Algorithmica最新文献

Quality diversity (QD) is a branch of evolutionary computation that gained increasing interest in recent years. The Map-Elites QD approach defines a feature space, i.e., a partition of the search space, and stores the best solution for each cell of this space. We study a simple QD algorithm in the context of pseudo-Boolean optimisation on the “number of ones” feature space, where the ith cell stores the best solution amongst those with a number of ones in ([(i-1)k, ik-1]). Here k is a granularity parameter (1 le k le n+1). We give a tight bound on the expected time until all cells are covered for arbitrary fitness functions and for all k and analyse the expected optimisation time of QD on OneMax and other problems whose structure aligns favourably with the feature space. On combinatorial problems we show that QD finds a ({(1-1/e)})-approximation when maximising any monotone sub-modular function with a single uniform cardinality constraint efficiently. Defining the feature space as the number of connected components of an edge-weighted graph, we show that QD finds a minimum spanning forest in expected polynomial time. We further consider QD’s performance on classes of transformed functions in which the feature space is not well aligned with the problem. The asymptotic performance is unaffected by transformations on easy functions like OneMax. Applying a worst-case transformation to a deceptive problem increases the expected optimisation time from (O(n^2 log n)) to an exponential time. However, QD is still faster than a (1+1) EA by an exponential factor.

Computing planar orthogonal drawings with the minimum number of bends is one of the most studied topics in Graph Drawing. The problem is known to be NP-hard, even when we want to test the existence of a rectilinear planar drawing, i.e., an orthogonal drawing without bends (Garg and Tamassia in SIAM J Comput 31(2):601–625, 2001). From the parameterized complexity perspective, the problem is fixed-parameter tractable when parameterized by the sum of three parameters: the number b of bends, the number k of vertices of degree at most two, and the treewidth (textsf{tw}) of the input graph (Di Giacomo et al. in J Comput Syst Sci 125:129–148, 2022). We improve this last result by showing that the problem remains fixed-parameter tractable when parameterized only by (b+k). As a consequence, rectilinear planarity testing lies in FPT parameterized by the number of vertices of degree at most two. We also prove that our choice of parameters is minimal, as deciding if an orthogonal drawing with at most b bends exists is already NP-hard when k is zero (i.e., the problem is para-NP-hard parameterized in k); hence, there is neither an FPT nor an XP algorithm parameterized only by the parameter k (unless P = NP). In addition, we prove that the problem is W[1]-hard parameterized by (k+textsf{tw}), complementing a recent result (Jansen et al. in Upward and orthogonal planarity are W[1]-hard parameterized by treewidth. CoRR, abs/2309.01264, 2023; in: Bekos MA, Chimani M (eds) Graph Drawing and Network Visualization, vol 14466, Springer, Cham, pp 203–217, 2023) that shows W[1]-hardness for the parameterization (b+textsf{tw}). As a consequence, we are able to trace a clear parameterized tractability landscape for the bend-minimum orthogonal planarity problem with respect to the three parameters b, k, and (textsf{tw}).

Understanding spatial correlation is vital in many fields including epidemiology and social science. Lee et al. (Stat Comput 31(4):51, 2021. https://doi.org/10.1007/s11222-021-10025-7) recently demonstrated that improved inference for areal unit count data can be achieved by carrying out modifications to a graph representing spatial correlations; specifically, they delete edges of the planar graph derived from border-sharing between geographic regions in order to maximise a specific objective function. In this paper, we address the computational complexity of the associated graph optimisation problem. We demonstrate that this optimisation problem is NP-hard; we further show intractability for two simpler variants of the problem. We follow these results with two parameterised algorithms that exactly solve the problem. The first is parameterised by both treewidth and maximum degree, while the second is parameterised by the maximum number of edges that can be removed and is also restricted to settings where the input graph has maximum degree three. Both of these algorithms solve not only the decision problem, but also enumerate all solutions with polynomial time precalculation, delay, and postcalculation time in respective restricted settings. For this problem, efficient enumeration allows the uncertainty in the spatial correlation to be utilised in the modelling. The first enumeration algorithm utilises dynamic programming on a tree decomposition of the input graph, and has polynomial time precalculation and linear delay if both the treewidth and maximum degree are bounded. The second algorithm is restricted to problem instances with maximum degree three, as may arise from triangulations of planar surfaces, but can output all solutions with FPT precalculation time and linear delay when the maximum number of edges that can be removed is taken as the parameter.

