A $2$-distance $k$-coloring of a graph is a proper $k$-coloring of the vertices where vertices at distance at most 2 cannot share the same color. We prove the existence of a $2$-distance ($\Delta+2$)-coloring for graphs with maximum average degree less than $\frac{8}{3}$ (resp. $\frac{14}{5}$) and maximum degree $\Delta\geq 6$ (resp. $\Delta\geq 10$). As a corollary, every planar graph with girth at least $8$ (resp. $7$) and maximum degree $\Delta\geq 6$ (resp. $\Delta\geq 10$) admits a $2$-distance $(\Delta+2)$-coloring.
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We consider the algorithmic complexity of recognizing bipartite temporal graphs. Rather than defining these graphs solely by their underlying graph or individual layers, we define a bipartite temporal graph as one in which every layer can be 2-colored in a way that results in few changes between any two consecutive layers. This approach follows the framework of multistage problems that has received a growing amount of attention in recent years. We investigate the complexity of recognizing these graphs. We show that this problem is NP-hard even if there are only two layers or if only one change is allowed between consecutive layers. We consider the parameterized complexity of the problem with respect to several structural graph parameters, which we transfer from the static to the temporal setting in three different ways. Finally, we consider a version of the problem in which we only restrict the total number of changes throughout the lifetime of the graph. We show that this variant is fixed-parameter tractable with respect to the number of changes.
Graph associahedra are generalized permutohedra arising as special cases of nestohedra and hypergraphic polytopes. The graph associahedron of a graph $G$ encodes the combinatorics of search trees on $G$, defined recursively by a root $r$ together with search trees on each of the connected components of $G-r$. In particular, the skeleton of the graph associahedron is the rotation graph of those search trees. We investigate the diameter of graph associahedra as a function of some graph parameters. We give a tight bound of $\Theta(m)$ on the diameter of trivially perfect graph associahedra on $m$ edges. We consider the maximum diameter of associahedra of graphs on $n$ vertices and of given tree-depth, treewidth, or pathwidth, and give lower and upper bounds as a function of these parameters. We also prove that the maximum diameter of associahedra of graphs of pathwidth two is $\Theta (n\log n)$. Finally, we give the exact diameter of the associahedra of complete split and of unbalanced complete bipartite graphs.
Let $G$ be a graph with vertex set $V$. {Two disjoint sets $V_1, V_2 \subseteq V$ form a coalition in $G$ if none of them is a dominating set of $G$ but their union $V_1\cup V_2$ is. A vertex partition $\Psi=\{V_1,\ldots, V_k\}$ of $V$ is called a coalition partition of $G$ if every set~$V_i\in \Psi$ is either a dominating set of $G$ with the cardinality $|V_i|=1$, or is not a dominating set but for some $V_j\in \Psi$, $V_i$ and $V_j$ form a coalition.} The maximum cardinality of a coalition partition of $G$ is called the coalition number of $G$, denoted by $\mathcal{C}(G)$. A $\mathcal{C}(G)$-partition is a coalition partition of $G$ with cardinality $\mathcal{C}(G)$. Given a coalition partition $\Psi=\{V_1, V_2,\ldots, V_r\}$ of~$G$, a coalition graph $CG(G, \Psi)$ is associated on $\Psi$ such that there is a one-to-one correspondence between its vertices and the members of $\Psi$. Two vertices of $CG(G, \Psi)$ are adjacent if and only if the corresponding sets form a coalition in $G$. In this paper, we first show that for any graph $G$ with $\delta(G)=1$, $\mathcal{C}(G)\leq 2\Delta(G)+2$, where $\delta(G)$ and $\Delta(G)$ are the minimum degree and the maximum degree of $G$, respectively. Moreover, we characterize all graphs~$G$ with $\delta(G)\leq 1$ and $\mathcal{C}(G)=n$, where $n$ is the number of vertices of $G$. Furthermore, we characterize all trees $T$ with $\mathcal{C}(T)=n$ and all trees $T$ with $\mathcal{C}(T)=n-1$. This solves partially one of the open problem posed in~\cite{coal0}. On the other hand, we theoretically and empirically determine the number of coalition graphs that can be defined by all coalition partitions of a given path $P_k$. Furthermore, we show that there is no universal coalition path, a path whose coalition partitions defines all possible coalition graphs. These solve two open problems posed by Haynes et al.~\cite{coal1}.
