Related papers: Complexity Analysis of 2-Heterogeneous Minimum Spa…
We define the problem segment cover as follows. We are given a set of pairs of sub-intervals of the unit interval. The problem asks if there is a choice of a single interval from each pair such that the union of the chosen intervals covers…
The geometric $\delta$-minimum spanning tree problem ($\delta$-MST) is the problem of finding a minimum spanning tree for a set of points in a normed vector space, such that no vertex in the tree has a degree which exceeds $\delta$, and the…
In the spanning-tree congestion problem ($\mathsf{STC}$), we are given a graph $G$, and the objective is to compute a spanning tree of $G$ that minimizes the maximum edge congestion. While $\mathsf{STC}$ is known to be $\mathbb{NP}$-hard,…
We study the {\em verification} problem in distributed networks, stated as follows. Let $H$ be a subgraph of a network $G$ where each vertex of $G$ knows which edges incident on it are in $H$. We would like to verify whether $H$ has some…
We investigate the computation of minimum-cost spanning trees satisfying prescribed vertex degree constraints: Given a graph $G$ and a constraint function $D$, we ask for a (minimum-cost) spanning tree $T$ such that for each vertex $v$, $T$…
The Minimum Spanning Tree problem (abbr. MSTP) is a well-known combinatorial optimization problem that has been extensively studied by the researchers in the field of evolutionary computing to theoretically analyze the optimization…
Constrained forest problems form a class of graph problems where specific connectivity requirements for certain cuts within the graph must be satisfied by selecting the minimum-cost set of edges. The prize-collecting version of these…
We introduce and study the complexity of Path Packing. Given a graph $G$ and a list of paths, the task is to embed the paths edge-disjoint in $G$. This generalizes the well known Hamiltonian-Path problem. Since Hamiltonian Path is…
In this paper, we study the problem of finding a minimum weight spanning tree that contains each vertex in a given subset $V_{\rm NT}$ of vertices as an internal vertex. This problem, called Minimum Weight Non-Terminal Spanning Tree,…
We present a simple, self-contained, linear reduction from 3-SAT to Treewidth. Specifically, it shows that 1.00005-approximating Treewidth is NP-hard, and solving Treewidth exactly requires $2^{\Omega(n)}$ time, unless the Exponential-Time…
The minimum-cost arborescence problem is a well-studied problem in the area of graph theory, with known polynomial-time algorithms for solving it. Previous literature introduced new variations on the original problem with different…
We introduce the NP-complete problem 3SAT_N and extend Tovey's results to a classification theorem for this problem. This theorem leads us to generalize the concept of truth assignments for SAT to aggressive truth assignments for 3SAT_N. We…
The Minimum Branch Vertices Spanning Tree problem aims to find a spanning tree $T$ in a given graph $G$ with the fewest branch vertices, defined as vertices with a degree three or more in $T$. This problem, known to be NP-hard, has…
We propose a new conjecture on hardness of low-degree $2$-CSP's, and show that new hardness of approximation results for Densest $k$-Subgraph and several other problems, including a graph partitioning problem, and a variation of the Graph…
We study the (non-uniform) quantified constraint satisfaction problem QCSP(H) as H ranges over partially reflexive forests. We obtain a complexity-theoretic dichotomy: QCSP(H) is either in NL or is NP-hard. The separating condition is…
In the minimum constraint removal ($MCR$), there is no feasible path to move from the starting point towards the goal and, the minimum constraints should be removed in order to find a collision-free path. It has been proved that $MCR$…
Minimal spanning trees on infinite vertex sets are investigated. A criterion for minimality of a spanning tree having a finite length is obtained, which generalizes the corresponding classical result for finite sets. It is given an analytic…
The standard proof of NP-Hardness of 3DM provides a power-$4$ reduction of 3SAT to 3DM. In this note, we provide a linear-time reduction. Under the exponential time hypothesis, this reduction improves the runtime lower bound from…
We show that {\sc Heegaard Genus $\leq g$}, the problem of deciding whether a triangulated 3-manifold admits a Heegaard splitting of genus less than or equal to $g$, is NP-hard. The result follows from a quadratic time reduction of the…
Given a set of colored points in the plane, we ask if there exists a crossing-free straight-line drawing of a spanning forest, such that every tree in the forest contains exactly the points of one color class. We show that the problem is…