The classic Impagliazzo–Nisan–Wigderson (INW) pseudorandom generator (PRG) (STOC ‘94) for space-bounded computation uses a seed of length (O(log n cdot log (nw/varepsilon )+log d)) to fool ordered branching programs of length n, width w, and alphabet size d to within error (varepsilon ). A series of works have shown that the analysis of the INW generator can be improved for the class of permutation branching programs or the more general regular branching programs, improving the (O(log ^2 n)) dependence on the length n to (O(log n)) or ({tilde{O}}(log n)). However, when also considering the dependence on the other parameters, these analyses still fall short of the optimal PRG seed length (O(log (nwd/varepsilon ))). In this paper, we prove that any “spectral analysis” of the INW generator requires seed length

to fool ordered permutation branching programs of length n, width w, and alphabet size d to within error (varepsilon ). By “spectral analysis” we mean an analysis of the INW generator that relies only on the spectral expansion of the graphs used to construct the generator; this encompasses all prior analyses of the INW generator. Our lower bound matches the upper bound of Braverman–Rao–Raz–Yehudayoff (FOCS 2010, SICOMP 2014) for regular branching programs of alphabet size (d=2) except for a gap between their (Oleft( log n cdot log log nright) ) term and our (Omega left( log n cdot log log min {n,d}right) ) term. It also matches the upper bounds of Koucký–Nimbhorkar–Pudlák (STOC 2011), De (CCC 2011), and Steinke (ECCC 2012) for constant-width ((w=O(1))) permutation branching programs of alphabet size (d=2) to within a constant factor. To fool permutation branching programs in the measure of spectral norm, we prove that any spectral analysis of the INW generator requires a seed of length (Omega left( log ncdot log log n+log ncdot log (1/varepsilon )right) ) when the width is at least polynomial in n ((w=n^{Omega (1)})), matching the recent upper bound of Hoza–Pyne–Vadhan (ITCS 2021) to within a constant factor.

As real-time systems are safety critical, guaranteeing a high reliability threshold is as important as meeting all deadlines. Periodic tasks are replicated to mitigate the negative impact of transient faults, which leads to redundancy and high energy consumption. On the other hand, energy saving is widely identified as increasingly relevant issues in real-time systems. In this paper, we formalize this challenging tri-criteria optimization problem, i.e., minimizing the expected energy consumption while enforcing the reliability threshold and meeting all task deadlines, and propose several mapping and scheduling heuristics to solve it. Specifically, a novel approach is designed to (i) map an arbitrary number of replicas onto processors, (ii) schedule each replica of each task instance on its assigned processor with less temporal overlap. The platform is composed of processing units with different characteristics, including speed profile, energy cost and fault rate. The heterogeneity of the computing platform makes the problem more complicated, because different mappings achieve different levels of reliability and consume different amounts of energy. Moreover, scheduling plays an important role in energy saving, as the expected energy consumption is the average over all failure scenarios. Once a task replica is successful, the other replicas of that task instance can be canceled, which calls for minimizing the overlap between any replica pair. Finally, to quantitatively analyze our methods, we derive a theoretical lower-bound for the expected energy consumption. Comprehensive experiments are conducted on a large set of execution scenarios and parameters. The comparison results reveal that our strategies perform better than the random baseline under almost all settings, with an average gain in energy consumption of more than 40%, and our best heuristic achieves an excellent performance: its energy saving is only 2% less than the lower-bound on average.

In a seminal paper in 2013, Witt showed that the (1+1) Evolutionary Algorithm with standard bit mutation needs time ((1+o(1))n ln n/p_1) to find the optimum of any linear function, as long as the probability (p_1) to flip exactly one bit is (Theta (1)). In this paper we investigate how this result generalizes if standard bit mutation is replaced by an arbitrary unbiased mutation operator. This situation is notably different, since the stochastic domination argument used for the lower bound by Witt no longer holds. In particular, starting closer to the optimum is not necessarily an advantage, and OneMax is no longer the easiest function for arbitrary starting positions. Nevertheless, we show that Witt’s result carries over if (p_1) is not too small, with different constraints for upper and lower bounds, and if the number of flipped bits has bounded expectation (chi ). Notably, this includes some of the heavy-tail mutation operators used in fast genetic algorithms, but not all of them. We also give examples showing that algorithms with unbounded (chi ) have qualitatively different trajectories close to the optimum.

