We consider the problem of counting the number of vertices reachable from each vertex in a digraph G, which is equal to computing all the out-degrees of the transitive closure of G. The current (theoretically) fastest algorithms run in quadratic time; however, Borassi has shown that this problem is not solvable in truly subquadratic time unless the Strong Exponential Time Hypothesis fails [Borassi

We study the collision detection of n moving balls of arbitrary sizes in 3-dimensional space. We improve both the storage and the event-handling time of the best existing KDS [1] by several logarithmic factors. Precisely, our KDS maintains O(n) certificates at the current time, uses O(nlog2⁡n) storage, handles each event in O(log3⁡n) time, and processes O(n2) events in the worst case, assuming that

A polygon is simple if it is a closed chain of straight line segments that do not self-intersect. Given a finite set P of input points in the Euclidean plane, the search for a simple polygon with vertex set P is a very well-known and studied computational problem, referred to as the simple polygonization. Its optimization version, that requires to find a simple polygon having either minimum or maximum

We consider the problem of reachability on One-Counter Systems (OCSs) and Pushdown Systems (PDSs). The problem has a well-known O(n3) bound on both models, while the bound is believed to be tight for PDSs. Here we establish new upper and lower bounds for reachability on OCSs and restricted PDSs. We show that the problem can be solved (i) in O(nω⋅log2⁡n) time on OCSs, and (ii) in O(n2) time on sparse

We show that weighted automata over the field of two elements can be exponentially more compact than non-deterministic finite state automata. To show this, we combine ideas from automata theory and communication complexity. However, weighted automata are also efficiently learnable in Angluin's minimal adequate teacher model in a number of queries that is polynomial in the size of the minimal weighted

At EUROCRYPT 2015, Dinur et al. proposed cube-attack-like cryptanalysis on reduced-round Keccak. The process of recovering the key is divided into the preprocessing and the online phase. The preprocessing phase is setting a look-up table by computing the cube sum of involved key bits. The online phase is computing the cube sum of auxiliary variables and recording the matching values in the table as

We study the number of simultaneous diagonal flips required to transform any triangulation into any other. The best known algorithm requires no more than 4×(2log⁡5453+2log⁡65)log⁡n+2≈327.1log⁡n simultaneous flips. This bound is asymptotically tight. In this paper, exploiting some newly discovered properties of blocking and blocked chords we bring the leading constant down to ≈85.8. We also show that

We show that there is no subexponential time algorithm for computing the exact solution of the maximum independent set problem in d-regular graphs, unless ETH fails. We expand our method to show that it helps to provide lowerbounds for other covering problems such as vertex cover and clique. We utilize the construction to show the NP-hardness of MIS on 5-regular planar graphs, closing the exact complexity

We study resource allocation problems in rooted trees in which demand values are given in the leaves. Single-type resources (weights) are to be assigned in the tree nodes such that the total weight in the rooted path from each leaf to the root equals its demand. The goal is to minimize the total costs of the allocated resources. It is known that the linear cost case, i.e., when the cost of a resource

We present another proof for the well-known small model property of two-variable logic. As far as we know, existing proofs of this property are based on a rather intricate model theoretic construction. In contrast, ours uses only simple combinatorial argument which we find more intuitive and direct.

Let P be a set of n points in the plane, no three in a line. The order type of P specifies, for every ordered triple, a positive or negative orientation; and the crossing type (for short, x-type) of P specifies, for every unordered pair of line segments spanned by P, whether they cross each other. Keller and Perles (2016) proved that the x-type of P can be reconstructed from the exchange graph G0(P)

We consider the complexity of finding a fixed point whose existence is guaranteed by an order-theoretic fixed point theorem, e.g., Caristi's fixed point theorem or Brøndsted's fixed point theorem. The problem of finding a fixed point guaranteed by some fixed point theorem is often used to characterize a complexity class. The best well-known such a problem is the problem of finding a Brouwer's fixed

Inspired by recent results for the so-called fair and local linear problems, we define two problems, Local Linear Set and Local Linear Vertex Cover. Here the task is to find a set of vertices X⊆V (which is a vertex cover) of the input graph (V,E) such that the local linear constraints ℓ(v)≤|X∩N(v)|≤u(v) hold for all v∈V; here, ℓ(v) and u(v) are on the input. We prove that both problems are in FPT for

In 1998, Beyer et al. described a nearest neighbor query as unstable if the query point has nearly identical distance from all points in the dataset. Subsequently, researchers have proven that, as data dimensionality goes to infinity, the probability of query instability approaches one for various kinds of data distributions, dataset size functions, and distance metrics. This paper addresses the problem

The problem of counting the number of spanning trees of a network built by a replacement procedure that yields a self-similar structure is considered. This problem has been receiving growing attention in the specialized literature in the recent years. One of the important measures of the global reliability of a network is the number of spanning trees. In this paper, we present a correction of the two

We show that the consistency problem for Statistical EL ontologies, defined by Peñaloza and Potyka, is -hard. Together with existing upper bounds, we conclude -completeness of the logic. Our proof goes via a reduction from the consistency problem for EL extended with negation of atomic concepts.