We study the problem of estimating the size of maximum matching and minimum vertex cover in sublinear time. Denoting the number of vertices by $n$ and the average degree in the graph by $\bar{d}$, we obtain the following results for both problems: * A multiplicative $(2+\epsilon)$-approximation that takes $\tilde{O}(n/\epsilon^2)$ time using adjacency list queries. * A multiplicative-additive $(2, \epsilon n)$-approximation in $\tilde{O}((\bar{d} + 1)/\epsilon^2)$ time using adjacency list queries. * A multiplicative-additive $(2, \epsilon n)$-approximation in $\tilde{O}(n/\epsilon^{3})$ time using adjacency matrix queries. All three results are provably time-optimal up to polylogarithmic factors culminating a long line of work on these problems. Our main contribution and the key ingredient leading to the bounds above is a new and near-tight analysis of the average query complexity of the randomized greedy maximal matching algorithm which improves upon a seminal result of Yoshida, Yamamoto, and Ito [STOC'09].
We show that the natural Glauber dynamics mixes rapidly and generates a random proper edge-coloring of a graph with maximum degree $\Delta$ whenever the number of colors is at least $q\geq (\frac{10}{3} + \epsilon)\Delta$, where $\epsilon>0$ is arbitrary and the maximum degree satisfies $\Delta \geq C$ for a constant $C = C(\epsilon)$ depending only on $\epsilon$. For edge-colorings, this improves upon prior work \cite{Vig99, CDMPP19} which show rapid mixing when $q\geq (\frac{11}{3}-\epsilon_0 ) \Delta$, where $\epsilon_0 \approx 10^{-5}$ is a small fixed constant. At the heart of our proof, we establish a matrix trickle-down theorem, generalizing Oppenheim's influential result, as a new technique to prove that a high dimensional simplical complex is a local spectral expander.
The multi-commodity flow-cut gap is a fundamental parameter that affects the performance of several divide \& conquer algorithms, and has been extensively studied for various classes of undirected graphs. It has been shown by Linial, London and Rabinovich and by Aumann and Rabani that for general $n$-vertex graphs it is bounded by $O(\log n)$ and the Gupta-Newman-Rabinovich-Sinclair conjecture asserts that it is $O(1)$ for any family of graphs that excludes some fixed minor. We show that the multicommodity flow-cut gap on \emph{directed} planar graphs is $O(\log^3 n)$. This is the first \emph{sub-polynomial} bound for any family of directed graphs of super-constant treewidth. We remark that for general directed graphs, it has been shown by Chuzhoy and Khanna that the gap is $\widetilde{\Omega}(n^{1/7})$, even for directed acyclic graphs. As a direct consequence of our result, we also obtain the first polynomial-time polylogarithmic-approximation algorithms for the Directed Non-Bipartite Sparsest-Cut, and the Directed Multicut problems for directed planar graphs, which extends the long-standing result for undirectd planar graphs by Rao (with a slightly weaker bound). At the heart of our result we investigate low-distortion quasimetric embeddings into \emph{directed} $\ell_1$. More precisely, we construct $O(\log^2 n)$-Lipschitz quasipartitions for the shortest-path quasimetric spaces of planar digraphs, which generalize the notion of Lipschitz partitions from the theory of metric embeddings. This construction combines ideas from the theory of bi-Lipschitz embeddings, with tools form data structures on directed planar graphs.
The metric dimension dim(G) of a graph $G$ is the minimum cardinality of a subset $S$ of vertices of $G$ such that each vertex of $G$ is uniquely determined by its distances to $S$. It is well-known that the metric dimension of a graph can be drastically increased by the modification of a single edge. Our main result consists in proving that the increase of the metric dimension of an edge addition can be amortized in the sense that if the graph consists of a spanning tree $T$ plus $c$ edges, then the metric dimension of $G$ is at most the metric dimension of $T$ plus $6c$. We then use this result to prove a weakening of a conjecture of Eroh et al. The zero forcing number $Z(G)$ of $G$ is the minimum cardinality of a subset $S$ of black vertices (whereas the other vertices are colored white) of $G$ such that all the vertices will turned black after applying finitely many times the following rule: a white vertex is turned black if it is the only white neighbor of a black vertex. Eroh et al. conjectured that, for any graph $G$, $dim(G)\leq Z(G) + c(G)$, where $c(G)$ is the number of edges that have to be removed from $G$ to get a forest. They proved the conjecture is true for trees and unicyclic graphs. We prove a weaker version of the conjecture: $dim(G)\leq Z(G)+6c(G)$ holds for any graph. We also prove that the conjecture is true for graphs with edge disjoint cycles, widely generalizing the unicyclic result of Eroh et al.
