A king in an n-vertex tournament graph G is a vertex that can reach any other vertex v with a path of length at most two. A kingdom is a data structure that given any vertex v returns such a path from a king to v in O(1) time. In this paper, we show how to maintain a kingdom while the tournament graph G undergoes updates. We consider both edge updates that flip the direction of an edge, and vertex updates that insert/delete vertices by activating/deactivating rows and columns of the graph’s adjacency matrix. For a single edge-flip, we show that after O(n^3/2) preprocessing time, we can maintain a kingdom in O(1) time following the edge-flip. With Õ(n^2) preprocessing time, we can support any constant number of edge-flips, vertex insertions, and vertex deletions in O(log n) time per operation. For an arbitrary number of edge-flips, we present a randomized algorithm that maintains a kingdom in O(log n) expected time following every edge-flip, and another algorithm that supports vertex insertions in O(√(n)) amortized time per insertion.
Given a set of results from range minimum queries (RMQs), our task is to construct a sequence that is consistent with the results of the queries. We study two types of RMQs: a value-based RMQ returns the minimum value and an index-based RMQ returns the index of the minimum. While the value-based version has been discussed informally in the context of competitive programming, the index-based version appears to be unexplored. In this paper, we provide a survey and unified analysis of the value-based version, and we propose efficient algorithms for the index-based version. These include algorithms for computing the lexicographically smallest consistent sequence and permutation, as well as an enumeration algorithm that outputs all consistent permutations with constant delay.
A simple graph is called a word-representable graph if there is a word over its vertex set such that any two vertices are adjacent in the graph if and only if they alternate in the word; such a word is called a word-representant of the graph. Srinivasan and Hariharasubramanian proved that a minimum length word-representant for a non-complete triangle-free circle graph on n vertices with at least one edge is of 2n-2 length. Further, they posed an open problem to find classes of word-representable graphs whose minimum length word-representants are of 2n-κ length, where n is the number of vertices and κ is the clique number of a graph. A graph is called a treelike comparability graph if it admits a transitive orientation such that the transitive reduction is a tree. When such a graph is also a permutation graph, it is called a treelike permutation graph. Recently, a subclass of treelike permutation graphs, viz., the class of double-arborescences, was established to be the first example satisfying the criterion given in the above-mentioned open problem. In this work, we devise a polynomial-time algorithm to construct a minimum length word-representant for a given treelike permutation graph and show that all treelike permutation graphs satisfy the criterion of the open problem.
The capacitated vehicle routing problem (c-VRP), which is also known as the k-tour cover problem, has been widely studied in many settings. In the Euclidean plane, a PTAS for constant capacity c was given more than 40 years ago, which has been improved and extended ever since. Here, we present arguably the easiest form of a PTAS possible. The algorithm simply chooses between greedy or complete enumeration. A direct corollary of this simplification is that the PTAS extends to basically any kind of order restriction, as long as the objective is to minimize total tour length. Furthermore, we show that when the metric is a tree, there is a clear separation in complexity between the c-VRP with and without order restrictions. We show that even for a fixed order, the c-VRP is APX-hard for arbitrary (non-constant) capacity c. This in contrast to the polynomial time approximation scheme for the standard c-VRP on trees.
We study the notion of full-compatibility: given two one-dimensional subshifts H and V, is there a two-dimensional subshift X such that H = {c_|ℤ×{0}| c ∈ X} and V ={c_|{0}×ℤ| c ∈ X} ? We show that this problem is decidable when both X and Y are nearest-neighbor SFTs but undecidable when at least one is allowed to be of slightly higher complexity. We also prove that the problem is undecidable for three nearest-neighbor SFTs combined in a three-dimensional subshift.
