Algorithmica最新文献

Over the past decades, various metrics have emerged in graph theory to grasp the complex nature of network vulnerability. In this paper, we study two specific measures: (weighted) vertex integrity (wVI) and (weighted) component order connectivity (wCOC). These measures not only evaluate the number of vertices that need to be removed to decompose a graph into fragments, but also take into account the size of the largest remaining component. The main focus of our paper is on kernelization algorithms tailored to both measures. We capitalize on the structural attributes inherent in different crown decompositions, strategically combining them to introduce novel kernelization algorithms that advance the current state of the field. In particular, we extend the scope of the balanced crown decomposition provided by Casel et al. [1] and expand the applicability of crown decomposition techniques. In summary, we improve the vertex kernel of VI from (p^3) to (3p^2), and of wVI from (p^3) to (3(p^2 + p^{1.5} p_ell )), where (p_ell < p) represents the weight of the heaviest component after removing a solution. For wCOC we improve the vertex kernel from (mathcal {O}(k^2W + kW^2)) to (3mu (k + sqrt{mu }W)), where (mu = max (k,W)). We also give a combinatorial algorithm that provides a 2kW vertex kernel in fixed-parameter tractable time when parameterized by r, where (r le k) is the size of a maximum ((W+1))-packing. We further show that the algorithm computing the 2kW vertex kernel for COC can be transformed into a polynomial algorithm for two special cases, namely when (W=1), which corresponds to the well-known vertex cover problem, and for claw-free graphs. In particular, we show a new way to obtain a 2k vertex kernel (or to obtain a 2-approximation) for the vertex cover problem by only using crown structures.

Several works have recently investigated the parameterized complexity of data completion problems, motivated by their applications in machine learning, and clustering in particular. Interestingly, these problems can be equivalently formulated as classical graph problems on induced subgraphs of powers of partially-defined hypercubes. In this paper, we follow up on this recent direction by investigating the Independent Set problem on this graph class, which has been studied in the data science setting under the name Diversity. We obtain a comprehensive picture of the problem’s parameterized complexity and establish its fixed-parameter tractability w.r.t. the solution size plus the power of the hypercube. Given that several such First Order Logic (FO) definable problems have been shown to be fixed-parameter tractable on the considered graph class, one may ask whether fixed-parameter tractability could be extended to capture all FO-definable problems. We answer this question in the negative by showing that FO model checking on induced subgraphs of hypercubes is as difficult as FO model checking on general graphs.

The (d)-Cut problem is to decide whether a graph has an edge cut such that each vertex has at most d neighbours on the opposite side of the cut. If (d=1), we obtain the intensively studied Matching Cut problem. The d-Cut problem has been studied as well, but a systematic study for special graph classes was lacking. We initiate such a study and consider classes of bounded diameter, bounded radius and H-free graphs. We prove that for all (dge 2), (d)-Cut is polynomial-time solvable for graphs of diameter 2, ((P_3+P_4))-free graphs and (P_5)-free graphs. These results extend known results for (d=1). However, we also prove several NP-hardness results for (d)-Cut that contrast known polynomial-time results for (d=1). Our results lead to full dichotomies for bounded diameter and bounded radius and to almost-complete dichotomies for H-free graphs.

Given a set P of n points that are moving in the plane, we consider the problem of computing a spanning tree for these moving points that does not change its combinatorial structure during the point movement. The objective is to minimize the bottleneck weight of the spanning tree (i.e., the largest Euclidean length of all edges) during the whole movement. The problem was solved in (O(n^2)) time previously [Akitaya, Biniaz, Bose, De Carufel, Maheshwari, Silveira, and Smid, WADS 2021]. In this paper, we present a new algorithm of (O(n^{4/3} log ^3 n)) time.

We propose a new, flexible approach for dynamically maintaining successful mutation rates in evolutionary algorithms using k-bit flip mutations. The algorithm adds successful mutation rates to an archive of promising rates that are favored in subsequent steps. Rates expire when their number of unsuccessful trials has exceeded a threshold, while rates currently not present in the archive can enter it in two ways: (i) via user-defined minimum selection probabilities for rates combined with a successful step or (ii) via a stagnation detection mechanism increasing the value for a promising rate after the current bit-flip neighborhood has been explored with high probability. For the minimum selection probabilities, we suggest different options, including heavy-tailed distributions. We conduct rigorous runtime analysis of the flexible evolutionary algorithm on the OneMax and Jump functions, on general unimodal functions, on minimum spanning trees, and on a class of hurdle-like functions with varying hurdle width that benefit particularly from the archive of promising mutation rates. In all cases, the runtime bounds are close to or even outperform the best known results for both stagnation detection and heavy-tailed mutations.

