Algorithmica最新文献

Population diversity is crucial in evolutionary algorithms as it helps with global exploration and facilitates the use of crossover. Despite many runtime analyses showing advantages of population diversity, we have no clear picture of how diversity evolves over time. We study how the population diversity of ((mu +1)) algorithms, measured by the sum of pairwise Hamming distances, evolves in a fitness-neutral environment. We give an exact formula for the drift of population diversity and show that it is driven towards an equilibrium state. Moreover, we bound the expected time for getting close to the equilibrium state. We find that these dynamics, including the location of the equilibrium, are unaffected by surprisingly many algorithmic choices. All unbiased mutation operators with the same expected number of bit flips have the same effect on the expected diversity. Many crossover operators have no effect at all, including all binary unbiased, respectful operators. We review crossover operators from the literature and identify crossovers that are neutral towards the evolution of diversity and crossovers that are not.

Since the CSP dichotomy conjecture has been established, a number of other dichotomy questions have attracted interest, including one for list homomorphism problems of signed graphs. Signed graphs arise naturally in many contexts, including for instance nowhere-zero flows for graphs embedded in non-orientable surfaces. The dichotomy classification is known for homomorphisms without list restrictions, so it is surprising that it is not known, or even conjectured, if lists are present since this usually makes the classifications easier to obtain. There is however a conjectured classification, due to Kim and Siggers, in the special case of “semi-balanced” signed graphs. These authors confirmed their conjecture for the class of reflexive signed graphs. As our main result we verify the conjecture for irreflexive signed graphs. For this purpose, we prove an extension result for two-directional ray graphs which is of independent interest and which leads to an analogous extension result for interval graphs. Moreover, we offer an alternative proof for the class of reflexive signed graphs, and a direct polynomial-time algorithm in the polynomial cases where the previous algorithms used algebraic methods of general CSP dichotomy theorems. For both reflexive and irreflexive cases the dichotomy classification depends on a result linking the absence of certain structures to the existence of a special ordering. The structures are used to prove the NP-completeness and the ordering is used to design polynomial algorithms.

A conflict-free coloring of a graph G is a (partial) coloring of its vertices such that every vertex u has a neighbor whose assigned color is unique in the neighborhood of u. There are two variants of this coloring, one defined using the open neighborhood and one using the closed neighborhood. For both variants, we study the problem of deciding whether the conflict-free coloring of a given graph G is at most a given number k.

In this work, we investigate the relation of clique-width and minimum number of colors needed (for both variants) and show that these parameters do not bound one another. Moreover, we consider specific graph classes, particularly graphs of bounded clique-width and types of intersection graphs, such as distance hereditary graphs, interval graphs and unit square and disk graphs. We also consider Kneser graphs and split graphs. We give (often tight) upper and lower bounds and determine the complexity of the decision problem on these graph classes, which improve some of the results from the literature. Particularly, we settle the number of colors needed for an interval graph to be conflict-free colored under the open neighborhood model, which was posed as an open problem.

Motivated by applications in DNA-nanotechnology, theoretical investigations in algorithmic tile-assembly have blossomed into a mature theory. In addition to computational universality, the abstract Tile Assembly Model (aTAM) was shown to be intrinsically universal (FOCS 2012), a strong notion of completeness where a single tile set is capable of simulating the full dynamics of all systems within the model; however, this construction fundamentally required non-deterministic tile attachments. This was confirmed necessary when it was shown that the class of directed aTAM systems, those where all possible sequences of tile attachments result in the same terminal assembly, is not intrinsically universal (FOCS 2016). Furthermore, it was shown that the non-cooperative aTAM, where tiles only need to match on 1 side to bind rather than 2 or more, is not intrinsically universal (SODA 2014) nor computationally universal (STOC 2017). Building on these results to further investigate the other dynamics, Hader et al. examined several tile-assembly models which varied across (1) the numbers of dimensions used, (2) how tiles diffused through space, and (3) whether each system is directed, and determined which models exhibited intrinsic universality (SODA 2020). In this paper we extend those results to provide direct comparisons of the various models against each other by considering intrinsic simulations between models. Our results show that in some cases, one model is strictly more powerful than another, and in others, pairs of models have mutually exclusive capabilities. This paper is a greatly expanded version of that which appeared in ICALP 2023.

A permutation (pi ) is a merge of a permutation (sigma ) and a permutation (tau ), if we can color the elements of (pi ) red and blue so that the red elements have the same relative order as (sigma ) and the blue ones as (tau ). We consider, for fixed hereditary permutation classes (mathcal {C}) and (mathcal {D}), the complexity of determining whether a given permutation (pi ) is a merge of an element of (mathcal {C}) with an element of (mathcal {D}). We develop general algorithmic approaches for identifying polynomially tractable cases of merge recognition. Our tools include a version of streaming recognizability of permutations via polynomially constructible nondeterministic automata, as well as a concept of bounded width decomposition, inspired by the work of Ahal and Rabinovich. As a consequence of the general results, we can provide nontrivial examples of tractable permutation merges involving commonly studied permutation classes, such as the class of layered permutations, the class of separable permutations, or the class of permutations avoiding a decreasing sequence of a given length. On the negative side, we obtain a general hardness result which implies, for example, that it is NP-complete to recognize the permutations that can be merged from two subpermutations avoiding the pattern 2413.

