A graph G is called a sum graph if there is a sum labeling of G, i.e., an injective function ℓ:V(G)→N such that for every u,v∈V(G) it holds that uv∈E(G) if and only if there exists a vertex w∈V(G) such that ℓ(u)+ℓ(v)=ℓ(w). We say that sum labeling ℓ is minimal if there is a vertex u∈V(G) such that ℓ(u)=1. In this paper, we show that if we relax the conditions (either allow non-injective labelings or consider graphs with loops) then there are sum graphs without a minimal labeling, which partially answers the question posed by Miller, Ryan and Smyth.
Given a connected graph G whose vertices are coloured in some way, a discordant voting process on G is as follows. At each step a pair of adjacent vertices with different colours interact, and one of the vertices changes its colour to match the other one. If eventually all vertices have the same colour, we say a consensus has been reached. A vertex is discordant if it has a discordant edge, i.e. a neighbour of a different colour. In the general discordant voting process, at each step a discordant vertex u is chosen uniformly at random, and then a discordant edge (u, v) is chosen from among the discordant edges of u, also uniformly at random. With probability beta vertex u adopts the colour of vertex v and with probability 1 - beta vertex v adopts the colour of vertex u. Let T be the number of steps needed to reach consensus. For the complete graph K-n with an initial colouring where half the vertices are red and half blue, then when beta = 0, ET = circle dot(n logn), whereas when beta = 1 then ET = circle dot(2(n)). The case beta = 1/2 corresponds to a simple random walk on a path with vertex set {0, 1,...,n} and has ET = n(2) / 4. We study the effect of varying beta from zero to one, thus revealing the detailed transition from ET = circle dot(n logn) to ET = circle dot(2(n)). In terms of beta, the transition from circle dot(n logn) to circle dot(n(2)) occurs in a scaling window of width O(1/n) around beta = 1/2. For any a > 1, there is an explicit value of beta = 1/2 + O(logn/n) for which ET = circle dot(n(a)). When beta > 1/2 constant, there is an explicit value a(beta) such that ET = circle dot(a(n)). (C) 2018 Elsevier B.V. All rights reserved.
A linear extension of a poset $P$ is a permutation of the elements of the set that respects the partial order. Let $L(P)$ denote the number of linear extensions. It is a #P complete problem to determine $L(P)$ exactly for an arbitrary poset, and so randomized approximation algorithms that draw randomly from the set of linear extensions are used. In this work, the set of linear extensions is embedded in a larger state space with a continuous parameter ?. The introduction of a continuous parameter allows for the use of a more efficient method for approximating $L(P)$ called TPA. Our primary result is that it is possible to sample from this continuous embedding in time that as fast or faster than the best known methods for sampling uniformly from linear extensions. For a poset containing $n$ elements, this means we can approximate $L(P)$ to within a factor of $1 + \epsilon$ with probability at least $1 - \delta$ using an expected number of random bits and comparisons in the poset which is at most $O(n^3(ln n)(ln L(P))\epsilon^{-2}\ln \delta^{-1}).$
This paper proposes a heuristic algorithm for designing real-world school transport schedules. It extends previously considered problem models by considering some important but hitherto overlooked features including the splitting and merging of routes, gauging vehicle dwell times, the selection of stopping points, and the minimisation of walking distances. We show that this formulation contains a number of interacting combinatorial subproblems including the time-constrained vehicle routing problem, set covering, and bin packing. As a result, a number of new and necessary algorithmic operators are proposed for this problem which are then used alongside other recognised heuristics. Primarily, the aim of this algorithm is to minimise the number of vehicles used by each school, though secondary issues concerning journey lengths and walking distances are also considered through the employment of suitable multiobjective techniques.
