Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2610.17152

Parameterized algorithms for representing graphs as intersections of threshold graphs

L. Sunil Chandran·Suraj Kumar Sahoo
View original source ↗Read source document ↗Export metadata ↓

Abstract

A graph is called a threshold graph if there is an assignment of real valued weights to its vertices such that two vertices have an edge between them if and only if the sum of the weights of the two vertices is above a certain `threshold value'. We will denote the class of all threshold graphs as $TH$. For a graph $G=(V,E)$, if for some integer $t\geq 0$, there are $t$ many threshold graphs $\{G_i=(V,E_i)\}_{i\in [t]}$ such that we have $E=E_1\cap E_2\cap \cdots \cap E_t$, then we say that the collection of threshold graphs $\{G_i=(V,E_i)\}_{i\in [t]}$ forms a $TH$-representation of $G$ of size $t$. \emph{The intersection dimension of $G$ with respect to threshold graphs} (denoted as $dim_{TH}(G)$) is the smallest non-negative integer $t$ such that there is a $TH$-representation of $G$ of size $t$. It is known that for a graph $G$ and integer $t>0$, the problem of deciding whether $dim_{TH}(G)\leq t$ is NP-complete even for $t=3$. In this paper, we study this problem in the framework of parameterized algorithms. We consider the vertex cover number and the cluster vertex deletion number as our parameters. We give a FPT algorithm for determining an optimal $TH$-representation of a graph when parameterized by its vertex cover number. Our main result is an efficient parameterized approximation scheme for this problem when parameterized by the cluster vertex deletion number of $G$.

Explore related subjects

cs.DS

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

L. Sunil Chandran, Suraj Kumar Sahoo. 2026-10-12. Parameterized algorithms for representing graphs as intersections of threshold graphs. https://arxiv.org/abs/2610.17152

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Adding an Edge in a $P_4$-sparse Graph

For a graph family~${\cal F}$, the minimum ${\cal F}$-completion problem asks for the minimum number of non-edges to be added to a given graph~$G$ so that the resulting graph belongs to the family~${\cal F}$. This problem is NP-complete for many subclasses of perfect graphs, while polynomial solutions are available for minimal completion sets or restricted input. We study the minimum $P_4$-sparse completion problem that arises after adding a prescribed non-edge to a \(P_4\)-sparse graph \(G\), with the resulting graph required to remain \(P_4\)-sparse. For the case in which $G$ consists of two connected components, each containing one endpoint of the added non-edge, we characterize the possible forms of an optimal solution. Combining this characterization with the solution of the other cases identified by the structural decomposition of \(P_4\)-sparse graphs, we obtain an algorithm for the general problem that runs in $O(n^2)$ time and $O(n)$ space.

cs.DS↗
arXiv2026

Optimal Extrapolation Bounds for Sparse Fourier Sums

We prove a sharp extrapolation bound for $k$-sparse Fourier sums with arbitrary real frequencies and no separation assumption. For every $g(t)=\sum_{j=1}^k v_je^{iλ_jt}$ with $λ_j\in\mathbb{R}$ and every $x\geq 1$, \begin{align*} |g(x)|\le k\exp(O(k\operatorname{arcosh} x))\|g\|_{L^2[-1,1]}. \end{align*} Near the endpoint at $x=1+δ$, this gives an exponential factor of $\exp(O(k\sqrtδ))$, improving on the $\exp(O(k^2\log k\cdotδ))$ estimate of Chen and Price (ICALP 2019), and the exponent is optimal up to a constant factor. Our theorem has algorithmic applications to recovering gapless Fourier-sparse signals: it improves the resolution of cluster-location algorithms and the sample complexity of Fourier interpolation algorithms. In addition, it gives sharp bounds on the error transfer of sparse Fourier models just outside the training interval.

cs.DS↗
arXiv2026

Stochastic Network Speed Scaling with Competitive Ratios Independent of the Network

In network speed scaling, jobs arrive over time at a network of servers whose speeds can be tuned, every job must be processed by the servers along one of its allowed routes, and the goal is to minimize the total flow time plus the total energy. For stochastic arrivals, the competitive ratio of Vaze and Nair depends on the network, through the lengths of the routes it uses. We show that this dependence can be removed: we give an algorithm, which routes the jobs by solving a convex program and runs every server at a fixed speed, whose competitive ratio depends only on the power functions; for $P(s)=s^2$, it is at most the golden ratio $φ\approx1.618$. The key idea is a lower bound on the optimal cost which, like the cost of our algorithm, is a sum over the servers of a function of each server's load, so the analysis reduces to a single server. We also show that the routing rule of Vaze and Nair can be a factor $Ω(L)$ away from optimal, where $L$ is the length of the longest route.

cs.DS↗