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In the thief orienteering problem an agent called a thief carries a knapsack of capacity W and has a time limit T to collect a set of items of total weight at most W and maximum profit along a simple path in a weighted graph (G = (V, E)) from a start vertex s to an end vertex t. There is a set I of items each with weight (w_{i}) and profit (p_{i}) that are distributed among (V{setminus }{s,t}). The time needed by the thief to travel an edge depends on the length of the edge and the weight of the items in the knapsack at the moment when the edge is traversed. There is a polynomial-time approximation scheme for a relaxed version of the thief orienteering problem on directed acyclic graphs that produces solutions that use time at most (T(1 + epsilon )) for any constant (epsilon > 0). We give a polynomial-time algorithm for transforming instances of the problem on 2-terminal series–parallel graphs into equivalent instances of the thief orienteering problem on directed acyclic graphs; therefore, yielding a polynomial-time approximation scheme for the relaxed version of the thief orienteering problem on this graph class.

A graph (G=(V,E)) is a star-k-pairwise compatibility graph (star-k-PCG) if there exists a weight function (w: V rightarrow mathbb {R}^+) and k mutually exclusive intervals (I_1, I_2, ldots I_k), such that there is an edge (uv in E) if and only if (w(u)+w(v) in bigcup _i I_i). These graphs are related to two important classes of graphs: pairwise compatibility graphs (PCGs) and multithreshold graphs. It is known that for any graph G there exists a k such that G is a star-k-PCG. Thus, for a given graph G it is interesting to know which is the minimum k such that G is a star-k-PCG. We define this minimum k as the star number of the graph, denoted by (gamma (G)). Here we investigate the star number of simple graph classes, such as graphs of small size, caterpillars, cycles and grids. Specifically, we determine the exact value of (gamma (G)) for all the graphs with at most 7 vertices. By doing so we show that the smallest graphs with star number 2 are only 4 and have exactly 5 vertices; the smallest graphs with star number 3 are only 3 and have exactly 7 vertices. Next, we provide a construction showing that the star number of caterpillars is one. Moreover, we show that the star number of cycles and two-dimensional grid graphs is 2 and that the star number of 4-dimensional grids is at least 3. Finally, we conclude with numerous open problems.

For (tge 3), (K_{1, t}) is called t-claw. A graph (G=(V, E)) is t-claw free if it does not contain t-claw as a vertex-induced subgraph. In minimum t-claw deletion problem (Min-t-Claw-Del), given a graph (G=(V, E)), it is required to find a vertex set S of minimum size such that (G[Vsetminus S]) is t-claw free. In a split graph, the vertex set is partitioned into two sets such that one forms a clique and the other forms an independent set. Every t-claw in a split graph has a center vertex in the clique partition. This observation motivates us to consider the minimum one-sided bipartite t-claw deletion problem (Min-t-OSBCD). Given a bipartite graph (G=(A cup B, E)), in Min-t-OSBCD it is asked to find a vertex set S of minimum size such that (G[(A cup B) {setminus } S]) has no t-claw with the center vertex in A. A primal-dual algorithm approximates Min-t-OSBCD within a factor of t. We prove that it is ({textsf{UGC}})-hard to approximate with a factor better than t. We also prove it is approximable within a factor of 2 for dense bipartite graphs. By using these results on Min-t-OSBCD, we prove that Min-t-Claw-Del is ({textsf{UGC}})-hard to approximate within a factor better than t, for split graphs. We also consider their complementary maximization problems and prove that they are ({textsf{APX}})-complete.

The piecewise complexity h(u) of a word is the minimal length of subwords needed to exactly characterise u. Its piecewise minimality index (rho (u)) is the smallest length k such that u is minimal among its order-k class ([u]_k) in Simon’s congruence. We initiate a study of these two descriptive complexity measures. Among other results, we provide efficient algorithms for computing h(u) and (rho (u)) for a given word u.

It was recently proved that any straight-line program (SLP) generating a given string can be transformed in linear time into an equivalent balanced SLP of the same asymptotic size. We generalize this proof to a general class of grammars we call generalized SLPs (GSLPs), which allow rules of the form (A rightarrow x) where x is any Turing-complete representation (of size |x|) of a sequence of symbols (potentially much longer than |x|). We then specialize GSLPs to so-called Iterated SLPs (ISLPs), which allow rules of the form (A rightarrow Pi _{i=k_1}^{k_2} B_1^{i^{c_1}}cdots B_t^{i^{c_t}}) of size (mathcal {O}(t)). We prove that ISLPs break, for some text families, the measure (delta ) based on substring complexity, a lower bound for most measures and compressors exploiting repetitiveness. Further, ISLPs can extract any substring of length (lambda ), from the represented text (T[1mathinner {.,.}n]), in time (mathcal {O}(lambda + log ^2 nlog log n)). This is the first compressed representation for repetitive texts breaking (delta ) while, at the same time, supporting direct access to arbitrary text symbols in polylogarithmic time. We also show how to compute some substring queries, like range minima and next/previous smaller value, in time (mathcal {O}(log ^2 n log log n)). Finally, we further specialize the grammars to run-length SLPs (RLSLPs), which restrict the rules allowed by ISLPs to the form (A rightarrow B^t). Apart from inheriting all the previous results with the term (log ^2 n log log n) reduced to the near-optimal (log n), we show that RLSLPs can exploit balancedness to efficiently compute a wide class of substring queries we call “composable”—i.e., (f(X cdot Y)) can be obtained from f(X) and f(Y). As an example, we show how to compute Karp-Rabin fingerprints of texts substrings in (mathcal {O}(log n)) time. While the results on RLSLPs were already known, ours are much simpler and require little precomputation time and extra data associated with the grammar.