A (partial) conflict-free coloring of a hypergraph ℋ is an assignment of colors to (a subset of) the vertex set of ℋ such that every hyperedge in ℋ has a vertex whose color is distinct from every other vertex in that hyperedge. The minimum number of colors required for such a coloring is known as the (partial) conflict-free chromatic number of ℋ . It is easy to see that the conflict-free chromatic number of a hypergraph is at most its partial conflict-free chromatic number plus one. Conflict-free coloring has also been studied on the open/closed neighborhood hypergraphs of a given graph under the name open/closed neighborhood conflict-free coloring. In this paper, we study partial and full list variants of conflict-free coloring where, for every vertex v, we are given a list of admissible colors L_v such that v is allowed to be colored only from L_v . It was shown by Pach and Tardos [Combinatorics, Probability and Computing, 2009] that for any constant ϵ > 0 , the closed-neighborhood conflict-free chromatic number of a graph G is at most O(ln ^2 + ϵ ) , where represents the maximum degree of G. Later, Glebov, Szabó, and Tardos [Combinatorics, Probability and Computing, 2014] showed that there exist graphs G that require (ln ^2 ) colors for a closed neighborhood conflict-free coloring. Bhyravarapu, Kalyanasundaram, and Mathew [Journal of Graph Theory, 2021] bridged the gap between the upper and the lower bound. They showed that the closed-neighborhood conflict-free chromatic number of any graph G is at most O(ln ^2 ) . In this paper, we extend the O(ln ^2 ) upper bound to the partial list variant of the closed-neighborhood conflict-free chromatic number. Further, we establish computational complexity results concerning the list open/closed-neighborhood conflict-free chromatic numbers.
In Duration-aware Pinwheel Scheduling (DAPS), each job J_i has a duration p_i (in days) and the objective is to find a schedule on a single machine such that each job J_i is completed at least once in every consecutive f_i days. This problem generalizes the well-known problem, (Ordinary) Pinwheel Scheduling, which assumes all jobs are unit-length (i.e., p_i = 1 ). Previous studies have shown that in order to schedule jobs J_1, … , J_n feasibly, the density of the instance, defined as p_1 / f_1 + … + p_n/f_n , must be ≤ 1 . Additionally, in the case of Ordinary Pinwheel Scheduling, every instance with density ≤ 5/6 is known to be always schedulable. However, it remains open whether a similar density threshold would lead to an efficient algorithm for deciding schedulability in DAPS. In this paper, we present a negative solution to this open problem under the assumption that , namely, we show that no constant density threshold leads to a polynomial-time algorithm for deciding schedulability in DAPS unless = . Furthermore, we demonstrate that another heuristic from Ordinary Pinwheel Scheduling fails to extend to DAPS. After that, we present an online scheduler which always schedule jobs feasibly for some DAPS instance class, by inventing a new measurement of schedulability of instances. This scheduler can be regarded as a 3-approximation algorithm for the optimization version of DAPS – known as Continuous Bamboo Garden Trimming on star graphs – which improves upon the previous best approximation ratio 3 + 2√(2) .
A set D of vertices of a graph is a defensive alliance if, for each element of D, the majority of its neighbors are in D. We consider the notion of local minimality in this paper. A defensive alliance D is called a locally minimal defensive alliance if removing any vertex v ∈ D destroys the defensive alliance property, i.e., D ∖{v} is no longer a defensive alliance [1]. Given an undirected graph G=(V,E) and an integer k ∈ℕ , we study Locally Minimal Defensive Alliance, where the goal is to check whether G has a locally minimal defensive alliance of size at least k. This problem is known to be NP-hard, but its parameterized complexity remains open until now. We enhance our understanding of the problem from the viewpoint of parameterized complexity by showing that (1) the problem admits a fixed-parameter tractable (FPT) algorithm on general graphs when parameterized by the solution size k, and (2) we also present a subexponential algorithm on planar graphs of minimum degree at least two using the tool of bidimensionality.
Previous work on parallel quantum algorithms has focused on minimizing the length of the critical path of the circuit (i.e., query span), motivated by the small decoherence times of qubits. In this paper, we study similar settings where queries to a quantum circuit can be made in parallel in each timestep. Rather than focus exclusively on query span, however, we are interested in also minimizing the total number of queries made across all parallel processes (i.e., query work). We give parallel quantum algorithms that solve the maxima set and convex hull problems for n lexicographically sorted points in the plane in Õ(√(n)) query span and Õ(√(nh)) query work, where h is the size of the output. These results therefore resolve a natural question of whether we can use quantum computers in parallel to improve the depth of a quantum algorithm without requiring more work than a sequential algorithm. Also, note that our work bounds are sublinear when h is O(n^1-ϵ) , for a small constant, ϵ >0 .
