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Reoptimization is a setting in which we are given a good approximate solution of an optimization problem instance and a local modification that slightly changes the instance. The main goal is that of finding a good approximate solution of the modified instance. We investigate one of the most studied scenarios in reoptimization known as Steiner tree reoptimization. Steiner tree reoptimization is a collection of strongly NP -hard optimization problems that are defined on top of the classical Steiner tree problem and for which several constant-factor approximation algorithms have been designed in the last decades. In this paper we improve upon all these results by developing a novel technique that allows us to design polynomial-time approximation schemes. Remarkably, prior to this paper, no approximation algorithm better than recomputing a solution from scratch was known for the elusive scenario in which the cost of a single edge decreases. Our results are best possible since none of the problems addressed in this paper admits a fully polynomial-time approximation scheme, unless P = NP [ABSTRACT FROM AUTHOR]

Given a graph, the general problem to cover the maximum number of vertices by a collection of vertex-disjoint long paths seems to escape 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 theoretical performance analyses are done via amortization and their practical performance is examined through simulation studies. [ABSTRACT FROM AUTHOR]

Parameterization above (or below) a guarantee is a successful concept in parameterized algorithms. The idea is that many computational problems admit "natural" guarantees bringing to algorithmic questions whether a better solution (above the guarantee) could be obtained efficiently. For example, for every boolean CNF formula on m clauses, there is an assignment that satisfies at least m/2 clauses. How difficult is it to decide whether there is an assignment satisfying more than m / 2 + k clauses? Or, if an n-vertex graph has a perfect matching, then its vertex cover is at least n/2. Is there a vertex cover of size at least n / 2 + k for some k ≥ 1 and how difficult is it to find such a vertex cover? The above guarantee paradigm has led to several exciting discoveries in the areas of parameterized algorithms and kernelization. We argue that this paradigm could bring forth fresh perspectives on well-studied problems in approximation algorithms. Our example is the longest cycle problem. One of the oldest results in extremal combinatorics is the celebrated Dirac's theorem from 1952. Dirac's theorem provides the following guarantee on the length of the longest cycle: for every 2-connected n-vertex graph G with minimum degree δ (G) ≤ n / 2 , the length of a longest cycle L is at least 2 δ (G) . Thus the "essential" part in finding the longest cycle is in approximating the "offset" k = L - 2 δ (G) . The main result of this paper is the above-guarantee approximation theorem for k. Informally, the theorem says that approximating the offset k is not harder than approximating the total length L of a cycle. In other words, for any (reasonably well-behaved) function f, a polynomial time algorithm constructing a cycle of length f(L) in an undirected graph with a cycle of length L, yields a polynomial time algorithm constructing a cycle of length 2 δ (G) + Ω (f (k)) . [ABSTRACT FROM AUTHOR]

k-center is one of the most popular clustering models. While it admits a simple 2-approximation in polynomial time in general metrics, the Euclidean version is NP-hard to approximate within a factor of 1.82, even in the plane, if one insists the dependence on k in the running time be polynomial. Without this restriction, a classic algorithm by Agarwal and Procopiuc [Algorithmica 2002] yields an O (n log k) + (1 / ϵ) O (2 d k 1 - 1 / d log k) -time (1 + ϵ) -approximation for Euclidean k-center, where d is the dimension. We show for a closely related problem, k-supplier, the double-exponential dependence on dimension is unavoidable if one hopes to have a sub-linear dependence on k in the exponent. We also derive similar algorithmic results to the ones by Agarwal and Procopiuc for both k-center and k-supplier. We use a relatively new tool, called Voronoi separator, which makes our algorithms and analyses substantially simpler. Furthermore we consider a well-studied generalization of k-center, called Non-uniform k-center (NUkC), where we allow different radii clusters. NUkC is NP-hard to approximate within any factor, even in the Euclidean case. We design a 2 O (k log k) n 2 time 3-approximation for NUkC in general metrics, and a 2 O ((k log k) / ϵ) d n time (1 + ϵ) -approximation for Euclidean NUkC. The latter time bound matches the bound for k-center. [ABSTRACT FROM AUTHOR]

