Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2405.10556

Parameterized Complexity of Dominating Set Variants in Almost Cluster and Split Graphs

Dishant Goyal·Ashwin Jacob·Kaushtubh Kumar·Diptapriyo Majumdar·Venkatesh Raman
View original source ↗Read source document ↗Export metadata ↓

Abstract

We consider structural parameterizations of the fundamental Dominating Set problem and its variants in the parameter ecology program. We give improved FPT algorithms and lower bounds under well-known conjectures for dominating set in graphs that are k vertices away from a cluster graph or a split graph. These are graphs in which there is a set of k vertices (called the modulator) whose deletion results in a cluster graph or a split graph. We also call k as the deletion distance (to the appropriate class of graphs). When parameterized by the deletion distance k to cluster graphs - we can find a minimum dominating set (DS) in 3^k n^{O(1)}-time. Within the same time, we can also find a minimum independent dominating set (IDS) or a minimum dominating clique (DC) or a minimum efficient dominating set (EDS) or a minimum total dominating set (TDS). We also show that most of these variants of dominating set do not have polynomial sized kernel. Additionally, we show that when parameterized by the deletion distance k to split graphs - IDS can be solved in 2^k n^{O(1)}-time and EDS can be solved in 3^{k/2}n^{O(1)}.

Explore related subjects

cs.DS

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Dishant Goyal, Ashwin Jacob, Kaushtubh Kumar, Diptapriyo Majumdar, Venkatesh Raman. 2024-05-17. Parameterized Complexity of Dominating Set Variants in Almost Cluster and Split Graphs. https://arxiv.org/abs/2405.10556

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

KEEP EXPLORING

Related papers

arXiv2026

Online Flexible Busy Time Scheduling on Heterogeneous Machines

We study the online busy time scheduling model on heterogeneous machines. In our setting, jobs with uniform processing time arrive online with a deadline that becomes known to the algorithm at the job's arrival time. An algorithm has access to machines, each with different associated capacities and costs. The goal is to schedule jobs on machines by their deadline, so that the total cost incurred by the scheduling algorithm is minimized. While busy time scheduling has been well-studied, relatively little is known when machines are heterogeneous (i.e., have different costs and capacities), despite this natural theoretical generalization being the most practical model for clients using cloud computing services. We make significant progress in understanding this model by designing a deterministic online algorithm with competitive ratio 8(2p-1)/p < 16 when all jobs have uniform processing time p. A randomized version of this algorithm is 4(2p-1)/(p \ln 2)-competitive against an oblivious adversary. For unit-processing-time jobs, we give lower bounds of 4 and e (where e is Euler's number) on the competitive ratio of deterministic and randomized online algorithms, respectively. For unit-processing-time jobs with agreeable deadlines, we provide a deterministic 2-competitive online algorithm and a matching lower bound.

cs.DS↗
arXiv2026

The Binary Tree Mechanism is Optimal for Differentially Private Continual Counting

Private continual counting is a fundamental problem in differential privacy: given a binary stream of length $n$, where each $1$ corresponds to the contribution of one individual, the goal is to release all running counts while protecting the privacy of each individual. For fixed privacy parameters, the standard binary tree mechanism achieves expected $\ell_\infty$ error $O(\log^{3/2} n)$ under approximate differential privacy and $O(\log^2 n)$ under pure differential privacy. Whether these dependences on the stream length are necessary has remained a central open problem. For fixed $\varepsilon\in(0,1)$, we prove a lower bound of $Ω(\log^{3/2} n)$ under approximate DP with sufficiently small fixed $δ>0$, and a lower bound of $Ω(\log^2 n)$ under pure DP. These bounds establish the optimality of the binary tree mechanism in both settings. The bounds hold for arbitrary mechanisms, even when the entire stream is available in advance. Both proofs use the same decomposition and accumulation of residual noise along a tree. As a consequence of the approximate-DP bound, we also obtain a largest-possible separation between hereditary discrepancy and private $\ell_\infty$ error for linear queries, showing that the known general upper bound in terms of hereditary discrepancy has the optimal dependence on the number of queries.

cs.DS↗
arXiv2026

Directed Hamiltonian-Cycle Parity in $O^*((3/2)^n)$ Deterministic Time and Polynomial Space

We give a deterministic algorithm that computes the parity of the number of Hamiltonian cycles in an $n$-vertex directed graph in $O(n^4(3/2)^n)$ time and $O(n^2)$ bits of working space, improving the $O^*(φ^n)$ bound of Björklund and Husfeldt. Their local-degree formula reduces the problem to a weighted sum over solutions of structured quadratic equations. We cover the corresponding ternary state space by binary subcubes, each inducing an affine system. The Kuang--Wang cover can be regenerated within the target bound; canonical ownership resolves its overlaps, while self-loop conditional expectations bound every affine solution visit. Rollback elimination shares the work across cover prefixes. The same cover gives a Las Vegas algorithm listing all $L$ solutions of $m$ affine product constraints in $N$ Boolean variables in expected time $\operatorname{poly}(N,m)((3/2)^m+L)$ and polynomial space. Finally, we show that complete enumeration can require $Ω((3/2)^n)$ visits even on strongly connected digraphs after an optimal self-loop choice. This is a limitation of the enumeration method, not a general lower bound for Hamiltonian-cycle parity.

cs.DS↗