In the coupon collector's problem with group drawings, a collector buys independent, identically distributed subsets of fixed size s≥2 out of a totality of n coupons. Let Wn,s denote the number of such subsets necessary to obtain each coupon at least once. We prove that, among all distributions that act on the class of all (ns) subsets of size s of the coupons, the uniform distribution minimizes the

Given two programs, say P and Q, a differentiator is a test suite T such that execution of P and Q on T produces different outcomes; this concept is used in mutation testing, where it is important to highlight semantic differences between non-equivalent mutants. Given a specification R and a program P, a detector is a test suite T such that execution of P on T disproves the correctness of P with respect

We define two algebra-based algorithms for solving the Minimum Spanning Tree problem with partially-ordered edges. The parametric structure we propose is a c-semiring, being able to represent different cost-metrics at the same time. We embed c-semirings into the Kruskal and Reverse-delete Kruskal algorithms (thus generalising them), and we suppose the edge costs to be partially ordered. C-semirings

Finding cohesive subgraphs is a fundamental problem for the analysis of graphs. Clique is a classical model of cohesive subgraph, but several alternative definitions have been given in the literature. In this paper we consider the γ-complete graph model, which is based on relaxing the degree constraint of the clique model, that is G[S] is a γ-complete graph, with γ∈(0,1], if and only if every vertex

Given a set ϒ of linear orders, we say that a poset Ha

For all k≥1, we show that deciding whether a graph is k-planar is NP-complete, extending the well-known fact that deciding 1-planarity is NP-complete. Furthermore, we show that the gap version of this decision problem is NP-complete. In particular, given a graph with local crossing number either at most k≥1 or at least 2k, we show that it is NP-complete to decide whether the local crossing number is

We study the two-opposite-facility location game on a line segment, where a mechanism determines the locations of a popular facility and an obnoxious facility for finitely many agents. The agents report their private locations, and try to maximize their own utilities depending on their distances to the facilities. We prove tight bounds on the inapproximability of both randomized and deterministic strategy-proof

In this paper, we introduce the longest previous overlapping factor array of a string – a variant of the longest previous factor array. We show that it can be computed in linear time in the length of the input string, via a reduction to the Max-variant of the Manhattan skyline problem Crochemore et al. (2014) [5].

We study the problem of exploring all vertices of an undirected weighted graph that is initially unknown to the searcher. An edge of the graph is only revealed when the searcher visits one of its endpoints. Beginning at some start node, the searcher's goal is to visit every vertex of the graph before returning to the start node on a tour as short as possible. We prove that the Nearest Neighbor algorithm's

The Bubble-sort graph BSn,n⩾2, is a Cayley graph over the symmetric group Symn generated by transpositions from the set {(12),(23),…,(n−1n)}. It is a bipartite graph containing all even cycles of length ℓ, where 4⩽ℓ⩽n!. We give an explicit combinatorial characterization of all its 4- and 6-cycles. Based on this characterization, we define generalized prisms in BSn,n⩾5, and present a new approach to

We provide a simple and direct proof of the exponential hardness of the KBKF formulas in the proof system Q-Resolution.

In the Bicluster Editing problem the input is a bipartite graph G and an integer k, and the goal is to decide whether G can be transformed into a bicluster graph by adding and removing at most k edges. In this paper we give an algorithm for Bicluster Editing whose running time is O⁎(2.636k).

A de Bruijn sequence of order n over a k-symbol alphabet is a circular sequence where each length-n sequence occurs exactly once. We present a way of extending de Bruijn sequences by adding a new symbol to the alphabet: the extension is performed by embedding a given de Bruijn sequence into another one of the same order, but over the alphabet with one more symbol, while ensuring that there are no long

A weighted point-availability time-dependent network is a list of temporal edges, where each temporal edge has an appearing time value, a travel time value, and a cost value. In this paper we consider the single source Pareto problem in weighted point-availability time-dependent networks, which consists of computing, for any destination d, all Pareto optimal pairs (t,c), where t and c are the arrival

We present the first o(n)-space polynomial-time algorithm for computing the length of a longest common subsequence. Given two strings of length n, the algorithm runs in O(n3) time with O(nlog1.5⁡n2log⁡n) bits of space.

