Greedy Localization and Color-Coding: Improved Matching and Packing Algorithms

The weighted m -d matching and m -set packing problems (m ≥ 3) are usually solved through approximation algorithms. In this paper, we define the parameterized versions of these problems and study their algorithms. For the weighted m -set packing problem, we develop a parameterized algorithm of running time O(12.8 mk n O(1)), which is based on the recently improved color-coding technology and dynamic programming. By refining the techniques, we also develop a more efficient parameterized algorithm of running time O(12.8(m − 1)k n O(1)) for the weighted m -d matching problem, which is a restricted version of the weighted m -set packing problem.

Packing problems have formed an important class of NP-hard problems. In this paper, we provide further theoretical study on the structure of the problems, and design improved algorithm for packing problems. For the 3-Set Packing problem, we present a deterministic algorithm of time O*(3.523k), which significantly improves the previous best algorithm of time O*(4.613k).

A parameterized algorithm for the hyperplane-cover problem

We present a deterministic near-linear time algorithm that computes the edge-connectivity and finds a minimum cut for a simple undirected unweighted graph G with n vertices and m edges. This is the first o(mn) time deterministic algorithm for the problem. In near-linear time we can also construct the classic cactus representation of all minimum cuts.The previous fastest deterministic algorithm by Gabow from STOC'91 took O(m+λ2 n), where λ is the edge connectivity, but λ could be Ω(n).At STOC'96 Karger presented a randomized near linear time Monte Carlo algorithm for the minimum cut problem. As he points out, there is no better way of certifying the minimality of the returned cut than to use Gabow's slower deterministic algorithm and compare sizes.Our main technical contribution is a near-linear time algorithm that contracts vertex sets of a simple input graph G with minimum degree δ, producing a multigraph G with ~O(m/δ) edges which preserves all minimum cuts of G with at least two vertices on each side.In our deterministic near-linear time algorithm, we will decompose the problem via low-conductance cuts found using PageRank a la Brin and Page (1998), as analyzed by Andersson, Chung, and Lang at FOCS'06. Normally such algorithms for low-conductance cuts are randomized Monte Carlo algorithms, because they rely on guessing a good start vertex. However, in our case, we have so much structure that no guessing is needed.

For the weighted rD-Matching problem, we present a deterministic parameterized algorithm with time complexity O *(4(r − 1)k ), improving the previous best upper bound O *(4 rk ). In particular, the algorithm can be applied to solve the unweighted 3D-Matching problem with time O *(16 k ), improving the previous best result O *(21.26 k ). For the weighted r-Set Packing problem, we present a deterministic parameterized algorithm with time complexity O *(2(2r − 1)k ), improving the previous best result O *(22rk ). The algorithm, when applied to the unweighted 3-Set Packing problem, has running time O *(32 k ), improving the previous best result O *(43.62 k ). Moreover, for the weighted rD-Matching and weighted r-Set Packing problems, we get a kernel of size O(k r ).

Approximation and online algorithms for multidimensional bin packing: A survey

A directed graph D is semicomplete if for every pair x, y of vertices of D, there is at least one arc between x and y. Thus, a tournament is a semicomplete digraph. In the Directed Component Order Connectivity (DCOC) problem, given a digraph D=(V,A) and a pair of natural numbers k and ell , we are to decide whether there is a subset X of V of size k such that the largest strongly connected component in D-X has at most ell vertices. Note that DCOC reduces to the Directed Feedback Vertex Set problem for ell =1. We study the parameterized complexity of DCOC for general and semicomplete digraphs with the following parameters: k, ell ,ell +k and n-ell . In particular, we prove that DCOC with parameter k on semicomplete digraphs can be solved in time O^*(2^{16k}) but not in time O^*(2^{o(k)}) unless the Exponential Time Hypothesis (ETH) fails. The upper bound O^*(2^{16k}) implies the upper bound O^*(2^{16(n-ell )}) for the parameter n-ell . We complement the latter by showing that there is no algorithm of time complexity O^*(2^{o({n-ell })}) unless ETH fails. Finally, we improve (in dependency on ell ) the upper bound of Göke, Marx and Mnich (2019) for the time complexity of DCOC with parameter ell +k on general digraphs from O^*(2^{O(kell log (kell ))}) to O^*(2^{O(klog (kell ))}). Note that Drange, Dregi and van ’t Hof (2016) proved that even for the undirected version of DCOC on split graphs there is no algorithm of running time O^*(2^{o(klog ell )}) unless ETH fails and it is a long-standing problem to decide whether Directed Feedback Vertex Set admits an algorithm of time complexity O^*(2^{o(klog k)}).

