arXiv subjects

David C. Kutner

Publications and source records attributed to David C. Kutner.

9 recordsLinked to original sources

arXiv2026

$O(n +f(k))$: Truly Linear FPT

Parameterized complexity has always been concerned with practical computing: by confining combinatorial explosion to a secondary parameter $k$, one can uncover why and how many NP-hard problems are effectively tackled in practice. Today, however, the scale of data has changed: scientists study Big Data, which is so large that even quadratic dependence in the total input size $n$ is unaffordable. Therefore, what constitutes a practical algorithm has also changed. Classically, parameterized complexity is blind to the difference between defining fixed parameter tractability multiplicatively (i.e. $f(k) \cdot n^c$) or additively (i.e. $f(k) + n^c$). But what if the constant $c$ is one and we require true linearity, is this distinction still inconsequential? Here, we define and explore Truly Linear FPT (TLFPT) -- that is $O(n)+f(k)$ -- and show that it is a strict subset of Linear FPT (LFPT) -- that is $O(n) \cdot f(k)$ -- via diagonalization. Populating TLFPT requires careful consideration of linear-time algorithmics and data structures. We meet many inhabitants of TLFPT: SAT, Vertex Cover, Min-Max Matching, $(n-k)$-Coloring, Diverse Pair of Matchings, $k$-Path, and $H$-Coloring. Our parameterizations are equally varied. Beyond classical parameters like solution size, we leverage two parameters, treedepth and BFS-width, which are particularly well-suited to the TLFPT regime. We do so by developing techniques based on depth- and breadth-first search. For parameterized complexity to be of service to the scientific community, we need to contend with Big Data. For sufficiently large inputs, FPT beyond linear may not suffice. Thus, there is a practical and theoretical need for more ambitious goals. TLFPT is a first step forward.

cs.CC↗

arXiv2026

Steiner Forest for $H$-Subgraph-Free Graphs

Our main result is a full classification, for every connected graph $H$, of the computational complexity of Steiner Forest on $H$-subgraph-free graphs. To obtain this dichotomy, we establish the following new algorithmic, hardness, and combinatorial results: Algorithms: We identify two new classes of graph-theoretical structures that make it possible to solve Steiner Forest in polynomial time. Roughly speaking, our algorithms handle the following cases: (1) a set $X$ of vertices of bounded size that are pairwise connected by subgraphs of treewidth $2$ or bounded size, possibly together with an independent set of arbitrary size that is connected to $X$ in an arbitrary way; (2) a set $X$ of vertices of arbitrary size that are pairwise connected in a cyclic manner by subgraphs of treewidth $2$ or bounded size. Hardness results: We show that Steiner Forest remains NP-complete for graphs with 2-deletion set number $3$. (The $c$-deletion set number is the size of a smallest cutset $S$ such that every component of $G-S$ has at most $c$ vertices.) Combinatorial results: To establish the dichotomy, we perform a delicate graph-theoretic analysis showing that if $H$ is a path or a subdivided claw, then excluding $H$ as a subgraph either yields one of the two algorithmically favourable structures described above, or yields a graph class for which NP-completeness of Steiner Forest follows from either our new hardness result or a previously known one. Along the way to classifying the hardness for excluded subgraphs, we establish a dichotomy for graphs with $c$-deletion set number at most $k$. Specifically, our results together with pre-existing ones show that Steiner Forest is polynomial-time solvable if (1) $c=1$ and $k\geq 0$, or (2) $c=2$ and $k\leq 2$, or (3) $c\geq 3$ and $k=1$, and is NP-complete otherwise.

math.CO↗

arXiv2025

Better late, then? The hardness of choosing delays to meet passenger demands in temporal graphs

In train networks, carefully-chosen delays may be beneficial for certain passengers, who would otherwise miss some connection. Given a simple (directed or undirected) temporal graph and a set of passengers (each specifying a starting vertex, an ending vertex, and a desired arrival time), we ask whether it is possible to delay some of the edges of the temporal graph to realize all the passengers' demands. We call this problem DelayBetter (DB), and study it along with two variants: in $δ$-DelayBetter, each delay must be of at most $δ$; in ($δ$-)Path DB, passengers also fully specify the vertices they should visit on their journey. On the positive side, we give a polynomial-time algorithm for Path DB and $δ$-Path DB, and obtain as a corollary a polynomial-time algorithm for DB and $δ$-DB on trees. We also provide an fpt algorithm for both problems parameterized by the size of the graph's Feedback Edge Set together with the number of passengers. On the negative side, we show NP-completeness of ($1$-)DB on bounded-degree temporal graphs even when the lifetime is $2$, and of ($10$-)DB on bounded-degree planar temporal graphs of lifetime $19$. Our results complement previous work studying reachability problems in temporal graphs with delaying operations. This is to our knowledge the first such problem in which the aim is to facilitate travel between specific points (as opposed to facilitating or impeding a broadcast from one or many sources).

