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The focus of this paper is to increase our understanding of the Longest Processing Time First (LPT) heuristic. LPT is a classical heuristic for the fundamental problem of uniform machine scheduling. For different machine speeds, LPT was first considered by Gonzalez et al (SIAM J. Computing, 1977). Since then, extensive work has been done to improve the approximation factor of the LPT heuristic. However, all known implementations of the LPT heuristic take $O(mn)$ time, where $m$ is the number of machines and $n$ is the number of jobs. In this work, we come up with the first near-linear time implementation for LPT. Specifically, the running time is $O((n+m)(\log^2{m}+\log{n}))$. Somewhat surprisingly, the result is obtained by mapping the problem to dynamic maintenance of lower envelope of lines, which has been well studied in the computational geometry community. Our second contribution is to analyze the performance of LPT for the Drones Warehouse Problem (DWP), which is a natural generalization of the uniform machine scheduling problem motivated by drone-based parcel delivery from a warehouse. In this problem, a warehouse has multiple drones and wants to deliver parcels to several customers. Each drone picks a parcel from the warehouse, delivers it, and returns to the warehouse (where it can also get charged). The speeds and battery lives of the drones could be different, and due to the limited battery life, each drone has a bounded range in which it can deliver parcels. The goal is to assign parcels to the drones so that the time taken to deliver all the parcels is minimized. We prove that the natural approach of solving this problem via the LPT heuristic has an approximation factor of $\phi$, where $\phi \approx 1.62$ is the golden ratio., Comment: To appear in FSTTCS 2024

Greenberger-Horne-Zeilinger (GHZ) states are quantum states involving at least three entangled particles. They are of fundamental interest in quantum information theory, and the construction of such states of high dimension has various applications in quantum communication and cryptography. They are of fundamental interest in quantum information theory, and the construction of such states of high dimension has various applications in quantum communication and cryptography. Krenn, Gu and Zeilinger discovered a correspondence between a large class of quantum optical experiments which produce GHZ states and edge-weighted edge-coloured multi-graphs with some special properties called the \emph{GHZ graphs}. On such GHZ graphs, a graph parameter called \emph{dimension} can be defined, which is the same as the dimension of the GHZ state produced by the corresponding experiment. Krenn and Gu conjectured that the dimension of any GHZ graph with more than $4$ vertices is at most $2$. An affirmative resolution of the Krenn-Gu conjecture has implications for quantum resource theory. On the other hand, the construction of a GHZ graph on a large number of vertices with a high dimension would lead to breakthrough results. In this paper, we study the existence of GHZ graphs from the perspective of the Krenn-Gu conjecture and show that the conjecture is true for graphs of vertex connectivity at most 2 and for cubic graphs. We also show that the minimal counterexample to the conjecture should be $4$-connected. Such information could be of great help in the search for GHZ graphs using existing tools like PyTheus. While the impact of the work is in quantum physics, the techniques in this paper are purely combinatorial, and no background in quantum physics is required to understand them., Comment: To appear in MFCS 2024

A class of graphs $\mathcal{G}$ is $\chi$-bounded if there exists a function $f$ such that $\chi(G) \leq f(\omega(G))$ for each graph $G \in \mathcal{G}$, where $\chi(G)$ and $\omega(G)$ are the chromatic and clique number of $G$, respectively. The square of a graph $G$, denoted as $G^2$, is the graph with the same vertex set as $G$ in which two vertices are adjacent when they are at a distance at most two in $G$. In this paper, we study the $\chi$-boundedness of squares of bipartite graphs and its subclasses. Note that the class of squares of graphs, in general, admit a quadratic $\chi$-binding function. Moreover there exist bipartite graphs $B$ for which $\chi\left(B^2\right)$ is $\Omega\left(\frac{\left(\omega\left(B^2\right)\right)^2 }{\log \omega\left(B^2\right)}\right)$. We first ask the following question: "What sub-classes of bipartite graphs have a linear $\chi$-binding function?" We focus on the class of convex bipartite graphs and prove the following result: for any convex bipartite graph $G$, $\chi\left(G^2\right) \leq \frac{3 \omega\left(G^2\right)}{2}$. Our proof also yields a polynomial-time $3/2$-approximation algorithm for coloring squares of convex bipartite graphs. We then introduce a notion called "partite testable properties" for the squares of bipartite graphs. We say that a graph property $P$ is partite testable for the squares of bipartite graphs if for a bipartite graph $G=(A,B,E)$, whenever the induced subgraphs $G^2[A]$ and $G^2[B]$ satisfies the property $P$ then $G^2$ also satisfies the property $P$. Here, we discuss whether some of the well-known graph properties like perfectness, chordality, (anti-hole)-freeness, etc. are partite testable or not. As a consequence, we prove that the squares of biconvex bipartite graphs are perfect., Comment: 22 pages, 5 figures

