Algorithmica最新文献

We consider the predecessor problem on the ultra-wide word RAM model of computation, which extends the word RAM model with ultrawords consisting of (w^2) bits (TAMC, 2015). The model supports arithmetic and boolean operations on ultrawords, in addition to scattered memory operations that access or modify w (potentially non-contiguous) memory addresses simultaneously. The ultra-wide word RAM model captures (and idealizes) modern vector processor architectures. Our main result is a simple, linear space data structure that supports predecessor in constant time and updates in amortized, expected constant time. This improves the space of the previous constant time solution that uses space in the order of the size of the universe. Our result holds even in a weaker model where ultrawords consist of (w^{1+epsilon }) bits for any (epsilon > 0 ). It is based on a new implementation of the classic x-fast trie data structure of Willard (Inform Process Lett 17(2):81–84, https://doi.org/10.1016/0020-0190(83)90075-3, 1983) combined with a new dictionary data structure that supports fast parallel lookups.

In this paper we study the classical problem of throughput maximization. In this problem we have a collection J of n jobs, each having a release time (r_j), deadline (d_j), and processing time (p_j). They have to be scheduled non-preemptively on m identical parallel machines. The goal is to find a schedule which maximizes the number of jobs scheduled entirely in their ([r_j,d_j]) window. This problem has been studied extensively (even for the case of (m=1)). Several special cases of the problem remain open. Bar-Noy et al. (Proceedings of the Thirty-First Annual ACM Symposium on Theory of Computing, May 1–4, 1999, Atlanta, Georgia, USA, pp. 622–631. ACM, 1999, https://doi.org/10.1145/301250.301420) presented an algorithm with ratio (1-1/(1+1/m)^m) for m machines, which approaches (1-1/e) as m increases. For (m=1), Chuzhoy et al. (42nd Annual Symposium on Foundations of Computer Science (FOCS) 2001, 14–17 October 2001, Las Vegas, Nevada, USA, pp. 348–356. IEEE Computer Society, 2001) presented an algorithm with approximation with ratio (1-frac{1}{e}-varepsilon ) (for any (varepsilon >0)). Recently Im et al. (SIAM J Discrete Math 34(3):1649–1669, 2020) presented an algorithm with ratio (1-1/e+varepsilon _0) for some absolute constant (varepsilon _0>0) for any fixed m. They also presented an algorithm with ratio (1-O(sqrt{log m/m})-varepsilon ) for general m which approaches 1 as m grows. The approximability of the problem for (m=O(1)) remains a major open question. Even for the case of (m=1) and (c=O(1)) distinct processing times the problem is open (Sgall in: Algorithms - ESA 2012 - 20th Annual European Symposium, Ljubljana, Slovenia, September 10–12, 2012. Proceedings, pp 2–11, 2012). In this paper we study the case of (m=O(1)) and show that if there are c distinct processing times, i.e. (p_j)’s come from a set of size c, then there is a randomized ((1-{varepsilon }))-approximation that runs in time (O(n^{mc^7{varepsilon }^{-6}}log T)), where T is the largest deadline. Therefore, for constant m and constant c this yields a PTAS. Our algorithm is based on proving structural properties for a near optimum solution that allows one to use a dynamic programming with pruning.

In this paper, we study the Harmless Set problem from a parameterized complexity perspective. Given a graph (G = (V,E)), a threshold function(~t~:~ V rightarrow {mathbb {N}}) and an integer k, we study Harmless Set, where the goal is to find a subset of vertices (S subseteq V) of size at least k such that every vertex (vin V) has fewer than t(v) neighbours in S. On the positive side, we obtain fixed-parameter algorithms for the problem when parameterized by the neighbourhood diversity, the twin-cover number and the vertex integrity of the input graph. We complement two of these results from the negative side. On dense graphs, we show that the problem is W[1]-hard parameterized by cluster vertex deletion number—a natural generalization of the twin-cover number. We show that the problem is W[1]-hard parameterized by a wide range of fairly restrictive structural parameters such as the feedback vertex set number, pathwidth, and treedepth—a natural generalization of the vertex integrity. We thereby resolve one open question stated by Bazgan and Chopin (Discrete Optim 14(C):170–182, 2014) concerning the complexity of Harmless Set parameterized by the treewidth of the input graph. We also show that Harmless Set for a special case where each vertex has the threshold set to half of its degree (the so-called Majority Harmless Set problem) is W[1]-hard when parameterized by the treewidth of the input graph. Given a graph G and an irredundant c-expression of G, we prove that Harmless Set can be solved in XP-time when parameterized by clique-width.