Compact directed acyclic word graphs (CDAWGs) (Blumer et al. in J ACM 34(3):578–595, 1987) are a fundamental data structure on strings with applications in text pattern searching, data compression, and pattern discovery. Intuitively, the CDAWG of a string T is obtained by merging isomorphic subtrees of the suffix tree (Weiner, in: Proceedings of the 14th annual symposium on switching and automata theory, pp 1–11, 1973) of the same string T, thus CDAWGs are a compact indexing structure. In this paper, we investigate the sensitivity of CDAWGs when a single character edit operation (insertion, deletion, or substitution) is performed at the left-end of the input string T, namely, we are interested in the worst-case increase in the size of the CDAWG after a left-end edit operation. We prove that if (textsf{e}) is the number of edges of the CDAWG for string T, then the number of new edges added to the CDAWG after a left-end edit operation on T does not exceed (textsf{e}). Further, we present a matching lower bound on the sensitivity of CDAWGs for left-end insertions, and almost matching lower bounds for left-end deletions and substitutions. We then generalize our lower-bound instance for left-end insertions to leftward online construction of the CDAWG, and show that it requires (Omega (n^2)) time for some string of length n.

Locality-sensitive hashing (LSH) is a fundamental algorithmic toolkit used by data scientists for approximate nearest neighbour search problems that have been used extensively in many large-scale data processing applications such as near-duplicate detection, nearest-neighbour search, clustering, etc. In this work, we aim to propose faster and space-efficient locality-sensitive hash functions for Euclidean distance and cosine similarity for tensor data. Typically, the naive approach for obtaining LSH for tensor data involves first reshaping the tensor into vectors, followed by applying existing LSH methods for vector data. However, this approach becomes impractical for higher-order tensors because the size of the reshaped vector becomes exponential in the order of the tensor. Consequently, the size of LSH’s parameters increases exponentially. To address this problem, we suggest two methods for LSH for Euclidean distance and cosine similarity, namely CP-E2LSH, TT-E2LSH, and CP-SRP, TT-SRP, respectively, building on CP and tensor train (TT) decompositions techniques. Our approaches are space-efficient and can be efficiently applied to low-rank CP or TT tensors. We provide a rigorous theoretical analysis of our proposal on their correctness and efficacy.

We introduce and discuss the Minimum Capacity-Preserving Subgraph (MCPS) problem: given a directed graph with edge capacities (textit{cap} ) and a retention ratio (alpha in (0,1)), find the smallest subgraph that, for each pair of vertices (u, v), preserves at least a fraction (alpha ) of a maximum u-v-flow’s value. This problem originates from the practical setting of reducing the power consumption in a computer network: it models turning off as many links as possible, while retaining the ability to transmit at least (alpha ) times the traffic compared to the original network. First we prove that MCPS is NP-hard already on a restricted set of directed acyclic graphs (DAGs) with unit edge capacities. Our reduction also shows that a closely related problem (which only considers the arguably most complicated core of the problem in the objective function) is NP-hard to approximate within a sublogarithmic factor already on DAGs. In terms of positive results, we present two algorithms that solve MCPS optimally on directed series-parallel graphs (DSPs): a simple linear-time algorithm for the special case of unit edge capacities and a cubic-time dynamic programming algorithm for the general case of non-uniform edge capacities. Further, we introduce the family of laminar series-parallel graphs (LSPs), a generalization of DSPs that also includes cyclic and very dense graphs. Their properties allow us to solve MCPS on LSPs by employing our DSP-algorithms as subroutines. In addition, we give a separate quadratic-time algorithm for MCPS on LSPs with unit edge capacities that also yields straightforward quadratic time algorithms for several related problems such as Minimum Equivalent Digraph and Directed Hamiltonian Cycle on LSPs.

Let G be a connected graph. For an edge (e=xy in E(G)), e is monitored by a vertex v if (d_G(v, y)ne d_{G-e}(v, y)) or (d_G(v, x)ne d_{G-e}(v, x)). A set M of vertices of a graph G is distance-edge-monitoring (DEM for short) set if every edge e of G is monitored by some vertex of M. A DEM set X for a graph G is called fault-tolerant DEM set if (Xsetminus {v}) is also DEM set for each v in X. Denote (operatorname {dem}(G)) and (operatorname {Fdem}(G)) the smallest size of DEM set and fault-tolerant DEM sets, respectively. In this paper, we first study the relation between (operatorname {Fdem}(G)) and (operatorname {dem}(G)) for a graph G. Next, we show that (2 le operatorname {Fdem}(G) le n) for any graph G with order n. Furthermore, the extremal graphs attaining lower and upper bounds are characterized. In the end, the exact values for some networks are given. Furthermore, it is shown that for (2le s<tle n), there exists a graph G of order n such that (operatorname {dem}(G)=s) and (operatorname {Fdem}(G)=t).