In this paper we introduce the notion of an optimal cover. Let M denote the maximum number of positions in w covered by any repeating substring of w. Then a longest (shortest) optimal cover u is a longest (shortest) repeating substring of w that covers M positions. The advantage of this notion is that it is not only applicable to all strings, but also that it does not share the deficiencies of the existing definitions of covers. We show that both the longest and the shortest optimal covers for a given string w of length n can be computed easily and efficiently in O(nlogn) time and O(n) space. We further show that the data structures used to compute optimal covers also compute the α-partial covers introduced in [10].
In this paper we present an algorithm to compute the Lyndon array of a string T of length n as a byproduct of the inversion of the Burrows–Wheeler transform of T. Our algorithm runs in linear time using only a stack in addition to the data structures used for Burrows–Wheeler inversion. We compare our algorithm with two other linear-time algorithms for Lyndon array construction and show that computing the Burrows–Wheeler transform and then constructing the Lyndon array is competitive compared to the known approaches. We also propose a new balanced parenthesis representation for the Lyndon array that uses 2n+o(n) bits of space and supports constant time access. This representation can be built in linear time using O(n) words of space, or in O(nlogn/loglogn) time using asymptotically the same space as T.
Document listing is a fundamental problem in information retrieval. The objective is to retrieve all documents from a document collection that are relevant to an input pattern. Several variations of this problem such as ranked document retrieval, document listing with two patterns and forbidden patterns have been studied. We introduce the problem of document retrieval with forbidden extension. Let D = {T-1, T-2,...,T-D} be a collection of D string documents of n characters in total, and P+ and P- be two query patterns, where P+ is a proper prefix of P-. We call P- as the forbidden extension of the included pattern P+ A forbidden extension query (P+, P-) asks to report all occ documents in D that contains P+ as a substring, but does not contain P- as one. A top-k forbidden extension query (P+, P-, k) asks to report those k documents among the occ documents that are most relevant to P+, where each document is given a unique fixed score (PageRank) and the relevance of a document is determined based on its score. We present a linear index (in words) with an O (vertical bar P-vertical bar + occ) query time for the document listing problem. For the top-k version of the problem, we achieve the following space-time trade-offs: center dot O(n) space (in words) and O ((vertical bar P-vertical bar log sigma + k) query time. center dot vertical bar CSA vertical bar + vertical bar CSA*vertical bar + D log n/D + O (n) bits and O (search(P-) + k . t(S)(A) center dot log(2+is an element of) n) query time, where is an element of > 0 is an arbitrarily small constant. center dot vertical bar CSA vertical bar + O (n log D) bits and O (search(P-) + (k + log D) log D) query time. Here sigma is the size of the alphabet set, CSA (of size vertical bar CSA vertical bar bits) is the compressed suffix array (CSA) of the concatenated text of all documents, CSA(d) is the CSA of T-d and vertical bar CSA*vertical bar = Sigma(D)(d=1) vertical bar CSA(d)vertical bar. Also, search(P-) is the time for pattern matching and t(S)(A) is the time to find suffix (or inverse suffix) array value. Published by Elsevier B.V.
The 0–1 Incremental Knapsack Problem (IKP) is a generalization of the standard 0–1 Knapsack Problem (KP) where the capacity grows over time periods and if an item is placed in the knapsack in a certain period, it cannot be removed afterwards. The problem calls for maximizing the sum of the profits over the whole time horizon. In this work, we consider the case with three time periods where we assume that each item can be packed also in the first time period. We propose an approximation algorithm with a tight approximation ratio of 3037. We strongly rely on Linear Programming (LP) to derive this bound showing how the proposed LP-based analysis can be seen as a valid alternative to more formal proof systems.
Given a vertex-coloured graph, a dominating set is said to be tropical if every colour of the graph appears at least once in the set. Here, we study minimum tropical dominating sets from structural and algorithmic points of view. First, we prove that the tropical dominating set problem is NP-complete even when restricted to a simple path. Then, we establish upper bounds related to various parameters of the graph such as minimum degree and number of edges. We also give an optimal upper bound for random graphs. Last, we give approximability and inapproximability results for general and restricted classes of graphs, and establish a FPT algorithm for interval graphs.