The partial representation extension problem generalizes the recognition problem for geometric intersection graphs. The input consists of a graph G, a subgraph (H subseteq G) and a representation (mathcal R') of H. The question is whether G admits a representation (mathcal R) whose restriction to H is (mathcal R'). We study this question for circle graphs, which are intersection graphs of chords of a circle. Their representations are called chord diagrams. We show that for a graph with n vertices and m edges the partial representation extension problem can be solved in (O((n + m) alpha (n + m))) time, thereby improving over an (O(n^3))-time algorithm by Chaplick et al. (J Graph Theory 91(4), 365–394, 2019). The main technical contributions are a canonical way of orienting chord diagrams and a novel compact representation of the set of all canonically oriented chord diagrams that represent a given circle graph G, which is of independent interest.

We extend known results on chordal graphs and distance-hereditary graphs to much larger graph classes by using only a common metric property of these graphs. Specifically, a graph is called (alpha _i)-metric ((iin {mathcal {N}})) if it satisfies the following (alpha _i)-metric property for every vertices u, w, v and x: if a shortest path between u and w and a shortest path between x and v share a terminal edge vw, then (d(u,x)ge d(u,v) + d(v,x)-i). Roughly, gluing together any two shortest paths along a common terminal edge may not necessarily result in a shortest path but yields a “near-shortest” path with defect at most i. It is known that (alpha _0)-metric graphs are exactly ptolemaic graphs, and that chordal graphs and distance-hereditary graphs are (alpha _i)-metric for (i=1) and (i=2), respectively. We show that an additive O(i)-approximation of the radius, of the diameter, and in fact of all vertex eccentricities of an (alpha _i)-metric graph can be computed in total linear time. Our strongest results are obtained for (alpha _1)-metric graphs, for which we prove that a central vertex can be computed in subquadratic time, and even better in linear time for so-called ((alpha _1,varDelta ))-metric graphs (a superclass of chordal graphs and of plane triangulations with inner vertices of degree at least 7). The latter answers a question raised in Dragan (Inf Probl Lett 154:105873, 2020), 2020). Our algorithms follow from new results on centers and metric intervals of (alpha _i)-metric graphs. In particular, we prove that the diameter of the center is at most (3i+2) (at most 3, if (i=1)). The latter partly answers a question raised in Yushmanov and Chepoi (Math Probl Cybernet 3:217–232, 1991).

In this paper we prove the following two results.

• We show that for any (C in {textsf {mVF}, textsf {mVP}, textsf {mVNP}}), (C = overline{C}). Here, (textsf {mVF}, textsf {mVP}), and (textsf {mVNP}) are monotone variants of (textsf {VF}, textsf {VP}), and (textsf {VNP}), respectively. For an algebraic complexity class C, (overline{C}) denotes the closure of C. For (textsf {mVBP}) a similar result was shown in Bläser et al. (in: 35th Computational Complexity Conference, CCC 2020. LIPIcs, vol 169, pp 21–12124, 2020. https://doi.org/10.4230/LIPIcs.CCC.2020.21). Here we extend their result by adapting their proof.

• We define polynomial families ({mathcal {P}(k)_n}_{n ge 0}), such that ({mathcal {P}(0)_n}_{n ge 0}) equals the determinant polynomial. We show that ({mathcal {P}(k)_n}_{n ge 0}) is (textsf {VBP}) complete for (k=1) and it becomes (textsf {VNP}) complete when (k ge 2). In particular, ({mathcal {P}(k)_n}) is (mathtt {Det^{ne k}_n(X)}), a polynomial obtained by summing over all signed cycle covers that avoid length k cycles. We show that (mathtt {Det^{ne 1}_n(X)}) is complete for (textsf {VBP}) and (mathtt {Det^{ne k}_n(X)}) is complete for (textsf {VNP}) for all (k ge 2) over any field (mathbb {F}).

Fungal automata are a variation of the two-dimensional sandpile automaton of Bak et al. (Phys Rev Lett 59(4):381–384, 1987. https://doi.org/10.1103/PhysRevLett.59.381). In each step toppling cells emit grains only to some of their neighbors chosen according to a specific update sequence. We show how to embed any Boolean circuit into the initial configuration of a fungal automaton with update sequence HV. In particular we give a constructor that, given the description B of a circuit, computes the states of all cells in the finite support of the embedding configuration in (O(log left| {B}right| )) space. As a consequence the prediction problem for fungal automata with update sequence HV is (textsf {P})-complete. This solves an open problem of Goles et al. (Phys Lett A 384(22):126541, 2020. https://doi.org/10.1016/j.physleta.2020.126541).

We present a new generalization of the bin covering problem that is known to be a strongly NP-hard problem. In our generalization there is a positive constant (varDelta ), and we are given a set of items each of which has a positive size. We would like to find a partition of the items into bins. We say that a bin is near exact covered if the total size of items packed into the bin is between 1 and (1+varDelta ). Our goal is to maximize the number of near exact covered bins. If (varDelta =0) or (varDelta >0) is given as part of the input, our problem is shown here to have no approximation algorithm with a bounded asymptotic approximation ratio (assuming that (Pne NP)). However, for the case where (varDelta >0) is seen as a constant, we present an asymptotic fully polynomial time approximation scheme (AFPTAS) that is our main contribution.