相关论文: Vertex Sparsifiers: New Results from Old Technique…
We consider the problem of testing graph cluster structure: given access to a graph $G=(V, E)$, can we quickly determine whether the graph can be partitioned into a few clusters with good inner conductance, or is far from any such graph?…
We consider the Hypergraph-$k$-cut problem. The input consists of a hypergraph $G=(V,E)$ with non-negative hyperedge-costs $c: E\rightarrow R_+$ and a positive integer $k$. The objective is to find a least-cost subset $F\subseteq E$ such…
We show how to find and efficiently maintain maximal k-edge-connected subgraphs in undirected graphs. In particular, we provide the following results. (1) A general framework for maintaining the maximal k-edge-connected subgraphs upon…
A natural requirement of many distributed structures is fault-tolerance: after some failures, whatever remains from the structure should still be effective for whatever remains from the network. In this paper we examine spanners of general…
An important open question in the area of vertex sparsification is whether $(1+\epsilon)$-approximate cut-preserving vertex sparsifiers with size close to the number of terminals exist. The work Chalermsook et al. (SODA 2021) introduced a…
A non-trivial minimum cut (NMC) sparsifier is a multigraph $\hat{G}$ that preserves all non-trivial minimum cuts of a given undirected graph $G$. We introduce a flexible data structure for fully dynamic graphs that can efficiently provide…
Graph sparsification is a powerful tool to approximate an arbitrary graph and has been used in machine learning over homogeneous graphs. In heterogeneous graphs such as knowledge graphs, however, sparsification has not been systematically…
The dual of a planar graph $G$ is a planar graph $G^*$ that has a vertex for each face of $G$ and an edge for each pair of adjacent faces of $G$. The profound relationship between a planar graph and its dual has been the algorithmic basis…
Mutual visibility in graphs requires pairs of vertices to be connected by shortest paths that avoid all other vertices of a prescribed set, a condition that is often overly restrictive. In this paper, we introduce a new variant, called…
Let $G=(V,E)$ be an undirected unweighted planar graph. Consider a vector storing the distances from an arbitrary vertex $v$ to all vertices $S = \{ s_1 , s_2 , \ldots , s_k \}$ of a single face in their cyclic order. The pattern of $v$ is…
Graph product structure theory expresses certain graphs as subgraphs of the strong product of much simpler graphs. In particular, an elegant formulation for the corresponding structural theorems involves the strong product of a path and of…
A subset $T \subseteq V$ of terminals is $k$-connected to a root $s$ in a directed/undirected graph $J$ if $J$ has $k$ internally-disjoint $vs$-paths for every $v \in T$; $T$ is $k$-connected in $J$ if $T$ is $k$-connected to every $s \in…
The $k$-Steiner-2NCS problem is as follows: Given a constant $k$, and an undirected connected graph $G = (V,E)$, non-negative costs $c$ on $E$, and a partition $(T, V-T)$ of $V$ into a set of terminals, $T$, and a set of non-terminals (or,…
X. Hou, H.-J. Lai, P. Li and C.-Q. Zhang [J. Graph Theory 69 (2012) 464-470] showed that for a simple graph $G$ with $|V(G)|\ge 44$, if $\min\{\delta(G),\delta(G^c)\}\ge 4$, then either $G$ or its complementary graph $G^c$ has a…
We study how to vertex-sparsify a graph while preserving both the graph's metric and structure. Specifically, we study the Steiner point removal (SPR) problem where we are given a weighted graph $G=(V,E,w)$ and terminal set $V' \subseteq V$…
We study the problem of distance-preserving graph compression for weighted paths and trees. The problem entails a weighted graph $G = (V, E)$ with non-negative weights, and a subset of edges $E^{\prime} \subset E$ which needs to be removed…
We show variants of spectral sparsification routines can preserve the total spanning tree counts of graphs, which by Kirchhoff's matrix-tree theorem, is equivalent to determinant of a graph Laplacian minor, or equivalently, of any SDDM…
We contribute an approach to the problem of locally computing sparse connected subgraphs of dense graphs. In this setting, given an edge in a connected graph $G = (V, E)$, an algorithm locally decides its membership in a sparse connected…
A cycle cover of a bridgeless graph $G$ is a collection of simple cycles in $G$ such that each edge $e$ appears on at least one cycle. The common objective in cycle cover computation is to minimize the total lengths of all cycles. Motivated…
Let $G$ be a graph with vertex set $V(G)$ and let $H:V(G)\rightarrow 2^N$ be a set function associating with $G$. An $H$-factor of graph $G$ is a spanning subgraphs $F$ such that $$d_F(v)\in H(v){4em}\hbox{for every}v\in V(G).$$ Let…