Domination and coloring are two classic problems in graph theory. The major focus of this paper is the CD-COLORING problem which combines the flavours of domination and colouring. Let $G$ be an undirected graph. A proper vertex coloring of $G$ is a $cd-coloring$ if each color class has a dominating vertex in $G$. The minimum integer $k$ for which there exists a $cd-coloring$ of $G$ using $k$ colors is called the cd-chromatic number, $\chi_{cd}(G)$. A set $S\subseteq V(G)$ is a total dominating set if any vertex in $G$ has a neighbor in $S$. The total domination number, $\gamma_t(G)$ of $G$ is the minimum integer $k$ such that $G$ has a total dominating set of size $k$. A set $S\subseteq V(G)$ is a $separated-cluster$ if no two vertices in $S$ lie at a distance 2 in $G$. The separated-cluster number, $\omega_s(G)$, of $G$ is the maximum integer $k$ such that $G$ has a separated-cluster of size $k$. In this paper, first we explore the connection between CD-COLORING and TOTAL DOMINATION. We prove that CD-COLORING and TOTAL DOMINATION are NP-Complete on triangle-free $d$-regular graphs for each fixed integer $d\geq 3$. We also study the relationship between the parameters $\chi_{cd}(G)$ and $\omega_s(G)$. Analogous to the well-known notion of `perfectness', here we introduce the notion of `cd-perfectness'. We prove a sufficient condition for a graph $G$ to be cd-perfect (i.e. $\chi_{cd}(H)= \omega_s(H)$, for any induced subgraph $H$ of $G$) which is also necessary for certain graph classes (like triangle-free graphs). Here, we propose a generalized framework via which we obtain several exciting consequences in the algorithmic complexities of special graph classes. In addition, we settle an open problem by showing that the SEPARATED-CLUSTER is polynomially solvable for interval graphs.

The most efficient automated way to construct a large class of quantum photonic experiments is via abstract representation of graphs with certain properties. While new directions were explored using Artificial intelligence and SAT solvers to find such graphs, it becomes computationally infeasible to do so as the size of the graph increases. So, we take an analytical approach and introduce the technique of local sparsification on experiment graphs, using which we answer a crucial open question in experimental quantum optics, namely whether certain complex entangled quantum states can be constructed. This provides us with more insights into quantum resource theory, the limitation of specific quantum photonic systems and initiates the use of graph-theoretic techniques for designing quantum physics experiments., Comment: 18 pages. Accepted in Quantum Journal

In this paper, we study the computational complexity of \textsc{$s$-Club Cluster Vertex Deletion}. Given a graph, \textsc{$s$-Club Cluster Vertex Deletion ($s$-CVD)} aims to delete the minimum number of vertices from the graph so that each connected component of the resulting graph has a diameter at most $s$. When $s=1$, the corresponding problem is popularly known as \sloppy \textsc{Cluster Vertex Deletion (CVD)}. We provide a faster algorithm for \textsc{$s$-CVD} on \emph{interval graphs}. For each $s\geq 1$, we give an $O(n(n+m))$-time algorithm for \textsc{$s$-CVD} on interval graphs with $n$ vertices and $m$ edges. In the case of $s=1$, our algorithm is a slight improvement over the $O(n^3)$-time algorithm of Cao \etal (Theor. Comput. Sci., 2018) and for $s \geq 2$, it significantly improves the state-of-the-art running time $\left(O\left(n^4\right)\right)$. We also give a polynomial-time algorithm to solve \textsc{CVD} on \emph{well-partitioned chordal graphs}, a graph class introduced by Ahn \etal (\textsc{WG 2020}) as a tool for narrowing down complexity gaps for problems that are hard on chordal graphs, and easy on split graphs. Our algorithm relies on a characterisation of the optimal solution and on solving polynomially many instances of the \textsc{Weighted Bipartite Vertex Cover}. This generalises a result of Cao \etal (Theor. Comput. Sci., 2018) on split graphs. We also show that for any even integer $s\geq 2$, \textsc{$s$-CVD} is NP-hard on well-partitioned chordal graphs.

