of the selection problem carried out since 1970s. The main trick of our algorithm is to simulate the comparisons and swaps performed by the AKS sorting network, the $n$-input sorting network of logarithmic depth discovered by Ajtai, Koml{ó}s and Szemer{é}di in 1983. We also show the universal time lower bound $\lg n$ for many basic data aggregation problems on ${\cal G}$, confirming the asymptotic time optimality of our parallel algorithm.
Consider an $m$-bit query $q$ to a bitwise trie $T$. A wildcard $*$ is an unspecified bit in $q$ for which the query asks the membership for both cases $*=0$ and $*=1$. It is common that such partial-match queries with wildcards are issued in tries. With uniformly random occurrences of $w$ wildcards in $q$ assumed, the obvious upper bound on the average number of traversal steps in $T$ is $2^w m$. We show that the average does not exceed \[ \frac{m+1}{w+1} \left( 2^{w+2} - 2 w - 4 \right) + m = O \left( \frac{2^w m}{w} \right), \] and equals the value exactly when $T$ includes all the $m$-bit keys as the worst case. Here the query $q$ performs with the naive backtracking algorithm in $T$. It is similarly shown that the average is $O \left( \frac{k^w m}{w} \right)$ in a general trie of maximum out-degree $k$. Our analysis for tries is extended to a distributed hash table (DHT), which is among the most frequently used decentralized data structures in networking. We show, under a natural probabilistic assumption for the largest class of DHTs, that the average number of hops required by an $m$-bit query $q$ to a DHT $D$ with random $w$ wildcards meets the same asymptotic bound. As a result, $q$ is answered with average $O \left( \frac{2^w m}{w} \right)$ hops rather than $Θ\left( 2^w m \right)$ in the four major DHTs Chord, Pastry, Tapestry and Kademlia. In addition, with a uniform key distribution for sufficiently many entries, we prove that a lookup request to the DHT Chord is answered correctly with $O(m)$ hops and probability $1 - 2^{-Ω(m)}$. To the author's knowledge, the probability $1 - 2^{-Ω(m)}$ of correct lookup in Chord has not been identified so far.
A sunflower with a core $Y$ is a family ${\cal B}$ of sets such that $U \cap U' = Y$ for each two different elements $U$ and $U'$ in ${\cal B}$. The well-known sunflower lemma states that a given family ${\cal F}$ of sets, each of cardinality at most $s$, includes a sunflower of cardinality $k$ if $|{\cal F}|> (k-1)^s s!$. Since Erdös and Rado proved it in 1960, it has not been known for more than half a century whether the sunflower bound $(k-1)^s s!$ can be improved asymptotically for any $k$ and $s$. It is conjectured that it can be reduced to $c_k^s$ for some real number $c_k>0$ depending only on $k$, which is called the sunflower conjecture. This paper shows that the general sunflower bound can be indeed reduced by an exponential factor: We prove that ${\cal F}$ includes a sunflower of cardinality $k$ if \[ |{\cal F|} \ge \left( \sqrt{10} -2 \right)^2 \left[ k \cdot \min \left( \frac{1}{\sqrt{10}-2}, \frac{c}{\log \min(k, s)} \right) \right]^s s!, \] for a constant $c>0$, and any $k \ge 2$ and $s \ge 2$. For instance, whenever $k \ge s^ε$ for a given constant $ε\in (0,1)$, the sunflower bound is reduced from $(k-1)^s s!$ to $(k-1)^s s! \cdot \left[ O \left( \frac{1}{\log s} \right) \right]^s$, achieving the reduction ratio of $\left[ O \left( \frac{1}{\log s} \right) \right]^s$. Also any ${\cal F}$ of cardinality at least $\left(\sqrt{10}-2 \right)^2 \left( \frac{k}{\sqrt{10}-2} \right)^s s!$ includes a sunflower of cardinality $k$, where $\frac{1}{\sqrt{10}-2}=0.8603796\ldots$. Our result demonstrates that the sunflower bound can be improved by a factor of less than a small constant to the power $s$, giving hope for further update.
The class P is in fact a proper sub-class of NP. We explore topological properties of the Hamming space 2^[n] where [n]={1, 2,..., n}. With the developed theory, we show: (i) a theorem that is closely related to Erdos and Rado's sunflower lemma, and claims a stronger statement in most cases, (ii) a new approach to prove the exponential monotone circuit complexity of the clique problem, (iii) NC \ne NP through the impossibility of a Boolean circuit with poly-log depth to compute cliques, based on the construction of (ii), and (iv) P \ne NP through the exponential circuit complexity of the clique problem, based on the construction of (iii). Item (i) leads to the existence of a sunflower with a small core in certain families of sets, which is not an obvious consequence of the sunflower lemma. In (iv), we show that any Boolean circuit computing the clique function CLIQUE_{n,k} (k=n^{1/4}) has a size exponential in n. Thus, we will separate P/poly from NP also. Razborov and Rudich showed strong evidence that no natural proof can prove exponential circuit complexity of a Boolean function. We confirm that the proofs for (iii) and (iv) are not natural.
In 1985, Razborov discovered a proof that the monotone circuit complexity of the clique problem is super-polynomial. Alon and Boppana improved the result into exponential lower bound exp(Ω(n / \log n)^{1/3})) of a monotone circuit C to compute cliques of size (1/4) (n / log n)^{2/3}, where n is the number of vertices in a graph. Both proofs are based on the method of approximations and Erdos and Rado's sunflower lemma. There has been an interest in further generalization of the proof scheme. In this paper, we present a new approach to show the exponential monotone complexity. Unlike the standard method, it dynamically constructs a counter example: Assuming a monotone circuit C of sub-exponential size to compute k-cliques c, an algorithm finds an edge set t containing no c in the disjunctive normal form constructed at the root of C. We call such t a shift. The proof shows that t is disjoint from an edge set z whose removal leaves no k-cliques. We explore the set theoretical nature of computation by Boolean circuits. We develop a theory by finding topological properties of the Hamming space 2^{[n]} where [n]={1, 2, ..., n}. A structural theorem is presented, which is closely related to the sunflower lemma and claims a stronger statement in most cases. The theory lays the foundation of the above shift method. It also shows the existence of a sunflower with small core in a family of sets, which is not an obvious consequence of the sunflower lemma. Lastly, we point out that the new methodology has potential to apply to a general circuit computing cliques due to the dynamic selection of t and z, and to improve the Alon-Boppana bound exp(Ω(n / \log n)^{1/3})).