This article is devoted to the study of the complexity of Power Edge Set (PES), a problem occurring when monitoring power networks. We show that PES remains NP-hard in bipartite planar graphs with bounded degree. This result is extended to unit disk graphs and grids with maximum degree three. To the best of our knowledge, this is the most restricted class of graphs known so far on which Power Edge Set is NP-complete. We also show that there is no 2o(n)-time algorithm for Power Edge Set, and there is no 2o(k)nO(1)-time algorithm for its parameterized version, even in bipartite planar subcubic graphs and unit disk graphs, assuming ETH. Finally, we show a polynomial-time algorithm for graphs of bounded treewidth (XP wrt. treewidth), such as series-parallel graphs or outerplanar graphs.
The context in which a substring appears is an important notion to identify – for example – its semantic meaning. However, existing definitions from stringology fail to model the context explicitly. We introduce here xkcd-repeats, a new family of repeats characterized by the number of different symbols at the left and right of their occurrences. These repeats include as special extreme cases the well-known classes of maximal and super-maximal repeats.We give sufficient and necessary condition to bound their number linearly in the size of the original string, and show an optimal algorithm that computes them in linear time – given a suffix array –, independent of the size of the alphabet, as well as two other algorithms that are faster in practice.We extend this in two ways: first we show how to generalize the notion of context. Instead of reducing it to the single symbol before and after an occurrence, we propose the notion of unbounded context, while keeping linear representation and computing time. Secondly, we provide a general framework which allows to compute these (and other) repeats incrementally; opening the door to streaming application.
Given a positive integer K, the weak Schur problem is to find the largest weakly sum-free K-partition of positive integers. Our goal is to increase known values of n for which the set {1,…,n} can be divided into sum-free sets, i.e. sets for which there are no three distinct elements x, y, z in the same set such that x+y=z.Our main contribution consists in designing a recursive deterministic algorithm which improves the results reported in the literature. It constructs a solution out of “perfect” or, at times, “perforated” series of integers using arbitrary rules. We call it the Deterministic Weak Schur Series (DWSS) algorithm. When analyzing our algorithm, we use the Gros sequence to bound from above the number of operations it executes. Consequently, we prove that its complexity is in O(33K). With DWSS, we discovered larger weakly sum-free partitions, breaking the literature records for K=8 to K=12.We next hybridize DWSS by stringing it together with a Monte Carlo-based method which explores the neighborhood of the solution obtained deterministically. We call it the Randomized Weak Schur Series (RWSS). As this algorithm requires more computational effort than DWSS, we are not able to compute partitions for K>10. We exceed our own DWSS results for K=8 to K=10.
This article gives a survey of combinatorial algorithms and methods for database security related to the work of Mirka Miller. The main contributions of Mirka Miller and coauthors to the security of statistical databases include the introduction of Static Audit Expert and theorems determining time complexity of its combinatorial algorithms, a polynomial time algorithm for deciding whether the maximum possible usability can be achieved in statistical database with a special class of answerable statistics, NP-completeness of similar problems concerning several other types of databases, sharp upper bounds on the number of compromise-free queries in certain categories of statistical databases, and analogous results on applications of Static Audit Expert for the prevention of relative compromise.
There exist many optimal (using single and multiple processors) and approximate solutions to the longest increasing subsequence (LIS) problem. Through this paper, we present the enhancement to the divide-and-conquer approach presented in paper [1]. An improved D&C algorithmic solution is proposed which outputs optimal solution in all cases. The proposed algorithm takes O(nlogn) time in best and average cases and o(nlog2n) time in worst case. The portion of the proposed solution can run in parallel using multiprocessors.