A list assignment $L$ of a graph $G$ is a function that assigns to every vertex $v$ of $G$ a set $L(v)$ of colors. A proper coloring $\alpha$ of $G$ is called an $L$-coloring of $G$ if $\alpha(v)\in L(v)$ for every $v\in V(G)$. For a list assignment $L$ of $G$, the $L$-recoloring graph $\mathcal{G}(G,L)$ of $G$ is a graph whose vertices correspond to the $L$-colorings of $G$ and two vertices of $\mathcal{G}(G,L)$ are adjacent if their corresponding $L$-colorings differ at exactly one vertex of $G$. A $d$-face in a plane graph is a face of length $d$. Dvo\v{r}\'ak and Feghali conjectured for a planar graph $G$ and a list assignment $L$ of $G$, that: (i) If $|L(v)|\geq 10$ for every $v\in V(G)$, then the diameter of $\mathcal{G}(G,L)$ is $O(|V(G)|)$. (ii) If $G$ is triangle-free and $|L(v)|\geq 7$ for every $v\in V(G)$, then the diameter of $\mathcal{G}(G,L)$ is $O(|V(G)|)$. In a recent paper, Cranston (European J. Combin. (2022)) has proved (ii). In this paper, we prove the following results. Let $G$ be a plane graph and $L$ be a list assignment of $G$. $\bullet$ If for every $3$-face of $G$, there are at most two $3$-faces adjacent to it and $|L(v)|\geq 10$ for every $v\in V(G)$, then the diameter of $\mathcal{G}(G,L)$ is at most $190|V(G)|$. $\bullet$ If for every $3$-face of $G$, there is at most one $3$-face adjacent to it and $|L(v)|\geq 9$ for every $v\in V(G)$, then the diameter of $\mathcal{G}(G,L)$ is at most $13|V(G)|$. $\bullet$ If the faces adjacent to any $3$-face have length at least $6$ and $|L(v)|\geq 7$ for every $v\in V(G)$, then the diameter of $\mathcal{G}(G,L)$ is at most $242|V(G)|$. This result strengthens the Cranston's result on (ii).

Krenn, Gu and Zeilinger initiated the study of PMValid edge-colourings because of its connection to a problem from quantum physics. A graph is defined to have a PMValid $k$-edge-colouring if it admits a $k$-edge-colouring (i.e. an edge colouring with $k$-colours) with the property that all perfect matchings are monochromatic and each of the $k$ colour classes contain at least one perfect matching. The matching index of a graph $G$, $\mu(G)$ is defined as the maximum value of $k$ for which $G$ admits a PMValid $k$-edge-colouring. It is easy to see that $\mu(G)\geq 1$ if and only if $G$ has a perfect matching (due to the trivial $1$-edge-colouring which is PMValid). Bogdanov observed that for all graphs non-isomorphic to $K_4$, $\mu(G)\leq 2$ and $\mu(K_4)=3$. However, the characterisation of graphs for which $\mu(G)=1$ and $\mu(G)=2$ is not known. In this work, we answer this question. Using this characterisation, we also give a fast algorithm to compute $\mu(G)$ of a graph $G$. In view of our work, the structure of PMValid $k$-edge-colourable graphs is now fully understood for all $k$. Our characterisation, also has an implication to the aforementioned quantum physics problem. In particular, it settles a conjecture of Krenn and Gu for a sub-class of graphs., Comment: 18 pages and 7 figures

Given a finite family $\mathcal{F}$ of graphs, we say that a graph $G$ is "$\mathcal{F}$-free" if $G$ does not contain any graph in $\mathcal{F}$ as a subgraph. A vertex-colored graph $H$ is called "rainbow" if no two vertices of $H$ have the same color. Given an integer $s$ and a finite family of graphs $\mathcal{F}$, let $\ell(s,\mathcal{F})$ denote the smallest integer such that any properly vertex-colored $\mathcal{F}$-free graph $G$ having $\chi(G)\geq\ell(s,\mathcal{F})$ contains an induced rainbow path on $s$ vertices. Scott and Seymour showed that $\ell(s,K)$ exists for every complete graph $K$. A conjecture of N. R. Aravind states that $\ell(s,C_3)=s$. The upper bound on $\ell(s,C_3)$ that can be obtained using the methods of Scott and Seymour setting $K=C_3$ are, however, super-exponential. Gy\'arf\'as and S\'ark\"ozy showed that $\ell(s,\{C_3,C_4\})=\mathcal{O}\big((2s)^{2s}\big)$. For $r\geq 2$, we show that $\ell(s,K_{2,r})\leq (r-1)(s-1)(s-2)/2+s$ and therefore, $\ell(s,C_4)\leq\frac{s^2-s+2}{2}$. This significantly improves Gy\'arf\'as and S\'ark\"ozy's bound and also covers a bigger class of graphs. We adapt our proof to achieve much stronger upper bounds for graphs of higher girth: we prove that $\ell(s,\{C_3,C_4,\ldots,C_{g-1}\})\leq s^{1+\frac{4}{g-4}}$, where $g\geq 5$. Moreover, in each case, our results imply the existence of at least $s!/2$ distinct induced rainbow paths on $s$ vertices. Along the way, we obtain some results on related problems on oriented graphs. For $r\geq 2$, let $\mathcal{B}_r$ denote the orientations of $K_{2,r}$ in which one vertex has out-degree or in-degree $r$. We show that every $\mathcal{B}_r$-free oriented graph $G$ having $\chi(G)\geq (r-1)(s-1)(s-2)+2s+1$ and every bikernel-perfect oriented graph $G$ with girth $g\geq 5$ having $\chi(G)\geq 2s^{1+\frac{4}{g-4}}$ contains every $s$ vertex oriented tree as an induced subgraph., Comment: 21 pages, 1 figure