We address long-standing open questions raised by Williamson, Goemans, Vazirani and Mihail pertaining to the design of approximation algorithms for problems in network design via the primal-dual method (Williamson et al. in Combinatorica 15(3):435–454, 1995. https://doi.org/10.1007/BF01299747). Williamson et al. prove an approximation ratio of two for connectivity augmentation problems where the connectivity requirements can be specified by uncrossable functions. They state: "Extending our algorithm to handle non-uncrossable functions remains a challenging open problem. The key feature of uncrossable functions is that there exists an optimal dual solution which is laminar... A larger open issue is to explore further the power of the primal-dual approach for obtaining approximation algorithms for other combinatorial optimization problems." Our main result proves that the primal-dual algorithm of Williamson et al. achieves an approximation ratio of 16 for a class of functions that generalizes the notion of an uncrossable function. There exist instances that can be handled by our methods where none of the optimal dual solutions has a laminar support. We present three applications of our main result to problems in the area of network design. (1) A 16 -approximation algorithm for augmenting a family of small cuts of a graph G. The previous best approximation ratio was O (log | V (G) |) . (2) A 16 · ⌈ k / u min ⌉ -approximation algorithm for the Cap-k-ECSS problem which is as follows: Given an undirected graph G = (V , E) with edge costs c ∈ Q ≥ 0 E and edge capacities u ∈ Z ≥ 0 E , find a minimum-cost subset of the edges F ⊆ E such that the capacity of any cut in (V, F) is at least k; u min (respectively, u max ) denotes the minimum (respectively, maximum) capacity of an edge in E, and w.l.o.g. u max ≤ k . The previous best approximation ratio was min (O (log | V |) , k , 2 u max) . (3) A 20 -approximation algorithm for the model of (p, 2)-Flexible Graph Connectivity. The previous best approximation ratio was O (log | V (G) |) , where G denotes the input graph. [ABSTRACT FROM AUTHOR]

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 Δ , 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 + Δ . Our goal is to maximize the number of near exact covered bins. If Δ = 0 or Δ > 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 P ≠ N P ). However, for the case where Δ > 0 is seen as a constant, we present an asymptotic fully polynomial time approximation scheme (AFPTAS) that is our main contribution. [ABSTRACT FROM AUTHOR]

We study the problem of approximating the eigenspectrum of a symmetric matrix A ∈ R n × n with bounded entries (i.e., ‖ A ‖ ∞ ≤ 1 ). We present a simple sublinear time algorithm that approximates all eigenvalues of A up to additive error ± ϵ n using those of a randomly sampled O ~ log 3 n ϵ 3 × O ~ log 3 n ϵ 3 principal submatrix. Our result can be viewed as a concentration bound on the complete eigenspectrum of a random submatrix, significantly extending known bounds on just the singular values (the magnitudes of the eigenvalues). We give improved error bounds of ± ϵ nnz (A) and ± ϵ ‖ A ‖ F when the rows of A can be sampled with probabilities proportional to their sparsities or their squared ℓ 2 norms respectively. Here nnz (A) is the number of non-zero entries in A and ‖ A ‖ F is its Frobenius norm. Even for the strictly easier problems of approximating the singular values or testing the existence of large negative eigenvalues (Bakshi, Chepurko, and Jayaram, FOCS '20), our results are the first that take advantage of non-uniform sampling to give improved error bounds. From a technical perspective, our results require several new eigenvalue concentration and perturbation bounds for matrices with bounded entries. Our non-uniform sampling bounds require a new algorithmic approach, which judiciously zeroes out entries of a randomly sampled submatrix to reduce variance, before computing the eigenvalues of that submatrix as estimates for those of A . We complement our theoretical results with numerical simulations, which demonstrate the effectiveness of our algorithms in practice. [ABSTRACT FROM AUTHOR]

In the Determinant Maximization problem, given an n × n positive semi-definite matrix A in Q n × n and an integer k, we are required to find a k × k principal submatrix of A having the maximum determinant. This problem is known to be NP-hard and further proven to be W[1]-hard with respect to k by Koutis (Inf Process Lett 100:8–13, 2006); i.e., a f (k) n O (1) -time algorithm is unlikely to exist for any computable function f. However, there is still room to explore its parameterized complexity in the restricted case, in the hope of overcoming the general-case parameterized intractability. In this study, we rule out the fixed-parameter tractability of Determinant Maximization even if an input matrix is extremely sparse or low rank, or an approximate solution is acceptable. We first prove that Determinant Maximization is NP-hard and W[1]-hard even if an input matrix is an arrowhead matrix; i.e., the underlying graph formed by nonzero entries is a star, implying that the structural sparsity is not helpful. By contrast, Determinant Maximization is known to be solvable in polynomial time on tridiagonal matrices (Al-Thani and Lee, in: LAGOS, 2021). Thereafter, we demonstrate the W[1]-hardness with respect to the rankr of an input matrix. Our result is stronger than Koutis' result in the sense that any k × k principal submatrix is singular whenever k > r . We finally give evidence that it is W[1]-hard to approximate Determinant Maximization parameterized by k within a factor of 2 - c k for some universal constant c > 0 . Our hardness result is conditional on the Parameterized Inapproximability Hypothesis posed by Lokshtanov et al. (in: SODA, 2020), which asserts that a gap version of Binary Constraint Satisfaction Problem is W[1]-hard. To complement this result, we develop an ε -additive approximation algorithm that runs in ε - r 2 · r O (r 3) · n O (1) time for the rank r of an input matrix, provided that the diagonal entries are bounded. [ABSTRACT FROM AUTHOR]