Blömer and Seifert [1] showed that SIVP2 is NP-hard to approximate by giving a reduction from CVP2 to SIVP2 for constant approximation factors as long as the CVP instance has a certain property. In order to formally define this requirement on the CVP instance, we introduce a new computational problem called the Gap Closest Vector Problem with Bounded Minima. We adapt the proof of [1] to show a reduction

In the Split to Block Vertex Deletion and Split to Threshold Vertex Deletion problems the input is a split graph G and an integer k, and the goal is to decide whether there is a set S of vertices of size at most k such that G−S is a block graph and G−S is a threshold graph, respectively. In this paper we give algorithms for these problems whose running times are O⁎(2.076k) and O⁎(1.619k), respectively

In the weighted minimum strongly connected spanning subgraph (WMSCSS ) problem we must purchase a minimum-cost strongly connected spanning subgraph of a digraph. We show that half-integral linear program (LP) solutions for WMSCSS can be efficiently rounded to integral solutions at a multiplicative 1.5 cost. This rounding matches a known 1.5 integrality gap lower bound for a half-integral instance.

A binary relation on graphs is recursively enumerable if and only if it can be computed by a formula of monadic second-order logic. The latter means that the formula defines a set of graphs, in the usual way, such that each “computation graph” in that set determines a pair consisting of an input graph and an output graph.

We investigate the coverability problem for a one-dimensional restriction of pushdown vector addition systems with states. We improve the lower complexity bound to PSpace, even in the acyclic case.

In 1996, Neville Robbins proved the amazing fact that the coefficient of Xn in the Fibonacci infinite product∏n≥2(1−XFn)=(1−X)(1−X2)(1−X3)(1−X5)(1−X8)⋯=1−X−X2+X4+⋯ is always either −1, 0, or 1. The same result was proved later by Federico Ardila using a different method. Meanwhile, in 2001, Jean Berstel gave a simple 4-state transducer that converts an “illegal” Fibonacci representation into a “legal”

If the bisection width of a 2-connected graph G of even order n is not less than n/2, i.e., if the graph satisfies the even cut condition, then G has at least 3n/2−2 edges. Here we characterize the 2-connected extremal graphs with 3n/2−2 edges for every n≥4. For n≥10 an extremal graph has two high-degree vertices, the other vertices have degree 2 and they are located on paths suspended between these

In the Kt-free Edge Deletion problem the input is a graph G and an integer k, and the goal is to decide whether there is a set of at most k edges of G whose removal results in a graph with no clique of size t. In this paper we give a kernel to this problem with O(kt−1) vertices and edges.

We study the isomorphism problem for random hypergraphs. We show that it is solvable in polynomial time for the binomial random k-uniform hypergraph Hn,p;k, for a wide range of p. We also show that it is solvable w.h.p. for random r-regular, k-uniform hypergraphs Hn,r;k,r=O(1).

The constrained longest common subsequence (CLCS) problem was introduced as a specific measure of similarity between molecules. It is a special case of the constrained sequence alignment problem and of the longest common subsequence (LCS) problem, which are both well-studied problems in the scientific literature. Finding similarities between sequences plays an important role in the fields of molecular

We present a distributed algorithm for solving Laplacian systems on strongly connected directed graphs, the first that can be analyzed in terms of the underlying graph's parameters. Our distributed solver works for a large and important class of Laplacian systems that we call “one-sink” Laplacian systems, which includes the important electrical flow computation problem. Specifically, given Dout as

A k-container of a graph G is a set of k internally disjoint paths between two distinct vertices. A k-container of G is a k⁎-container if it contains all vertices of G. Graph G is k⁎-connected if there exists a k⁎-container between any two distinct vertices. A k-connected graph G is super spanning connected if G is k⁎-connected for every r with 1≤r≤k. This study demonstrates that the split-star network

Given a set P of points in the plane, and a point r∈P identified as the root of P, the rooted y−Monotone Minimum Spanning Tree (rooted y−MMST) of P is the connected spanning geometric graph of P in which all the vertices are connected to the root by some y−monotone path and the sum of the Euclidean lengths of its edges is the minimum. We give a linear-time algorithm that draws any rooted tree as a

A subset S of vertices in a graph G with vertex set V and edge set E is a dominating set of G if every vertex of V∖S is adjacent to a vertex in S. The minimum cardinality of a dominating set is the dominating number γ(G) of G. G is a 3-γ-critical graph if γ(G)=3 and γ(G+e)≤2 for e∉E; G is bicritical if G−u−v contains a perfect matching for every pair of distinct vertices u,v∈V. In this paper, we characterize