Interval scheduling is a basic algorithmic problem and a classical task in combinatorial optimization. We develop techniques for partitioning and grouping jobs based on their starting/ending times, enabling us to view an instance of interval scheduling on many jobs as a union of multiple interval scheduling instances, each containing only a few jobs. Instantiating these techniques in a dynamic setting produces several new results. For ((1+varepsilon ))-approximation of job scheduling of n jobs on a single machine, we develop a fully dynamic algorithm with (O(nicefrac {log {n}}{varepsilon })) update and (O(log {n})) query worst-case time. Our techniques are also applicable in a setting where jobs have weights. We design a fully dynamic deterministic algorithm whose worst-case update and query times are (text {poly} (log n,frac{1}{varepsilon })). This is the first algorithm that maintains a ((1+varepsilon ))-approximation of the maximum independent set of a collection of weighted intervals in (text {poly} (log n,frac{1}{varepsilon })) time updates/queries. This is an exponential improvement in (1/varepsilon ) over the running time of an algorithm of Henzinger, Neumann, and Wiese  [SoCG, 2020]. Our approach also removes all dependence on the values of the jobs’ starting/ending times and weights.

We show that, for planar point sets, the number of non-crossing Hamiltonian paths is polynomially bounded in the number of non-crossing paths, and the number of non-crossing Hamiltonian cycles (polygonalizations) is polynomially bounded in the number of surrounding cycles. As a consequence, we can list the non-crossing Hamiltonian paths or the polygonalizations, in time polynomial in the output size, by filtering the output of simple backtracking algorithms for non-crossing paths or surrounding cycles respectively. We do not assume that the points are in general position. To prove these results we relate the numbers of non-crossing structures to two easily-computed parameters of the point set: the minimum number of points whose removal results in a collinear set, and the number of points interior to the convex hull. These relations also lead to polynomial-time approximation algorithms for the numbers of structures of all four types, accurate to within a constant factor of the logarithm of these numbers.

Let X be an n-element point set in the k-dimensional unit cube ([0,1]^k) where (k ge 2). According to an old result of Bollobás and Meir (Oper Res Lett 11:19–21, 1992) , there exists a cycle (tour) (x_1, x_2, ldots , x_n) through the n points, such that (left( sum _{i=1}^n |x_i - x_{i+1}|^k right) ^{1/k} le c_k), where (|x-y|) is the Euclidean distance between x and y, and (c_k) is an absolute constant that depends only on k, where (x_{n+1} equiv x_1). From the other direction, for every (k ge 2) and (n ge 2), there exist n points in ([0,1]^k), such that their shortest tour satisfies (left( sum _{i=1}^n |x_i - x_{i+1}|^k right) ^{1/k} = 2^{1/k} cdot sqrt{k}). For the plane, the best constant is (c_2=2) and this is the only exact value known. Bollobás and Meir showed that one can take (c_k = 9 left( frac{2}{3} right) ^{1/k} cdot sqrt{k}) for every (k ge 3) and conjectured that the best constant is (c_k = 2^{1/k} cdot sqrt{k}), for every (k ge 2). Here we significantly improve the upper bound and show that one can take (c_k = 3 sqrt{5} left( frac{2}{3} right) ^{1/k} cdot sqrt{k}) or (c_k = 2.91 sqrt{k} (1+o_k(1))). Our bounds are constructive. We also show that (c_3 ge 2^{7/6}), which disproves the conjecture for (k=3). Connections to matching problems, power assignment problems, related problems, including algorithms, are discussed in this context. A slightly revised version of the Bollobás–Meir conjecture is proposed.

We consider standard T-interval dynamic networks, under the synchronous timing model and the broadcast CONGEST model. In a T-interval dynamic network, the set of nodes is always fixed and there are no node failures. The edges in the network are always undirected, but the set of edges in the topology may change arbitrarily from round to round, as determined by some adversary and subject to the following constraint: For every T consecutive rounds, the topologies in those rounds must contain a common connected spanning subgraph. Let (H_r) to be the maximum (in terms of number of edges) such subgraph for round r through (r+T-1). We define the backbone diameter d of a T-interval dynamic network to be the maximum diameter of all such (H_r)’s, for (rge 1). We use n to denote the number of nodes in the network. Within such a context, we consider a range of fundamental distributed computing problems including Count/Max/Median/Sum/LeaderElect/Consensus/ConfirmedFlood. Existing algorithms for these problems all have time complexity of (Omega (n)) rounds, even for (T=infty ) and even when d is as small as O(1). This paper presents a novel approach/framework, based on the idea of massively parallel aggregation. Following this approach, we develop a novel deterministic Count algorithm with (O(d^3 log ^2 n)) complexity, for T-interval dynamic networks with (T ge ccdot d^2 log ^2n). Here c is a (sufficiently large) constant independent of d, n, and T. To our knowledge, our algorithm is the very first such algorithm whose complexity does not contain a (Theta (n)) term. This paper further develops novel algorithms for solving Max/Median/Sum/LeaderElect/Consensus/ConfirmedFlood, while incurring (O(d^3 text{ polylog }(n))) complexity. Again, for all these problems, our algorithms are the first ones whose time complexity does not contain a (Theta (n)) term.