A well-known consequence of Brooks’ Theorem is that 3-Colouring, a.k.a. CSP(K_3) , is solvable in polynomial time for subcubic graphs. We begin our investigation by observing that QCSP(K_3) , a.k.a. Quantified 3-Colouring, is Pspace-complete on subcubic planar graphs. This allows us to give a complexity classification for bounded alternation _i - QCSP(K_3) ( i≥ 3 , odd) for ℋ -topological-minor-free graphs. For all such (maybe infinite) sets ℋ , we delineate between those for which _i - QCSP(K_3) is solvable in polynomial time, and those for which it is _i-2^P -hard. We continue our investigation of QCSP(K_3) on ℋ -subgraph-free graphs, contrasting with both CSP(K_3) , as well as the recently introduced C123-framework of [10]. Next, we turn our attention to P_5 - and P_4 -free graphs. We prove that QCSP(H) is in NP , for all finite graphs H, when the input is restricted to P_5 -free graphs. For QCSP(K_3) , on P_4 -free graphs, we are able to improve this to a polynomial time algorithm. Finally, we propose a certain template, the line graph of a subdivided star L(K^r_1,4) , so that QCSP(L(K^r_1,4)) behaves curiously on P_m -free graphs. When m is small, this problem is solvable in polynomial time. For some medium m, it is NP -complete, while for large enough m, it is Pspace-complete.
We present algorithms for computing invariants of K-theoretic persistent homology, extending beyond additive summaries in persistent homology in computational geometry. Building on fundamental theorems (i.e., interval calculus, Betti–exterior identity, and integral formula), we provide algorithms for compressed representations, output-sensitive enumerations, and fast statistics for concurrency layers. We establish complexity results and demonstrate practical advantages on graph filtrations.
We rigorously measure the reverse mathematical strengths of fundamental analytic principles for two neural architectures, Transformer and MLP (MultiLayer Perceptron). We prove that the standard compactness principle for Transformer is equivalent (over _0 ) to _0 . For ReLU MLPs on compact domains, we prove that the standard universal approximation theorem for MLPs (with rational parameters) is equivalent to _0 . These results mean that the transformer compactness strictly exceeds the MLP universality in the reverse mathematical hierarchy. This paves the way for a systematic program of reverse mathematical classification of neural networks, bridging proof theory, computable analysis, and machine learning in a novel, interdisciplinary manner.
Given a set of n colored points P ⊂ℝ^d we wish to store P such that, given some query region Q, we can efficiently report the colors of the points appearing in the query region, along with their frequencies. This is the color frequency reporting problem. We study the case where query regions Q are axis-aligned boxes or dominance ranges. If Q contains k colors, the main goal is to achieve “strictly output-sensitive” query time O(f(n) + k) . We show that, for every s ∈{2,… ,n} , there exists a simple O(nslog _s n) -size data structure for points in ℝ^2 that allows frequency reporting queries in O(log n + klog _s n) time. Furthermore, we give a lower bound for the weighted version of the problem in the arithmetic model, proving that with O(m) space one can not achieve query times better than ( ϕlog (n / ϕ )/log (2m / n)) , where ϕ is the number of possible colors. This means that our data structure is near-optimal. We extend these results to higher dimensions as well. Finally, we give an O(n^1+ε + m log n + K) -time algorithm that can answer m dominance queries ℝ^2 with total output complexity K, while using only linear working space.
We introduce an asymptotically optimal representation of the Manacher array of a string that supports constant-time access. The approach relies on the combinatorial properties of palindromes, yielding a compact yet efficient structure. This work fits within the broader study of compressed text indexing and highlights structural aspects of palindromic substrings that may inspire further algorithmic applications.
We show that every two-way deterministic finite automaton (2dfa) that solves one-way liveness on height h has (h^2) states. This implies a quadratic lower bound for converting one-way nondeterministic finite automata to 2dfas, which asymptotically matches Chrobak’s well-known lower bound for this conversion on unary languages. In contrast to Chrobak’s simple proof, which relies on a 2dfa’s inability to differentiate between any two sufficiently distant locations in a unary input, our argument works on alphabets of arbitrary size and is structured around a main lemma that is general enough to potentially be reused elsewhere.