cs.DS↗

arXiv2025

Generalising the maximum independent set algorithm via Boolean networks

A simple greedy algorithm to find a maximal independent set (MIS) in a graph starts with the empty set and visits every vertex, adding it to the set if and only if none of its neighbours are already in the set. In this paper, we consider (the complexity of decision problems related to) the generalisation of this MIS algorithm wherein any starting set is allowed. Two main approaches are leveraged. Firstly, we view the MIS algorithm as a sequential update of a Boolean network according to a permutation of the vertex set. Secondly, we introduce the concept of a constituency of a graph: a set of vertices that is dominated by an independent set. Recognizing a constituency is NP-complete, a fact we leverage repeatedly in our investigation. Our contributions are multiple: we establish that deciding whether all maximal independent sets can be reached from some configuration is coNP-complete; that fixing words (which reach a MIS from any starting configuration) and fixing permutations (briefly, permises) are coNP-complete to recognize; and that permissible graphs (graphs with a permis) are coNP-hard to recognize. We also exhibit large classes of permissible and non-permissible graphs, notably near-comparability graphs which may be of independent interest. Lastly, we extend our study to digraphs, where we search for kernels. Since the natural generalisation of our approach may not necessarily find a kernel, we introduce two further Boolean networks for digraphs: one always finds an independent set, and the other always finds a dominating set.

cs.DS↗

arXiv2025

Reconfigurable routing in data center networks

A hybrid network is a static (electronic) network that is augmented with optical switches. The Reconfigurable Routing Problem (RRP) in hybrid networks is the problem of finding settings for the optical switches augmenting a static network so as to achieve optimal delivery of some given workload. The problem has previously been studied in various scenarios with both tractability and NP-hardness results obtained. However, the data center and interconnection networks to which the problem is most relevant are almost always such that the static network is highly structured (and often node-symmetric) whereas all previous results assume that the static network can be arbitrary (which makes existing computational hardness results less technologically relevant and also easier to obtain). In this paper, and for the first time, we prove various intractability results for RRP where the underlying static network is highly structured, for example consisting of a hypercube, and also extend some existing tractability results.

cs.CC↗

arXiv2024

Payment Scheduling in the Interval Debt Model

The network-based study of financial systems has received considerable attention in recent years but has seldom explicitly incorporated the dynamic aspects of such systems. We consider this problem setting from the temporal point of view and introduce the Interval Debt Model (IDM) and some scheduling problems based on it, namely: Bankruptcy Minimization/Maximization, in which the aim is to produce a payment schedule with at most/at least a given number of bankruptcies; Perfect Scheduling, the special case of the minimization variant where the aim is to produce a schedule with no bankruptcies (that is, a perfect schedule); and Bailout Minimization, in which a financial authority must allocate a smallest possible bailout package to enable a perfect schedule. We show that each of these problems is NP-complete, in many cases even on very restricted input instances. On the positive side, we provide for Perfect Scheduling a polynomial-time algorithm on (rooted) out-trees although in contrast we prove NP-completeness on directed acyclic graphs, as well as on instances with a constant number of nodes (and hence also constant treewidth). When we allow non-integer payments, we show by a linear programming argument that the problem Bailout Minimization can be solved in polynomial time.

cs.DM↗

arXiv2024

Temporal Reachability Dominating Sets: contagion in temporal graphs

Given a population with dynamic pairwise connections, we ask if the entire population could be (indirectly) infected by a small group of $k$ initially infected individuals. We formalise this problem as the Temporal Reachability Dominating Set (TaRDiS}) problem on temporal graphs. We provide positive and negative parameterized complexity results in four different parameters: the number $k$ of initially infected, the lifetime $τ$ of the graph, the number of locally earliest edges in the graph, and the treewidth of the footprint graph $\mathcal{G}_\downarrow$. We additionally introduce and study the MaxMinTaRDiS problem, where the aim is to schedule connections between individuals so that at least $k$ individuals must be infected for the entire population to become fully infected. We classify three variants of the problem: Strict, Nonstrict, and Happy. We show these to be coNP-complete, NP-hard, and $Σ_2^P$-complete, respectively. Interestingly, we obtain hardness of the Nonstrict variant by showing that a natural restriction is exactly the well-studied Distance-3 Independent Set problem on static graphs.

cs.DM↗

arXiv2023

Words fixing the kernel network and maximum independent sets in graphs

The simple greedy algorithm to find a maximal independent set of a graph can be viewed as a sequential update of a Boolean network, where the update function at each vertex is the conjunction of all the negated variables in its neighbourhood. In general, the convergence of the so-called kernel network is complex. A word (sequence of vertices) fixes the kernel network if applying the updates sequentially according to that word. We prove that determining whether a word fixes the kernel network is coNP-complete. We also consider the so-called permis, which are permutation words that fix the kernel network. We exhibit large classes of graphs that have a permis, but we also construct many graphs without a permis.

cs.DM↗

arXiv2022

Vibration-based communication for deafblind people

Deafblind people have both hearing and visual impairments, which makes communication with other people often dependent on expensive technologies e.g., Braille displays, or on caregivers acting as interpreters. This paper presents Morse I/O (MIO), a vibrotactile interface for Android, evaluated through experiments and interviews with deafblind participants. MIO was shown to enable consistent text entry and recognition after only a few hours of practice. The participants were willing to continue using the interface, although there were perceived difficulties in learning to use it. Overall, MIO is a cost-effective, portable interface for deafblind people without access to Braille displays or similar.

cs.HC↗