For a set ({mathcal {Q}}) of points in the plane and a real number (delta ge 0), let ({mathbb {G}}_delta ({mathcal {Q}})) be the graph defined on ({mathcal {Q}}) by connecting each pair of points at distance at most (delta ).We consider the connectivity of ({mathbb {G}}_delta ({mathcal {Q}})) in the best scenario when the location of a few of the points is uncertain, but we know for each uncertain point a line segment that contains it. More precisely, we consider the following optimization problem: given a set ({mathcal {P}}) of (n-k) points in the plane and a set ({mathcal {S}}) of k line segments in the plane, find the minimum (delta ge 0) with the property that we can select one point (p_sin s) for each segment (sin {mathcal {S}}) and the corresponding graph ({mathbb {G}}_delta ( {mathcal {P}}cup { p_smid sin {mathcal {S}}})) is connected. It is known that the problem is NP-hard. We provide an algorithm to exactly compute an optimal solution in ({{,mathrm{{mathcal {O}}},}}(f(k) n log n)) time, for a computable function (f(cdot )). This implies that the problem is FPT when parameterized by k. The best previous algorithm uses ({{,mathrm{{mathcal {O}}},}}((k!)^k k^{k+1}cdot n^{2k})) time and computes the solution up to fixed precision.

The leafage of a chordal graph G is the minimum integer (ell ) such that G can be realized as an intersection graph of subtrees of a tree with (ell ) leaves. We consider structural parameterization by the leafage of classical domination and cut problems on chordal graphs. Fomin, Golovach, and Raymond [ESA 2018, Algorithmica 2020] proved, among other things, that Dominating Set on chordal graphs admits an algorithm running in time (2^{mathcal {O}(ell ^2)} cdot n^{mathcal {O}(1)}). We present a conceptually much simpler algorithm that runs in time (2^{mathcal {O}(ell )} cdot n^{mathcal {O}(1)}). We extend our approach to obtain similar results for Connected Dominating Set and Steiner Tree. We then consider the two classical cut problems MultiCut with Undeletable Terminals and Multiway Cut with Undeletable Terminals. We prove that the former is W[1]-hard when parameterized by the leafage and complement this result by presenting a simple (n^{mathcal {O}(ell )})-time algorithm. To our surprise, we find that Multiway Cut with Undeletable Terminals on chordal graphs can be solved, in contrast, in (n^{{{mathcal {O}}}(1)})-time.

In offline scheduling models, jobs are given with their exact processing times. In their online counterparts, jobs arrive in sequence together with their processing times and the scheduler makes irrevocable decisions on how to execute each of them upon its arrival. We consider a semi-online variant which has equally rich application background, called scheduling with testing, where the exact processing time of a job is revealed only after a required testing operation is finished, or otherwise the job has to be executed for a given possibly over-estimated length of time. For multiprocessor scheduling with testing to minimize the total job completion time, we present several first approximation algorithms with constant competitive ratios for various settings, including a (2 varphi )-competitive algorithm for the non-preemptive general testing case and a ((0.0382 + 2.7925 (1 - frac{1}{2,m})))-competitive randomized algorithm, when the number of machines (m ge 37) or otherwise 2.7925-competitive, where (varphi = (1 + sqrt{5}) / 2 < 1.6181) is the golden ratio and m is the number of machines, a ((3.5 - frac{3}{2,m}))-competitive algorithm allowing job preemption when (m ge 3) or otherwise 3-competitive, and a ((varphi + frac{varphi + 1}{2} (1 - frac{1}{,}m)))-competitive algorithm for the non-preemptive uniform testing case when (m ge 5) or otherwise ((varphi + 1))-competitive. Our results improve three previous best approximation algorithms for the single machine scheduling with testing problems, respectively.

