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Given a graph, the general problem to cover the maximum number of vertices by a collection of vertex-disjoint long paths seemingly escapes from the literature. A path containing at least k vertices is considered long. When k ≤ 3, the problem is polynomial time solvable; when k is the total number of vertices, the problem reduces to the Hamiltonian path problem, which is NP-complete. For a fixed k ≥ 4, the problem is NP-hard and the best known approximation algorithm for the weighted set packing problem implies a k-approximation algorithm. To the best of our knowledge, there is no approximation algorithm directly designed for the general problem; when k = 4, the problem admits a 4-approximation algorithm which was presented recently. We propose the first (0.4394 k + O(1))-approximation algorithm for the general problem and an improved 2-approximation algorithm when k = 4. Both algorithms are based on local improvement, and their performance analyses are done via amortization., LIPIcs, Vol. 241, 47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022), pages 53:1-53:14

We design the first efficient sensitivity oracles and dynamic algorithms for a variety of parameterized problems. Our main approach is to modify the algebraic coding technique from static parameterized algorithm design, which had not previously been used in a dynamic context. We particularly build off of the "extensor coding" method of Brand, Dell and Husfeldt [STOC'18], employing properties of the exterior algebra over different fields. For the k-Path detection problem for directed graphs, it is known that no efficient dynamic algorithm exists (under popular assumptions from fine-grained complexity). We circumvent this by designing an efficient sensitivity oracle, which preprocesses a directed graph on n vertices in 2^k poly(k) n^{ω+o(1)} time, such that, given 𝓁 updates (mixing edge insertions and deletions, and vertex deletions) to that input graph, it can decide in time 𝓁² 2^kpoly(k) and with high probability, whether the updated graph contains a path of length k. We also give a deterministic sensitivity oracle requiring 4^k poly(k) n^{ω+o(1)} preprocessing time and 𝓁² 2^{ω k + o(k)} query time, and obtain a randomized sensitivity oracle for the task of approximately counting the number of k-paths. For k-Path detection in undirected graphs, we obtain a randomized sensitivity oracle with O(1.66^k n³) preprocessing time and O(𝓁³ 1.66^k) query time, and a better bound for undirected bipartite graphs. In addition, we present the first fully dynamic algorithms for a variety of problems: k-Partial Cover, m-Set k-Packing, t-Dominating Set, d-Dimensional k-Matching, and Exact k-Partial Cover. For example, for k-Partial Cover we show a randomized dynamic algorithm with 2^k poly(k)polylog(n) update time, and a deterministic dynamic algorithm with 4^k poly(k)polylog(n) update time. Finally, we show how our techniques can be adapted to deal with natural variants on these problems where additional constraints are imposed on the solutions., LIPIcs, Vol. 229, 49th International Colloquium on Automata, Languages, and Programming (ICALP 2022), pages 9:1-9:19

The role of an actor in a social network is identified through a set of measures called centrality. Degree centrality, betweenness centrality, closeness centrality, and clustering coefficient are the most frequently used metrics to compute the node centrality. Their computational complexity in some cases makes unfeasible, when not practically impossible, their computations. For this reason, we focused on two alternative measures, WERW-Kpath and Game of Thieves, which are at the same time highly descriptive and computationally affordable. Our experiments show that a strong correlation exists between WERW-Kpath and Game of Thieves and the classical centrality measures. This may suggest the possibility of using them as useful and more economic replacements of the classical centrality measures.

