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

Fici et al. defined a word to be a k-power if it is the concatenation of k consecutive identical blocks, and an r-antipower if it is the concatenation of r pairwise distinct blocks of the same size. They defined N(k,r) as the smallest ℓ such that every binary word of length ℓ contains either a k-power or an r-antipower. In this note we obtain some new upper and lower bounds on N(k,r). We also consider

We obtain a polynomial-time deterministic (2ee−1+ϵ)-approximation algorithm for maximizing symmetric submodular functions under a budget constraint. Although there exist randomized algorithms with better expected performance, our algorithm achieves the best known factor achieved by a deterministic algorithm, improving on the previously known factor of 6. Furthermore, it is simple, combining two elegant

Given an unpredictable Boolean function f:{0,1}n→{0,1}, the standard Yao's XOR lemma is a statement about the unpredictability of computing ⊕i∈[k]f(xi) given x1,...,xk∈{0,1}n, whereas the Selective XOR lemma is a statement about the unpredictability of computing ⊕i∈Sf(xi) given x1,...,xk∈{0,1}n and S⊆{1,...,k}. We give a reduction from the Selective XOR lemma to the standard XOR lemma. Our reduction

We explore global verifiability; discovering that voting systems vulnerable to attack can be proven to satisfy that security notion, whereas many secure systems cannot. We conclude that current definitions are unsuitable for the analysis of voting systems, fuelling the exploration for a suitable definition.

A dealer-free and non-interactive dynamic threshold secret sharing scheme has been proposed by Harn and Hsu in Information Processing Letters in 2015. In this scheme, a (t,n) secret sharing scheme in secret sharing phase can turn into a (m,n) scheme in secret reconstruction phase, where m is the number of participating shareholders. It has been claimed that the secrecy of shares and the secrecy of

In turn-based games played on graphs, the constrained existence problem of equilibria is a well studied problem. This problem is already known to be decidable in P for weak subgame perfect equilibria in Boolean games with Büchi objectives. In this paper, we prove that this problem is P-complete.

The known linear-time kernelizations for d-Hitting Set guarantee linear worst-case running times using a quadratic-size data structure (that is not fully initialized). Getting rid of this data structure, we show that problem kernels of asymptotically optimal size O(kd) for d-Hitting Set are computable in linear time and space. Additionally, we experimentally compare the linear-time kernelizations for

A vertex v is a Grundy vertex with respect to a proper k-coloring c of a graph G=(V,E) if v has a neighbor of color j for every j (1≤j

We consider two natural problems about nondeterministic finite automata (NFA). First, given an NFA M of n states, and a length ℓ, does M accept a word of length ℓ? We show that the classic problem of triangle-free graph recognition reduces to this problem, and give an O(nω(log⁡n)1+ϵlog⁡ℓ)-time algorithm to solve it, where ω is the optimal exponent for matrix multiplication. Second, provided L(M) is

We study the puzzle game Buttons and Scissors in which the goal is to remove all buttons from an n×m grid by a series of horizontal and vertical cuts. We show that the corresponding decision problem has an algorithm with time complexity 2O(k2log⁡k)+(n+m)O(1), where k is an upper bound on the number of cuts.

QUIXO is a two-player strategic board game played on a 5×5 game board. Players alternately move cubes on the game board, and the player who first made a line of five cubes with the player's symbol wins the game. In this paper, we prove EXPTIME-completeness of deciding the winner on generalized QUIXO played on an n×n game board.

We present a simple reduction of the problem of nondecreasing paths (with minimal last edge weight) in a directed edge-weighted graph to a reachability problem in a directed unweighted graph. The reduction yields an alternative simple method of solving the single-source nondecreasing paths problem in almost linear time. If the edge weights are integers then our algorithm can be implemented in O((n+m)log⁡log⁡n)

The skyline of a set P of points consists of the “best” points with respect to minimization or maximization of the attribute values. A point p dominates another point q if p is as good as q in all dimensions and it is strictly better than q in at least one dimension. In this work, we focus on the 2-d space and provide expected performance guarantees for dynamic (insertions and deletions) 3-sided range

We prove that the constructive weighted coalitional manipulation problem for the Schulze voting rule can be solved in polynomial time for an unbounded number of candidates and an unbounded number of manipulators.

In this paper, we consider a scheduling problem on multiple speed-scalable processors, where the objective is to minimize a weighted sum of total completion times of jobs and total energy consumption on processors. We propose a simple algorithm for the scheduling problem that runs faster than the existing algorithm.

We consider two sub-classes of fully dynamic algorithms for maintaining a maximal matching, which we define as lazy algorithms and eager algorithms. We prove a conditional lower bound which is sub-linear in the number of edges for the update time of lazy algorithms and eager algorithms. Our result is conditioned on the well known conjecture about the hardness of triangle detection in a graph.

There are two options for the leasing problem, buying at a high price or leasing at a low price. Once you choose to purchase the resources, you lose the opportunity to invest in other financial assets for additional benefits, such as a risk-free return of the interest rate. Thus, opportunity cost is an important factor to consider in the leasing problem. Accounting for opportunity cost, this paper

Detecting and eliminating logic hazards in Boolean circuits is a fundamental problem in logic circuit design. We show that there is no O(3(1−ϵ)npoly(s)) time algorithm, for any ϵ>0, that detects logic hazards in Boolean circuits of size s on n variables under the assumption that the strong exponential time hypothesis is true. This lower bound holds even when the input circuits are restricted to be

In 2015, Chou and Orlandi presented an oblivious transfer protocol that already drew a lot of attention both from theorists and practitioners due to its extreme simplicity and high efficiency. Chou and Orlandi claimed that their protocol is universally composable secure (UC-secure) in the random oracle model under dynamic corruptions. UC-security is a very strong security guarantee that assures that

In this paper, we consider the all best swap edges problem in a distributed environment. We are given a 2-edge connected positively weighted network X, where all communication is routed through a rooted spanning tree T of X. If a tree edge e={x,y} fails, the communication network will be disconnected. However, since X is 2-edge connected, communication can be restored by replacing e by non-tree edge

Given a constant number d of n×n matrices with total m non-infinity entries, we show how to construct in (essentially optimal) O˜(mn+n2) time a data structure that can compute in (essentially optimal) O˜(n2) time the distance product of these matrices after incrementing the value (possibly to infinity) of a constant number k of entries. Our result is obtained by designing an oracle for single source

A blockchain is designed to be a self-sufficient decentralised ledger: a peer verifying the validity of past transactions only needs to download the blockchain (the ledger) and nothing else. However, it might be of interest to make two different blockchains interoperable, i.e., to allow one to transmit information from one blockchain to another blockchain. In this paper, we give a formalisation of

In 2014, Mesnager proposed two open problems in [4] on the construction of bent functions. One problem has been settled by Tang et al. in 2017. However, the other is still outstanding, which is solved in this letter by considering a class of PS− vectorial bent functions.

We revisit the notions of cryptographic anonymity and share unpredictability in secret sharing, introducing more systematic and fine grained definitions. We derive tight negative and positive results characterizing access structures with respect to the generalized definitions.

In the sequential setting, a decades-old fundamental result in online algorithms states that if there is a c-competitive randomized online algorithm against an adaptive, offline adversary, then there is a c-competitive deterministic algorithm. The adaptive, offline adversary is the strongest adversary among the ones usually considered, so the result states that if one has to be competitive against