In this paper, we consider the problem of finding a regression in a version control system (VCS), such as git. The set of versions is modelled by a directed acyclic graph (DAG) where vertices represent versions of the software, and arcs are the changes between different versions. We assume that somewhere in the DAG, a bug was introduced, which persists in all of its subsequent versions. It is possible to query a vertex to check whether the corresponding version carries the bug. Given a DAG and a bugged vertex, the Regression Search Problem consists in finding the first vertex containing the bug in a minimum number of queries in the worst-case scenario. This problem is known to be NP-complete. We study the algorithm used in git to address this problem, known as git bisect. We prove that in a general setting, git bisect can use an exponentially larger number of queries than an optimal algorithm. We also consider the restriction where all vertices have indegree at most 2 (i.e. where merges are made between at most two branches at a time in the VCS), and prove that in this case, git bisect is a (frac{1}{log _2(3/2)})-approximation algorithm, and that this bound is tight. We also provide a better approximation algorithm for this case. Finally, we give an alternative proof of the NP-completeness of the Regression Search Problem, via a variation with bounded indegree.

Stochastic optimization has experienced significant growth in recent decades, with the increasing prevalence of variance reduction techniques in stochastic optimization algorithms to enhance computational efficiency. In this paper, we introduce two projection-free stochastic approximation algorithms for maximizing diminishing return (DR) submodular functions over convex constraints, building upon the Stochastic Path Integrated Differential EstimatoR (SPIDER) and its variants. Firstly, we present a SPIDER Continuous Greedy (SPIDER-CG) algorithm for the monotone case that guarantees a ((1-e^{-1})text {OPT}-varepsilon ) approximation after (mathcal {O}(varepsilon ^{-1})) iterations and (mathcal {O}(varepsilon ^{-2})) stochastic gradient computations under the mean-squared smoothness assumption. For the non-monotone case, we develop a SPIDER Frank–Wolfe (SPIDER-FW) algorithm that guarantees a (frac{1}{4}(1-min _{xin mathcal {C}}{Vert xVert _{infty }})text {OPT}-varepsilon ) approximation with (mathcal {O}(varepsilon ^{-1})) iterations and (mathcal {O}(varepsilon ^{-2})) stochastic gradient estimates. To address the practical challenge associated with a large number of samples per iteration, we introduce a modified gradient estimator based on SPIDER, leading to a Hybrid SPIDER-FW (Hybrid SPIDER-CG) algorithm, which achieves the same approximation guarantee as SPIDER-FW (SPIDER-CG) algorithm with only (mathcal {O}(1)) samples per iteration. Numerical experiments on both simulated and real data demonstrate the efficiency of the proposed methods.

The maximum independent set problem is one of the most important problems in graph algorithms and has been extensively studied in the line of research on the worst-case analysis of exact algorithms for NP-hard problems. In the weighted version, each vertex in the graph is associated with a weight and we are going to find an independent set of maximum total vertex weight. Many reduction rules for the unweighted version have been developed that can be used to effectively reduce the input instance without loss the optimality. However, it seems that reduction rules for the weighted version have not been systemically studied. In this paper, we design a series of reduction rules for the maximum weighted independent set problem and also design a fast exact algorithm based on the reduction rules. By using the measure-and-conquer technique to analyze the algorithm, we show that the algorithm runs in (O^*(1.1443^{(0.624x-0.872)n'})) time and polynomial space, where (n') is the number of vertices of degree at least 2 and x is the average degree of these vertices in the graph. When the average degree is small, our running-time bound beats previous results. For example, on graphs with the minimum degree at least 2 and average degree at most 3.68, our running time bound is better than that of previous polynomial-space algorithms for graphs with maximum degree at most 4.

A natural generalization of the recognition problem for a geometric graph class is the problem of extending a representation of a subgraph to a representation of the whole graph. A related problem is to find representations for multiple input graphs that coincide on subgraphs shared by the input graphs. A common restriction is the sunflower case where the shared graph is the same for each pair of input graphs. These problems translate to the setting of comparability graphs where the representations correspond to transitive orientations of their edges. We use modular decompositions to improve the runtime for the orientation extension problem and the sunflower orientation problem to linear time. We apply these results to improve the runtime for the partial representation problem and the sunflower case of the simultaneous representation problem for permutation graphs to linear time. We also give the first efficient algorithms for these problems on circular permutation graphs.