Hypercontractive Inequality for Pseudo-Boolean Functions of Bounded Fourier Width

A function $f:\ \{-1,1\}^n\rightarrow \mathbb{R}$ is called pseudo-Boolean. It is well-known that each pseudo-Boolean function $f$ can be written as $f(x)=\sum_{I\in {\cal F}}\hat{f}(I)\chi_I(x),$ where ${\cal F}\subseteq \{I:\ I\subseteq [n]\}$,$[n]=\{1,2,...,n\}$, and$\chi_I(x)=\prod_{i\in I}x_i$and$\hat{f}(I)$are non-zero reals. The degree of$f$is$\max \{|I|:\ I\in {\cal F}\}$and the width of$f$is the minimum integer$\rho$such that every$i\in [n]$appears in at most$\rho$sets in$\cal F$. For$i\in [n]$, let$\mathbf{x}_i$be a random variable taking values 1 or -1 uniformly and independently from all other variables$\mathbf{x}_j$,$j\neq i.$Let$\mathbf{x}=(\mathbf{x}_1,...,\mathbf{x}_n)$. The$p$-norm of$f$is$||f||_p=(\mathbb E[|f(\mathbf{x})|^p])^{1/p}$for any$p\ge 1$. It is well-known that$||f||_q\ge ||f||_p$whenever$q> p\ge 1$. However, the higher norm can be bounded by the lower norm times a coefficient not directly depending on$f$: if$f$is of degree$d$and$q> p>1$then$ ||f||_q\le (\frac{q-1}{p-1})^{d/2}||f||_p.$This inequality is called the Hypercontractive Inequality. We show that one can replace$d$by$\rho$in the Hypercontractive Inequality for each$q> p\ge 2$as follows:$ ||f||_q\le ((2r)!\rho^{r-1})^{1/(2r)}||f||_p,$where$r=\lceil q/2\rceil$. For the case$q=4$and$p=2$, which is important in many applications, we prove a stronger inequality:$ ||f||_4\le (2\rho+1)^{1/4}||f||_2.

Transversals in 44-Uniform Hypergraphs

Let $H$ be a $3$-regular $4$-uniform hypergraph on $n$ vertices. The transversal number $\tau(H)$ of $H$ is the minimum number of vertices that intersect every edge. Lai and Chang [J. Combin. Theory Ser. B 50 (1990), 129--133] proved that $\tau(H) \le 7n/18$. Thomass\'{e} and Yeo [Combinatorica 27 (2007), 473--487] improved this bound and showed that $\tau(H) \le 8n/21$. We provide a further improvement and prove that $\tau(H) \le 3n/8$, which is best possible due to a hypergraph of order eight. More generally, we show that if $H$ is a $4$-uniform hypergraph on $n$ vertices and $m$ edges with maximum degree $\Delta(H) \le 3$, then $\tau(H) \le n/4 + m/6$, which proves a known conjecture. We show that an easy corollary of our main result is that the total domination number of a graph on $n$ vertices with minimum degree at least~4 is at most $3n/7$, which was the main result of the Thomass\'{e}-Yeo paper [Combinatorica 27 (2007), 473--487].Comment: 41 page

Kernels for Below-Upper-Bound Parameterizations of the Hitting Set and Directed Dominating Set Problems

In the {\sc Hitting Set} problem, we are given a collection $\cal F$ of subsets of a ground set $V$ and an integer $p$, and asked whether $V$ has a $p$-element subset that intersects each set in $\cal F$. We consider two parameterizations of {\sc Hitting Set} below tight upper bounds: $p=m-k$ and $p=n-k$. In both cases $k$ is the parameter. We prove that the first parameterization is fixed-parameter tractable, but has no polynomial kernel unless coNP$\subseteq$NP/poly. The second parameterization is W[1]-complete, but the introduction of an additional parameter, the degeneracy of the hypergraph $H=(V,{\cal F})$, makes the problem not only fixed-parameter tractable, but also one with a linear kernel. Here the degeneracy of $H=(V,{\cal F})$ is the minimum integer $d$ such that for each $X\subset V$ the hypergraph with vertex set $V\setminus X$ and edge set containing all edges of $\cal F$ without vertices in $X$, has a vertex of degree at most $d.$ In {\sc Nonblocker} ({\sc Directed Nonblocker}), we are given an undirected graph (a directed graph) $G$ on $n$ vertices and an integer $k$, and asked whether $G$ has a set $X$ of $n-k$ vertices such that for each vertex $y\not\in X$ there is an edge (arc) from a vertex in $X$ to $y$. {\sc Nonblocker} can be viewed as a special case of {\sc Directed Nonblocker} (replace an undirected graph by a symmetric digraph). Dehne et al. (Proc. SOFSEM 2006) proved that {\sc Nonblocker} has a linear-order kernel. We obtain a linear-order kernel for {\sc Directed Nonblocker}

Parameterized Rural Postman Problem

The Directed Rural Postman Problem (DRPP) can be formulated as follows: given a strongly connected directed multigraph $D=(V,A)$ with nonnegative integral weights on the arcs, a subset $R$ of $A$ and a nonnegative integer $\ell$, decide whether $D$ has a closed directed walk containing every arc of $R$ and of total weight at most $\ell$. Let $k$ be the number of weakly connected components in the the subgraph of $D$ induced by $R$. Sorge et al. (2012) ask whether the DRPP is fixed-parameter tractable (FPT) when parameterized by $k$, i.e., whether there is an algorithm of running time $O^*(f(k))$ where $f$ is a function of $k$ only and the $O^*$ notation suppresses polynomial factors. Sorge et al. (2012) note that this question is of significant practical relevance and has been open for more than thirty years. Using an algebraic approach, we prove that DRPP has a randomized algorithm of running time $O^*(2^k)$ when $\ell$ is bounded by a polynomial in the number of vertices in $D$. We also show that the same result holds for the undirected version of DRPP, where $D$ is a connected undirected multigraph

Out-degree reducing partitions of digraphs

Let $k$ be a fixed integer. We determine the complexity of finding a $p$-partition $(V_1, \dots, V_p)$ of the vertex set of a given digraph such that the maximum out-degree of each of the digraphs induced by $V_i$, ($1\leq i\leq p$) is at least $k$ smaller than the maximum out-degree of $D$. We show that this problem is polynomial-time solvable when $p\geq 2k$ and ${\cal NP}$-complete otherwise. The result for $k=1$ and $p=2$ answers a question posed in \cite{bangTCS636}. We also determine, for all fixed non-negative integers $k_1,k_2,p$, the complexity of deciding whether a given digraph of maximum out-degree $p$ has a $2$-partition $(V_1,V_2)$ such that the digraph induced by $V_i$ has maximum out-degree at most $k_i$ for $i\in [2]$. It follows from this characterization that the problem of deciding whether a digraph has a 2-partition $(V_1,V_2)$ such that each vertex $v\in V_i$ has at least as many neighbours in the set $V_{3-i}$ as in $V_i$, for $i=1,2$ is ${\cal NP}$-complete. This solves a problem from \cite{kreutzerEJC24} on majority colourings.Comment: 11 pages, 1 figur