In the Non-Uniform $k$-Center problem, a generalization of the famous $k$-center clustering problem, we want to cover the given set of points in a metric space by finding a placement of balls with specified radii. In $t$-NU$k$C Problem, we assume that the number of distinct radii is equal to $t$, and we are allowed to use $k_i$ balls of radius $r_i$, for $1 \le i \le t$. This problem was introduced by Chakrabarty et al. [ACM Trans. Alg. 16(4):46:1-46:19], who showed that a constant approximation for $t$-NU$k$C is not possible if $t$ is unbounded. On the other hand, they gave a bicriteria approximation that violates the number of allowed balls as well as the given radii by a constant factor. They also conjectured that a constant approximation for $t$-NU$k$C should be possible if $t$ is a fixed constant. Since then, there has been steady progress towards resolving this conjecture -- currently, a constant approximation for $3$-NU$k$C is known via the results of Chakrabarty and Negahbani [IPCO 2021], and Jia et al. [To appear in SOSA 2022]. We push the horizon by giving an $O(1)$-approximation for the Non-Uniform $k$-Center for $4$ distinct types of radii. Our result is obtained via a novel combination of tools and techniques from the $k$-center literature, which also demonstrates that the different generalizations of $k$-center involving non-uniform radii, and multiple coverage constraints (i.e., colorful $k$-center), are closely interlinked with each other. We hope that our ideas will contribute towards a deeper understanding of the $t$-NU$k$C problem, eventually bringing us closer to the resolution of the CGK conjecture.
We revisit the (block-angular) min-max resource sharing problem, which is a well-known generalization of fractional packing and the maximum concurrent flow problem. It consists of finding an $\ell_{\infty}$-minimal element in a Minkowski sum $\mathcal{X}= \sum_{C \in \mathcal{C}} X_C$ of non-empty closed convex sets $X_C \subseteq \mathbb{R}^{\mathcal{R}}_{\geq 0}$, where $\mathcal{C}$ and $\mathcal{R}$ are finite sets. We assume that an oracle for approximate linear minimization over $X_C$ is given. In this setting, the currently fastest known FPTAS is due to M\"uller, Radke, and Vygen. For $\delta \in (0,1]$, it computes a $\sigma(1+\delta)$-approximately optimal solution using $\mathcal{O}((|\mathcal{C}|+|\mathcal{R}|)\log |\mathcal{R}| (\delta^{-2} + \log \log |\mathcal{R}|))$ oracle calls, where $\sigma$ is the approximation ratio of the oracle. We describe an extension of their algorithm and improve on previous results in various ways. Our FPTAS, which, as previous approaches, is based on the multiplicative weight update method, computes close to optimal primal and dual solutions using $\mathcal{O}\left(\frac{|\mathcal{C}|+ |\mathcal{R}|}{\delta^2} \log |\mathcal{R}|\right)$ oracle calls. We prove that our running time is optimal under certain assumptions, implying that no warm-start analysis of the algorithm is possible. A major novelty of our analysis is the concept of local weak duality, which illustrates that the algorithm optimizes (close to) independent parts of the instance separately. Interestingly, this implies that the computed solution is not only approximately $\ell_{\infty}$-minimal, but among such solutions, also its second-highest entry is approximately minimal. We prove that this statement cannot be extended to the third-highest entry.
Given a graph $G$ of degree $k$ over $n$ vertices, we consider the problem of computing a near maximum cut or a near minimum bisection in polynomial time. For graphs of girth $L$, we develop a local message passing algorithm whose complexity is $O(nkL)$, and that achieves near optimal cut values among all $L$-local algorithms. Focusing on max-cut, the algorithm constructs a cut of value $nk/4+ n\mathsf{P}_\star\sqrt{k/4}+\mathsf{err}(n,k,L)$, where $\mathsf{P}_\star\approx 0.763166$ is the value of the Parisi formula from spin glass theory, and $\mathsf{err}(n,k,L)=o_n(n)+no_k(\sqrt{k})+n \sqrt{k} o_L(1)$ (subscripts indicate the asymptotic variables). Our result generalizes to locally treelike graphs, i.e., graphs whose girth becomes $L$ after removing a small fraction of vertices. Earlier work established that, for random $k$-regular graphs, the typical max-cut value is $nk/4+ n\mathsf{P}_\star\sqrt{k/4}+o_n(n)+no_k(\sqrt{k})$. Therefore our algorithm is nearly optimal on such graphs. An immediate corollary of this result is that random regular graphs have nearly minimum max-cut, and nearly maximum min-bisection among all regular locally treelike graphs. This can be viewed as a combinatorial version of the near-Ramanujan property of random regular graphs.
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