We combine ideas from distance sensitivity oracles (DSOs) and fixed-parameter tractability (FPT) to design sensitivity oracles for FPT graph problems. An oracle with sensitivity f for an FPT problem �� on a graph G with parameter k preprocesses G in time O(g(f,k) ��� poly(n)). When queried with a set F of at most f edges of G, the oracle reports the answer to the �� - with the same parameter k - on the graph G-F, i.e., G deprived of F. The oracle should answer queries in a time that is significantly faster than merely running the best-known FPT algorithm on G-F from scratch. We design sensitivity oracles for the k-Path and the k-Vertex Cover problem. Our first oracle for k-Path has size O(k^{f+1}) and query time O(f min{f, log(f) + k}). We use a technique inspired by the work of Weimann and Yuster [FOCS 2010, TALG 2013] on distance sensitivity problems to reduce the space to O(({f+k}/f)^f ({f+k}/k)^k fk���log(n)) at the expense of increasing the query time to O(({f+k}/f)^f ({f+k}/k)^k f min{f,k}���log(n)). Both oracles can be modified to handle vertex-failures, but we need to replace k with 2k in all the claimed bounds. Regarding k-Vertex Cover, we design three oracles offering different trade-offs between the size and the query time. The first oracle takes O(3^{f+k}) space and has O(2^f) query time, the second one has a size of O(2^{f+k��+k}) and a query time of O(f+k��); finally, the third one takes O(fk+k��) space and can be queried in time O(1.2738^k + f). All our oracles are computable in time (at most) proportional to their size and the time needed to detect a k-path or k-vertex cover, respectively. We also provide an interesting connection between k-Vertex Cover and the fault-tolerant shortest path problem, by giving a DSO of size O(poly(f,k) ��� n) with query time in O(poly(f,k)), where k is the size of a vertex cover. Following our line of research connecting fault-tolerant FPT and shortest paths problems, we introduce parameterization to the computation of distance preservers. We study the problem, given a directed unweighted graph with a fixed source s and parameters f and k, to construct a polynomial-sized oracle that efficiently reports, for any target vertex v and set F of at most f edges, whether the distance from s to v increases at most by an additive term of k in G-F. The oracle size is O(2^k k�����n), while the time needed to answer a query is O(2^k f^�� k^��), where �� < 2.373 is the matrix multiplication exponent. The second problem we study is about the construction of bounded-stretch fault-tolerant preservers. We construct a subgraph with O(2^{fk+f+k} k ��� n) edges that preserves those s-v-distances that do not increase by more than k upon failure of F. This improves significantly over the O��(f n^{2-1/(2^f)}) bound in the unparameterized case by Bodwin et al. [ICALP 2017]., LIPIcs, Vol. 215, 13th Innovations in Theoretical Computer Science Conference (ITCS 2022), pages 23:1-23:18

We present randomized algorithms for some well-studied, hard combinatorial problems: the k-path problem, the p-packing of q-sets problem, and the q-dimensional p-matching problem. Our algorithms solve these problems with high probability in time exponential only in the parameter (k, p, q) and using polynomial space; the constant bases of the exponentials are significantly smaller than in previous works. For example, for the k-path problem the improvement is from 2 to 1.66. We also show how to detect if a d-regular graph admits an edge coloring with $d$ colors in time within a polynomial factor of O(2^{(d-1)n/2}). Our techniques build upon and generalize some recently published ideas by I. Koutis (ICALP 2009), R. Williams (IPL 2009), and A. Bj\"orklund (STACS 2010, FOCS 2010).

We study the problem of counting the isomorphic occurrences of a k-vertex pattern graph P as a subgraph in an n-vertex host graph G. Our specific interest is on algorithms for subgraph counting that are sensitive to the maximum degree Delta of the host graph. Assuming that the pattern graph P is connected and admits a vertex balancer of size b, we present an algorithm that counts the occurrences of P in G in O ((2 Delta-2)^{(k+b)/2} 2^{-b} n/(Delta) k^2 log n) time. We define a balancer as a vertex separator of P that can be represented as an intersection of two equal-size vertex subsets, the union of which is the vertex set of P, and both of which induce connected subgraphs of P. A corollary of our main result is that we can count the number of k-vertex paths in an n-vertex graph in O((2 Delta-2)^{floor[k/2]} n k^2 log n) time, which for all moderately dense graphs with Delta