In their ground-breaking paper on grammar-based compression, Charikar et al. (2005) gave a separation between straight-line programs (SLPs) and Lempel–Ziv '77 (LZ77): they described an infinite family of strings such that the size of the smallest SLP generating a string of length n in that family, is an Ω(logn/loglogn)-factor larger than the size of the LZ77 parse of that string. However, the strings in that family have run-length SLPs (RLSLPs) — i.e., SLPs in which we can indicate many consecutive copies of a symbol by only one copy with an exponent — as small as their LZ77 parses. In this paper we modify Charikar et al.'s proof to obtain the same Ω(logn/loglogn)-factor separation between RLSLPs and LZ77.
In this paper, we show a very close relationship between the compaction, vertex-compaction, and retraction problems for reflexive and bipartite graphs. The relationships that we present relate to a long-standing open problem concerning whether any pair of these problems are polynomially equivalent for every graph. The relationships we present also relate to the constraint satisfaction problem, providing evidence that similar to the compaction and retraction problems, it is also likely to be difficult to give a complete computational complexity classification of the vertex-compaction problem for every reflexive or bipartite graph. In this paper, we however give a complete computational complexity classification of the vertex-compaction problem for all graphs, including even partially reflexive graphs, with four or fewer vertices, by giving proofs based on mostly just knowing the computational complexity classification results of the compaction problem for such graphs determined earlier by the author. Our results show that the compaction, vertex-compaction, and retraction problems are polynomially equivalent for every graph with four or fewer vertices.
Indeterminate strings have received considerable attention in the recent past; see for example [1] and [3]. This attention is due to their applicability in bioinformatics, and to the natural correspondence with undirected graphs. One aspect of this correspondence is the fact that the minimum alphabet size of indeterminates representing any given undirected graph equals the size of the minimal clique cover of this graph. This paper first considers a related problem proposed in [3]: characterize Θn(m), which is the size of the largest possible minimal clique cover (i.e., an exact upper bound), and hence alphabet size of the corresponding indeterminate, of any graph on n vertices and m edges. We provide improvements to the known upper bound for Θn(m) in section 3.3. [3] also presents an algorithm which finds clique covers in polynomial time. We build on this result with a heuristic for vertex sorting which significantly improves their algorithm's results, particularly in dense graphs.
The pattern scan order is a major factor affecting the performance of string matching algorithms. Depending on the pattern scan order, one can reduce the number of comparisons in a window or increase the shift length. Classical algorithms for string matching determine the pattern scan order only using the characteristics of a text and a pattern. However, if we additionally use the scan results at the time we determine each scan position of the pattern, we can improve the performance of string matching. In this paper we propose new pattern-scan-order algorithms that maximize shift lengths using scan results. We present the theoretical analysis and experimental results that these algorithms run faster than previous algorithms on average.
We consider the task to compute the pathwidth of a graph which has been shown to be equivalent to the vertex separation problem. The latter is naturally modeled as a linear ordering problem w.r.t. the vertices of the graph. Mixed-integer programs proposed so far express linear orders using either position or set assignment variables. As we show, the lower bound on the pathwidth obtained when solving their linear programming relaxations is zero for any directed graph. We then present a new formulation based on conventional linear ordering variables and a slightly different perspective on the problem that sustains stronger lower bounds. An experimental evaluation of three mixed-integer programs, each representing one of the different modeling schemes, displays their potentials and limitations when used to solve the problem to optimality.
In this paper we consider a scenario where a user wants to outsource her documents to the cloud, so that she can later reliably delegate (to the cloud) pattern matching operations on these documents. We propose an efficient solution to this problem that relies on the homomorphic MAC for polynomials proposed by Catalano and Fiore in [14]. Our main contribution are new methods to express pattern matching operations (both in their exact and approximate variants) as low degree polynomials, i.e. polynomials whose degree solely depends on the size of the pattern. To better assess the practicality of our schemes, we propose a concrete implementation that further optimizes the efficiency of the homomorphic MAC from [14]. Our implementation shows that the proposed protocols are extremely efficient for the client, while remaining feasible at server side.
