(2/2/3)(2/2/3)-SAT problem and its applications in dominating set problems

The satisfiability problem is known to be $\mathbf{NP}$-complete in general and for many restricted cases. One way to restrict instances of $k$-SAT is to limit the number of times a variable can be occurred. It was shown that for an instance of 4-SAT with the property that every variable appears in exactly 4 clauses (2 times negated and 2 times not negated), determining whether there is an assignment for variables such that every clause contains exactly two true variables and two false variables is $\mathbf{NP}$-complete. In this work, we show that deciding the satisfiability of 3-SAT with the property that every variable appears in exactly four clauses (two times negated and two times not negated), and each clause contains at least two distinct variables is $\mathbf{NP}$-complete. We call this problem $(2/2/3)$-SAT. For an $r$-regular graph $G = (V,E)$ with $r\geq 3$, it was asked in [Discrete Appl. Math., 160(15):2142--2146, 2012] to determine whether for a given independent set $T$ there is an independent dominating set $D$ that dominates $T$ such that $T \cap D =\varnothing$? As an application of $(2/2/3)$-SAT problem we show that for every $r\geq 3$, this problem is $\mathbf{NP}$-complete. Among other results, we study the relationship between 1-perfect codes and the incidence coloring of graphs and as another application of our complexity results, we prove that for a given cubic graph $G$ deciding whether $G$ is 4-incidence colorable is $\mathbf{NP}$-complete

Upper bounds for the 2-hued chromatic number of graphs in terms of the independence number

A 2-hued coloring of a graph $G$ (also known as conditional $(k, 2)$-coloring and dynamic coloring) is a coloring such that for every vertex $v\in V(G)$ of degree at least $2$, the neighbors of $v$ receive at least $2$ colors. The smallest integer $k$ such that $G$ has a 2-hued coloring with $k$ colors, is called the {\it 2-hued chromatic number} of $G$ and denoted by $\chi_2(G)$. In this paper, we will show that if $G$ is a regular graph, then $\chi_{2}(G)- \chi(G) \leq 2 \log _{2}(\alpha(G)) +\mathcal{O}(1)$ and if $G$ is a graph and $\delta(G)\geq 2$, then $\chi_{2}(G)- \chi(G) \leq 1+\lceil \sqrt[\delta -1]{4\Delta^{2}} \rceil ( 1+ \log _{\frac{2\Delta(G)}{2\Delta(G)-\delta(G)}} (\alpha(G)) )$ and in general case if $G$ is a graph, then $\chi_{2}(G)- \chi(G) \leq 2+ \min \lbrace \alpha^{\prime}(G),\frac{\alpha(G)+\omega(G)}{2}\rbrace$.Comment: Dynamic chromatic number; conditional (k, 2)-coloring; 2-hued chromatic number; 2-hued coloring; Independence number; Probabilistic metho

The inapproximability for the (0,1)-additive number

An {\it additive labeling} of a graph $G$ is a function $\ell :V(G) \rightarrow\mathbb{N}$, such that for every two adjacent vertices $v$ and $u$ of $G$, $\sum_{w \sim v}\ell(w)\neq \sum_{w \sim u}\ell(w)$ ($x \sim y$ means that $x$ is joined to $y$). The {\it additive number} of $G$, denoted by $\eta(G)$, is the minimum number $k$ such that $G$ has a additive labeling $\ell :V(G) \rightarrow \mathbb{N}_k$. The {\it additive choosability} of a graph $G$, denoted by $\eta_{\ell}(G)$, is the smallest number $k$ such that $G$ has an additive labeling for any assignment of lists of size $k$ to the vertices of $G$, such that the label of each vertex belongs to its own list. Seamone (2012) \cite{a80} conjectured that for every graph $G$, $\eta(G)= \eta_{\ell}(G)$. We give a negative answer to this conjecture and we show that for every $k$ there is a graph $G$ such that $\eta_{\ell}(G)- \eta(G) \geq k$. A {\it $(0,1)$-additive labeling} of a graph $G$ is a function $\ell :V(G) \rightarrow\{0,1\}$, such that for every two adjacent vertices $v$ and $u$ of $G$, $\sum_{w \sim v}\ell(w)\neq \sum_{w \sim u}\ell(w)$. A graph may lack any $(0,1)$-additive labeling. We show that it is $\mathbf{NP}$-complete to decide whether a $(0,1)$-additive labeling exists for some families of graphs such as perfect graphs and planar triangle-free graphs. For a graph $G$ with some $(0,1)$-additive labelings, the $(0,1)$-additive number of $G$ is defined as $\sigma_{1} (G) = \min_{\ell \in \Gamma}\sum_{v\in V(G)}\ell(v)$ where $\Gamma$ is the set of $(0,1)$-additive labelings of $G$. We prove that given a planar graph that admits a $(0,1)$-additive labeling, for all $\varepsilon >0$, approximating the $(0,1)$-additive number within $n^{1-\varepsilon}$ is $\mathbf{NP}$-hard.Comment: 14 pages, 3 figures, Discrete Mathematics & Theoretical Computer Scienc