Erase-limited memory, such as flash memory and phase change memory (PCM), has limitations on the number of times that any memory cell can be erased. The Start-Gap algorithm has shown a significant ability in practice to distribute updates across the cells of an erase-limited memory, but it has heretofore not been characterized in terms of its competitive ratio against an optimal offline algorithm that

Let P be a set of n points in R3 amid a bounded number of obstacles. Consider the metric space M=(P,dM) where dM(p,q) is the geodesic distance of p and q, i.e., the length of a shortest path from p to q avoiding obstacles. When obstacles are axis-parallel boxes, we prove that M admits an 83-spanner with O(nlog3⁡n) edges. In other words, let S be a complete graph on n vertices where each node corresponds

We use an elementary argument building on group actions to prove that the selection-free steady state genetic algorithm analyzed by Sutton and Witt (GECCO 2019) takes an expected number of Ω(2n/n) iterations to find any particular target search point. This bound is valid for all population sizes μ. Our result improves over the previous lower bound of Ω(exp⁡(nδ/2)) valid for population sizes μ=O(n1/2−δ)

In parameterized string matching, a string comprises static and parameterized symbols. Two strings are said to be matched if there exists a one-to-one mapping of parameterized symbols onto itself such that it transforms one string into the other. Traditionally, to construct a full-text index for this problem, suffixes are transformed in a manner referred to as previous occurrence encoding. Although

We show that the number of length-n words over a k-letter alphabet having no even palindromic prefix is the same as the number of length-n unbordered words, by constructing an explicit bijection between the two sets. A slightly different but analogous result holds for those words having no odd palindromic prefix. Using known results on borders, we get an asymptotic enumeration for the number of words

A geometric graph is a graph whose vertices are points in general position in the plane and its edges are straight line segments joining these points. In this paper we give an O(n2log⁡n) time algorithm to compute the number of pairs of edges that cross in a geometric graph on n vertices. For layered graphs and convex geometric graphs the algorithm takes O(n2) time.

In this paper, we introduce two new notions of S-continuous information systems and S-abstract bases which provide concrete representations of stably continuous semi-lattices. We shed light on how the states of an S-continuous information system and the round ideal completion of an S-abstract basis form a stably continuous semi-lattice, respectively. Then we obtain two representation theorems for stably

The complexity of computing the Fourier transform is a longstanding open problem. Very recently, Ailon (2013, 2014, 2015) showed in a collection of papers that, roughly speaking, a speedup of the Fourier transform computation implies numerical ill-condition. The papers also quantify this tradeoff. The main method for proving these results is via a potential function called quasi-entropy, reminiscent

The strong matching preclusion number of a graph is the minimum number of vertices and edges whose deletion results in a graph that has neither perfect matchings nor almost perfect matchings. The class of data center networks has been proposed as a good class of models in the design of computer networks. Such a data center network can be viewed as a server-centric interconnection network structure

The non-uniform dispersion process on the infinite integer line is a synchronous process where n particles are placed at the origin initially, and any particle not exclusively occupying an integer site will move at the next time step to the right adjacent integer with probability pn and to the left with probability 1−pn independently. We characterize the longest distance from the origin when the dispersion

We consider a single machine lot scheduling problem. A number of customer orders of different sizes may be processed in the same lot. We consider first the setting that splitting orders between consecutive lots is allowed. We focus on minimizing the number of tardy orders. A polynomial time solution algorithm is introduced for this problem. We then study the extension to minimizing the weighted number

A secret sharing scheme is a cryptographic technique to protect a secret from loss and leakage by dividing it into shares. A ramp secret sharing scheme can improve efficiency in terms of the information ratio by allowing partial information about the secret which is composed of several sub-secrets to leak out. The notion of strong security has been introduced to control the amount of information on

This paper extends the problem of 2-dimensional palindrome search into the area of approximate matching. Using the Hamming distance as the measure, we search for 2D palindromes that allow up to k mismatches. We consider two different definitions of 2D palindromes and describe efficient algorithms for both of them. The first definition implies a square, while the second definition (also known as a centrosymmetric

The length a(n) of the longest common subsequence of the nth Thue-Morse word and its bitwise complement is studied. An open problem suggested by Jean Berstel in 2006 is to find a formula for a(n). In this paper we prove new lower bounds on a(n) by explicitly constructing a common subsequence between the Thue-Morse words and their bitwise complement. We obtain the lower bound a(n)=2n(1−o(1)), saying

In the restricted Santa Claus problem we are given resources R and players P. Every resource j∈R has a value vj and every player i∈P desires a set of resources R(i). We are interested in distributing the resources to players that desire them. The quality of a solution is measured by the least happy player, i.e., the lowest sum of resource values. This value should be maximized. The local search algorithm