The NP-complete $k$-Path problem asks whether a given undirected graph has a (simple) path of length at least $k$. We prove that $k$-Path has polynomial-size Turing kernels when restricted to planar graphs, graphs of bounded degree, claw-free graphs, or to $K_{3,t}$-minor-free graphs for some constant $t$. This means that there is an algorithm that, given a $k$-Path instance $(G,k)$ belonging to one of these graph classes, computes its answer in polynomial time when given access to an oracle that solves $k$-Path instances of size polynomial in $k$ in a single step. The difficulty of $k$-Path can therefore be confined to subinstances whose size is independent of the total input size, but is bounded by a polynomial in the parameter $k$ alone. These results contrast existing superpolynomial lower bounds for the sizes of traditional kernels for the $k$-Path problem on these graph classes: there is no polynomial-time algorithm that reduces any instance $(G,k)$ to a single, equivalent instance $(G',k')$ of size polynomial in $k$ unless $NP \subseteq coNP/poly$. The same positive and negative results apply to the $k$-Cycle problem, which asks for the existence of a cycle of length at least $k$. Our kernelization schemes are based on a new methodology called Decompose-Query-Reduce., 39 pages, 8 figures

The notion of Turing kernelization investigates whether a polynomial-time algorithm can solve an NP-hard problem, when it is aided by an oracle that can be queried for the answers to bounded-size subproblems. One of the main open problems in this direction is whether $$k$$ -Path admits a polynomial Turing kernel: can a polynomial-time algorithm determine whether an undirected graph has a simple path of length k, using an oracle that answers queries of size $$k^{\mathscr {O} (1)}$$ ? We show this can be done when the input graph avoids a fixed graph H as a topological minor, thereby significantly generalizing an earlier result for bounded-degree and $$K_{3,t}$$ -minor-free graphs. Moreover, we show that $$k$$ -Path even admits a polynomial Turing kernel when the input graph is not H-topological-minor-free itself, but contains a known vertex modulator of size bounded polynomially in the parameter, whose deletion makes it so. To obtain our results, we build on the graph minors decomposition to show that any H-topological-minor-free graph that does not contain a k-path, has a separation that can safely be reduced after communication with the oracle.

An edge-ordering of a graph G=(V,E) is a one-to-one function f from E to a subset of the set of positive integers. A path P in G is called an f-ascent if f increases along the edge sequence of P. The height h(f) of f is the maximum length of an f-ascent in G.In this paper we deal with computational problems concerning finding ascents in graphs. We prove that for a given edge-ordering f of a graph G the problem of determining the value of h(f) is NP-hard. In particular, the problem of deciding whether there is an f-ascent containing all the vertices of G is NP-complete. We also study several variants of this problem, discuss randomized and deterministic approaches and provide an algorithm for the finding of ascents of order at least k in graphs of order n in running time O(4knO(1)).

Günümüzde ağ kaynaklarının gelişmesi ve yaygınlaşmasıyla doğru orantılı olarak, ağ üzerinde çalışan uygulamaların çeşitliliğinin artması ve kullanıcıların sayıca fazla olması, var olan fiziksel ağ yapısının etkin bir şekilde kullanılmasını gerektirmektedir.Bu tez çalışmasında öncelikle literatürde görevdeş ağlarda mantıksal katman kurulumu ile ilgili bir çalışma olan Bölgesel-Farkındalı Paylaşımlı Ağ Kurulumu olan ?mOverlay?, yöntemi temel alınarak, kümeleme başarım ölçütleri geliştirilmiş ve bu ölçütlerin literatürde mevcut olan kümeleme ?doğruluğu? ve ?hassasiyeti? ölçütleri ile kıyaslaması yapılmıştır. Daha sonra küme oluşturma üzerine çalışılmış ve farklı lider seçme kriterleri ve farklı dağıtık küme bölünme yöntemleri önerilerek bu yöntemler merkezi olan ve global bilgi kullananK-Means ile kıyaslanmıştır. Today, the growth and widespread use of network resources in accordance with the increasing diversity of applications running on the network as well as increasing number of users require efficient use of existing physical network structureIn this thesis, we first develop alternative performance metrics for clustering and compare them with two metrics from the literature, namely, ?accuracy? and ?correctedness? by using a distributed locality-aware overlay network called ?mOverlay? from the literature. Then cluster formation methods are developed in which alternative leader selection and cluster split policies are proposed and their performance is compared with that of K-Means algorithm from the literature. 73