Sigma Partitioning: Complexity and Random Graphs

A $\textit{sigma partitioning}$ of a graph $G$ is a partition of the vertices into sets $P_1, \ldots, P_k$ such that for every two adjacent vertices $u$ and $v$ there is an index $i$ such that $u$ and $v$ have different numbers of neighbors in $P_i$. The $\textit{ sigma number}$ of a graph $G$, denoted by $\sigma(G)$, is the minimum number $k$ such that $G$ has a sigma partitioning $P_1, \ldots, P_k$. Also, a $\textit{ lucky labeling}$ of a graph $G$ is a function $\ell :V(G) \rightarrow \mathbb{N}$, such that for every two adjacent vertices $v$ and $u$ of $G$, $\sum_{w \sim v}\ell(w)\neq \sum_{w \sim u}\ell(w)$ ($x \sim y$ means that $x$ and $y$ are adjacent). The $\textit{ lucky number}$ of $G$, denoted by $\eta(G)$, is the minimum number $k$ such that $G$ has a lucky labeling $\ell :V(G) \rightarrow \mathbb{N}_k$. It was conjectured in [Inform. Process. Lett., 112(4):109--112, 2012] that it is $\mathbf{NP}$-complete to decide whether $\eta(G)=2$ for a given 3-regular graph $G$. In this work, we prove this conjecture. Among other results, we give an upper bound of five for the sigma number of a uniformly random graph

Neural Approximate Dynamic Programming for the Ultra-fast Order Dispatching Problem

Same-Day Delivery (SDD) services aim to maximize the fulfillment of online orders while minimizing delivery delays but are beset by operational uncertainties such as those in order volumes and courier planning. Our work aims to enhance the operational efficiency of SDD by focusing on the ultra-fast Order Dispatching Problem (ODP), which involves matching and dispatching orders to couriers within a centralized warehouse setting, and completing the delivery within a strict timeline (e.g., within minutes). We introduce important extensions to ultra-fast ODP such as order batching and explicit courier assignments to provide a more realistic representation of dispatching operations and improve delivery efficiency. As a solution method, we primarily focus on NeurADP, a methodology that combines Approximate Dynamic Programming (ADP) and Deep Reinforcement Learning (DRL), and our work constitutes the first application of NeurADP outside of the ride-pool matching problem. NeurADP is particularly suitable for ultra-fast ODP as it addresses complex one-to-many matching and routing intricacies through a neural network-based VFA that captures high-dimensional problem dynamics without requiring manual feature engineering as in generic ADP methods. We test our proposed approach using four distinct realistic datasets tailored for ODP and compare the performance of NeurADP against myopic and DRL baselines by also making use of non-trivial bounds to assess the quality of the policies. Our numerical results indicate that the inclusion of order batching and courier queues enhances the efficiency of delivery operations and that NeurADP significantly outperforms other methods. Detailed sensitivity analysis with important parameters confirms the robustness of NeurADP under different scenarios, including variations in courier numbers, spatial setup, vehicle capacity, and permitted delay time