In this paper, we study the dominating set problem in RDV graphs, a graph class that lies between interval graphs and chordal graphs and defined as the vertex-intersection graphs of downward paths in a rooted tree. It was shown in a previous paper that adjacency queries in an RDV graph can be reduced to the question whether a horizontal segment intersects a vertical segment. This was then used to find a maximum matching in an n-vertex RDV graph, using priority search trees, in O(nlog n) time, i.e., without even looking at all edges. In this paper, we show that if additionally we also use a ray shooting data structure, we can also find a minimum dominating set in an RDV graph O(nlog n) time (presuming a linear-sized representation of the graph is given).. The same idea can also be used for a new proof to find a minimum dominating set in an interval graph in O(n) time.
We consider basic communication tasks in arbitrary radio networks: k-broadcasting and k-gathering. In the case of k-broadcasting, messages from k sources have to get to all nodes in the network. The goal of k-gathering is to collect messages from k source nodes in a designated sink node. We consider these problems in the framework of distributed algorithms with advice. Krisko and Miller showed in 2021 that the optimal size of advice for k-broadcasting is (min (log , log k)) , where is equal to the maximum degree of a vertex of the input communication graph. We show that the same bound (min (log , log k)) on the size of optimal labeling scheme holds also for the k-gathering problems. Moreover, we design fast algorithms for both problems with asymptotically optimal size of advice. For k-gathering, our algorithm works in at most D+k rounds, where D is the diameter of the communication graph. This time bound is optimal even for centralized algorithms. We apply the k-gathering algorithm for k-broadcasting to achieve an algorithm working in time O(D+log ^2 n+k) rounds. We also exhibit a logarithmic time complexity gap between distributed algorithms with advice of optimal size and distributed algorithms with distinct arbitrary labels.
The 3-admissibility of a graph is a promising measure to identify real-world networks that have an algorithmically favourable structure. We design an algorithm that decides whether the 3-admissibility of an input graph G is at most p in time O(m p^7) and space O(n p^3) , where m is the number of edges in G and n the number of vertices. To the best of our knowledge, this is the first explicit algorithm to compute the 3-admissibility. The linear dependence on the input size in both time and space complexity, coupled with an ‘optimistic’ design philosophy for the algorithm itself, makes this algorithm practicable, as we demonstrate with an experimental evaluation on a corpus of 217 real-world networks. Our experimental results show, surprisingly, that the 3-admissibility of most real-world networks is not much larger than the 2-admissibility, despite the fact that the former has better algorithmic properties than the latter.
In the solution discovery problem for a search problem on graphs, we are given an initial placement of k tokens on the vertices of a graph and asked whether this placement can be transformed into a feasible solution by applying a small number of modifications. In this paper, we study the computational complexity of solution discovery for several fundamental vertex-subset problems on graphs, namely Vertex Cover Discovery, Independent Set Discovery, Dominating Set Discovery, and Feedback Vertex Set Discovery. We first present XP algorithms for all four problems parameterized by clique-width. We then prove that Vertex Cover Discovery, Independent Set Discovery, and Feedback Vertex Set Discovery are NP-complete for chordal graphs and graphs of diameter 2, which have unbounded clique-width. In contrast to these hardness results, we show that all three problems can be solved in polynomial time on split graphs. Furthermore, we design an FPT algorithm for Feedback Vertex Set Discovery parameterized by the number of tokens.
We study the online unweighted bipartite matching problem in the random arrival order model, with n offline and n online vertices, in the learning-augmented setting: The algorithm is provided with untrusted predictions of the types (neighborhoods) of the online vertices. We build upon the work of Choo et al. (ICML 2024, pp. 8762–8781) who proposed an approach that uses a prefix of the arrival sequence as a sample to determine whether the predictions are close to the true arrival sequence and then either follows the predictions or uses a known baseline algorithm that ignores the predictions and is β -competitive. Their analysis is limited to the case that the optimal matching has size n, i.e., every online vertex can be matched. We generalize their approach and analysis by removing any assumptions on the size of the optimal matching while only requiring that the size of the predicted matching is at least α n for any constant 0 < α≤ 1 . Our learning-augmented algorithm achieves (1-o(1)) -consistency and (β -o(1)) -robustness. Additionally, we show that the competitive ratio degrades smoothly between consistency and robustness with increasing prediction error.