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 φ -competitive algorithm for the non-preemptive general testing case and a (0.0382 + 2.7925 (1 - 1 2 m)) -competitive randomized algorithm, when the number of machines m ≥ 37 or otherwise 2.7925-competitive, where φ = (1 + 5) / 2 < 1.6181 is the golden ratio and m is the number of machines, a (3.5 - 3 2 m) -competitive algorithm allowing job preemption when m ≥ 3 or otherwise 3-competitive, and a (φ + φ + 1 2 (1 - 1 m)) -competitive algorithm for the non-preemptive uniform testing case when m ≥ 5 or otherwise (φ + 1) -competitive. Our results improve three previous best approximation algorithms for the single machine scheduling with testing problems, respectively. [ABSTRACT FROM AUTHOR]

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 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. [ABSTRACT FROM AUTHOR]

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 - 1 e - ε (for any ε > 0 ). Recently Im et al. (SIAM J Discrete Math 34(3):1649–1669, 2020) presented an algorithm with ratio 1 - 1 / e + ε 0 for some absolute constant ε 0 > 0 for any fixed m. They also presented an algorithm with ratio 1 - O (log m / m) - ε 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 - ε) -approximation that runs in time O (n m c 7 ε - 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. [ABSTRACT FROM AUTHOR]

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) OPT - ε approximation after O (ε - 1) iterations and O (ε - 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 1 4 (1 - min x ∈ C ‖ x ‖ ∞) OPT - ε approximation with O (ε - 1) iterations and O (ε - 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 O (1) samples per iteration. Numerical experiments on both simulated and real data demonstrate the efficiency of the proposed methods. [ABSTRACT FROM AUTHOR]

In this work, we introduce a multi-vehicle (or multi-postman) extension of the classical Mixed Rural Postman Problem, which we call the Min–Max Mixed Rural Postmen Cover Problem (MRPCP). The MRPCP is defined on a mixed graph G = (V , E , A) , where V is the vertex set, E denotes the (undirected) edge set and A represents the (directed) arc set. Let F ⊆ E ( H ⊆ A ) be the set of required edges (required arcs). There is a nonnegative weight associated with each edge and arc. The objective is to determine no more than k closed walks to cover all the required edges in F and all the required arcs in H such that the weight of the maximum weight closed walk is minimized. By replacing closed walks with (open) walks in the MRPCP, we obtain the Min–Max Mixed Rural Postmen Walk Cover Problem (MRPWCP). The Min–Max Mixed Chinese Postmen Cover Problem (MCPCP) is a special case of the MRPCP where F = E and H = A . The Min–Max Stacker Crane Cover Problem (SCCP) is another special case of the MRPCP where F = ∅ and H = A For the MRPCP with the input graph satisfying the weakly symmetric condition, i.e., for each arc there exists a parallel edge whose weight is not greater than this arc, we devise a 27 4 -approximation algorithm. This algorithm achieves an approximation ratio of 33 5 for the SCCP with the weakly symmetric condition. Moreover, we obtain the first 5-approximation algorithm (4-approximation algorithm) for the MRPWCP (MCPCP) with the weakly symmetric condition. [ABSTRACT FROM AUTHOR]

In the 70 s, Berge introduced 1-extendable graphs (also called B-graphs), which are graphs where every vertex belongs to a maximum independent set. Motivated by an application in the design of wireless networks, we study the computational complexity of 1-extendability, the problem of deciding whether a graph is 1-extendable. We show that, in general, 1-extendability cannot be solved in 2 o (n) time assuming the Exponential Time Hypothesis, where n is the number of vertices of the input graph, and that it remains NP-hard in subcubic planar graphs and in unit disk graphs (which is a natural model for wireless networks). Although 1-extendability seems to be very close to the problem of finding an independent set of maximum size (a.k.a.Maximum Independent Set), we show that, interestingly, there exist 1-extendable graphs for which Maximum Independent Set is NP-hard. Finally, we investigate a parameterized version of 1-extendability. [ABSTRACT FROM AUTHOR]