It is well-known that the 2-Thief-Necklace-Splitting problem reduces to the discrete Ham Sandwich problem. In fact, this reduction was crucial in the proof of the (textsf{PPA})-completeness of the Ham Sandwich problem [Filos-Ratsikas and Goldberg, STOC’19]. Recently, a variant of the Ham Sandwich problem called (alpha )-Ham Sandwich has been studied, in which the point sets are guaranteed to be well-separated [Steiger and Zhao, DCG’10]. The complexity of this search problem remains unknown, but it is known to lie in the complexity class (textsf{UEOPL}) [Chiu, Choudhary and Mulzer, ICALP’20]. We define the analogue of this well-separation condition in the necklace splitting problem — a necklace is n-separable, if every subset A of the n types of jewels can be separated from the types ([n]setminus A) by at most n separator points. Since this version of necklace splitting reduces to (alpha )-Ham Sandwich in a solution-preserving way it follows that instances of this version always have unique solutions. We furthermore provide two FPT algorithms: The first FPT algorithm solves 2-Thief-Necklace-Splitting on ((n-1+ell ))-separable necklaces with n types of jewels and m total jewels in time (2^{O(ell log ell )}+O(m^2)). In particular, this shows that 2-Thief-Necklace-Splitting is polynomial-time solvable on n-separable necklaces. Thus, attempts to show hardness of (alpha )-Ham Sandwich through reduction from the 2-Thief-Necklace-Splitting problem cannot work. The second FPT algorithm tests ((n-1+ell ))-separability of a given necklace with n types of jewels in time (2^{O(ell ^2)}cdot n^4). In particular, n-separability can thus be tested in polynomial time, even though testing well-separation of point sets is (textsf{coNP})-complete [Bergold et al., SWAT’22].

Lettericity is a graph parameter responsible for many attractive structural properties. In particular, graphs of bounded lettericity have bounded linear clique-width and they are well-quasi-ordered by induced subgraphs. The latter property implies that any hereditary class of graphs of bounded lettericity can be described by finitely many forbidden induced subgraphs. This, in turn, implies, in a non-constructive way, polynomial-time recognition of such classes. However, no constructive algorithms and no specific bounds on the size of forbidden graphs are available up to date. In the present paper, we develop an algorithm that recognizes n-vertex graphs of lettericity at most k in time (f(k) cdot n^3) and show that any minimal graph of lettericity more than k has at most (2^{O(k^2log k)}) vertices.

The vertex connectivity of a graph G is the size of the smallest set of vertices S such that (G setminus S) is disconnected. For the class of planar graphs, the problem of vertex connectivity is well-studied, both from structural and algorithmic perspectives. Let G be a plane embedded graph, and (Lambda (G)) be an auxiliary graph obtained by inserting a face vertex inside each face and connecting it to all vertices of G incident with the face. If S is a minimal vertex cut of G, then there exists a cycle of length 2|S| whose vertices alternate between vertices of S and face vertices. This structure facilitates the designing of a linear-time algorithm to find minimum vertex cuts of planar graphs. In this paper, we attempt a similar approach for the class of 1-plane graphs—these are graphs with a drawing on the plane where each edge is crossed at most once. We consider different classes of 1-plane graphs based on the subgraphs induced by the endpoints of crossings. For 1-plane graphs where the endpoints of every crossing induce the complete graph (K_4), we show that the structure of minimum vertex cuts is identical to that in plane graphs, as mentioned above. For 1-plane graphs where the endpoints of every crossing induce at least three edges (i.e., one edge apart from the crossing pair of edges), we show that for any minimal vertex cut S, there exists a cycle of diameter O(|S|) in (Lambda (G)) such that all vertices of S are in the neighbourhood of the cycle. This structure enables us to design a linear time algorithm to compute the vertex connectivity of all such 1-plane graphs.

We say that a (multi)graph ( user2{G} = (user2{V},user2{E}) ) has geometric thickness t if there exists a straight-line drawing ( user2{varphi }:user2{V} to mathbb{R}^{{mathbf{2}}} ) and a t-coloring of its edges where no two edges sharing a point in their relative interior have the same color. The Geometric Thickness problem asks whether a given multigraph has geometric thickness at most t. This problem was shown to be NP-hard for ( user2{t} = mathbf{2} ) (Durocher et al. Comput Geom 56:1–18, 2016. https://doi.org/10.1016/j.comgeo.2016.03.003). In this paper, we settle the computational complexity of Geometric Thickness by showing that it is (exists mathbb {R})-complete already for thickness 30. Moreover, our reduction shows that the problem is (exists mathbb {R})-complete for 4392-planar graphs, where a graph is k-planar if it admits a topological drawing with at most k crossings per edge. In the course of our paper we answer previous questions on geometric thickness and on other related problems, in particular that simultaneous graph embeddings of 31 edge-disjoint graphs and pseudo-segment stretchability with chromatic number 30 are (exists mathbb {R})-complete.

Let d be a (well-behaved) shortest-path metric defined on a path-connected subset of (mathbb {R}^2) and let (mathcal {D}={D_1,ldots,D_n}) be a set of geodesic disks with respect to the metric d. We prove that (mathcal {G}^{times }(mathcal {D})), the intersection graph of the disks in (mathcal {D}), has a clique-based separator consisting of (O(n^{3/4+varepsilon })) cliques. This significantly extends the class of objects whose intersection graphs have small clique-based separators. Our clique-based separator yields an algorithm for q-Coloring that runs in time (2^{O(n^{3/4+varepsilon })}), assuming the boundaries of the disks (D_i) can be computed in polynomial time. We also use our clique-based separator to obtain a simple, efficient, and almost exact distance oracle for intersection graphs of geodesic disks. Our distance oracle uses (O(n^{7/4+varepsilon })) storage and can report the hop distance between any two nodes in (mathcal {G}^{times }(mathcal {D})) in (O(n^{3/4+varepsilon })) time, up to an additive error of one. So far, distance oracles with an additive error of one that use subquadratic storage and sublinear query time were not known for such general graph classes.