In this paper, we study the following robust optimization problem. Given a set family representing feasibility and candidate objective functions, we choose a feasible set, and then an adversary chooses one objective function, knowing our choice. The goal is to find a randomized strategy (i.e., a probability distribution over the feasible sets) that maximizes the expected objective value in the worst case. This problem is fundamental in wide areas such as artificial intelligence, machine learning, game theory, and optimization. To solve the problem, we provide a general framework based on the dual linear programming problem. In the framework, we utilize the ellipsoid algorithm with the approximate separation algorithm. We prove that there exists an α -approximation algorithm for our robust optimization problem if there exists an α -approximation algorithm for finding a (deterministic) feasible set that maximizes a nonnegative linear combination of the candidate objective functions. Using our result, we provide approximation algorithms for the max–min fair randomized allocation problem and the maximum cardinality robustness problem with a knapsack constraint. [ABSTRACT FROM AUTHOR]

In the general max–min fair allocation problem, there are m players and n indivisible resources, each player has his/her own utilities for the resources, and the goal is to find an assignment that maximizes the minimum total utility of resources assigned to a player. The problem finds many natural applications such as bandwidth distribution in telecom networks, processor allocation in computational grids, and even public-sector decision making. We introduce an over-estimation strategy to design approximation algorithms for this problem. When all utilities are positive, we obtain an approximation ratio of c 1 - ϵ , where c is the maximum ratio of the largest utility to the smallest utility of any resource. When some utilities are zero, we obtain an approximation ratio of (1 + 3 c ^ + O (δ c ^ 2)) , where c ^ is the maximum ratio of the largest utility to the smallest positive utility of any resource. [ABSTRACT FROM AUTHOR]

We consider the robust version of the classic k-center clustering problem, where we wish to remove up to z points (outliers), so as to be able to cluster the remaining points in k clusters with minimum maximum radius. We study such a problem under the fully dynamic adversarial model, where points can be inserted or deleted arbitrarily. In this setting, the main goal is to design algorithms that maintain a high quality solution at any point in time, while requiring a "small" amortized cost, i.e. a "small" number of operations per insertion or deletion, on average. In our work, we provide the first constant bi-criteria approximation algorithm for such a problem with its amortized cost being independent of both z and the size of the current input. We also complement our positive result with a lower bound showing that any constant (non bi-criteria) approximation algorithm has amortized cost at least linear in z. Finally, we conduct an in-depth experimental analysis of our algorithm on Twitter, Flickr, and Air-Quality datasets showing the effectiveness of our approach. [ABSTRACT FROM AUTHOR]

We prove that Simulated Annealing with an appropriate cooling schedule computes arbitrarily tight constant-factor approximations to the minimum spanning tree problem in polynomial time. This result was conjectured by Wegener (Automata, Languages and Programming, ICALP, Berlin, 2005). More precisely, denoting by n , m , w max , and w min the number of vertices and edges as well as the maximum and minimum edge weight of the MST instance, we prove that simulated annealing with initial temperature T 0 ≥ w max and multiplicative cooling schedule with factor 1 - 1 / ℓ , where ℓ = ω (m n ln (m)) , with probability at least 1 - 1 / m computes in time O (ℓ (ln ln (ℓ) + ln (T 0 / w min))) a spanning tree with weight at most 1 + κ times the optimum weight, where 1 + κ = (1 + o (1)) ln (ℓ m) ln (ℓ) - ln (m n ln (m)) . Consequently, for any ϵ > 0 , we can choose ℓ in such a way that a (1 + ϵ) -approximation is found in time O ((m n ln (n)) 1 + 1 / ϵ + o (1) (ln ln n + ln (T 0 / w min))) with probability at least 1 - 1 / m . In the special case of so-called (1 + ϵ) -separated weights, this algorithm computes an optimal solution (again in time O ((m n ln (n)) 1 + 1 / ϵ + o (1) (ln ln n + ln (T 0 / w min))) ), which is a significant speed-up over Wegener's runtime guarantee of O (m 8 + 8 / ϵ) . Our tighter upper bound also admits the result that in some situations a hybridization of simulated annealing and the (1 + 1) EA can lead to stronger runtime guarantees than either algorithm alone. [ABSTRACT FROM AUTHOR]