This paper gives the first o(n2.5) deterministic algorithm for the maximum flow problem in any undirected unit-capacity graph with no parallel edges. In an n-vertex, m-edge graph with maximum flow value v, our running time is Õ(n9/4v1/8) = Õ(n2.375). Note that v ≤ n for simple unit-capacity graphs. The previous deterministic algorithms [Karger and Levine 1998] achieve O(m+nv3/2) and O(nm2/3v1/6) time bound, which are both O(n2.5) for dense simple graphs and v = Θ(n).

In this paper, we deal with a f0od packing problem in an automated food packing system. The packing system has n hoppers, where an item is thrown into each hopper. An integer weight ui, and a priority γi are associated with each item i, where any wi is assumed to be bounded by an integer wmax. Given a set I of n items in the hoppers, the packing system chooses a subset I' (⊂e;I) of items, and puts them into a package with a specified target weight B. After that, the packing system updates the set I by taking the union of the remaining items in I-I' and new items thrown into empty hoppers. Repeating these operations, it makes a large number of packages one by one. In the packing system, an item may be left in a hopper for a long time until it is chosen. The priority γi is introduced to avoid such a situation, if possible. The packing problem is formulated as a lexicographic bi-criteria discrete optimization problem, where the first objective is to minimize Σi∈Iu'i such that Σi∈I'wi≥B must be satisfied, and the second is to maximize Σi∈Iγi.In this paper, we propose an O (n2wmax) time algorithm based on dynamic programming to the packing problem. We also examine the performance of the proposed approach by means of numerical experiments, and report the results.

We revisit a classical problem in computational geometry that has been studied since the 1980s: in the rectangle enclosure problem we want to report all k enclosing pairs of n input rectangles in 2D. We present the first deterministic algorithm that takes O(nlogn + k) worst-case time and O(n) space in the word-RAM model. This improves previous deterministic algorithms with O((nlogn + k)loglogn) running time. We achieve the result by derandomizing the algorithm of Chan, Larsen and Pătraşcu [SoCG’11] that attains the same time complexity but in expectation.The 2D rectangle enclosure problem is related to the offline dominance range reporting problem in 4D, and our result leads to the currently fastest deterministic algorithm for offline dominance reporting in any constant dimension d ≥ 4.A key tool behind Chan et al.’s previous randomized algorithm is shallow cuttings for 3D dominance ranges. Recently, Afshani and Tsakalidis [SODA’14] obtained a deterministic O(nlogn)-time algorithm to construct such cuttings. We first present an improved deterministic construction algorithm that runs in O(nloglogn) time in the word-RAM; this result is of independent interest. Many additional ideas are then incorporated, including a linear-time algorithm for merging shallow cuttings and an algorithm for an offline tree point location problem.KeywordsQuery PointDeterministic AlgorithmInput PointOuter CornerPlanar SeparatorThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Massively Parallel Computation (MPC) is an emerging model which distills core aspects of distributed and parallel computation. It was developed as a tool to solve (typically graph) problems in systems where input is distributed over many machines with limited space. Recent work has focused on the regime in which machines have sublinear (in n, number of nodes in the input graph) space, with randomized algorithms presented for the fundamental problems of Maximal Matching and Maximal Independent Set. There are, however, no prior corresponding deterministic algorithms. A major challenge in the sublinear space setting is that the local space of each machine may be too small to store all the edges incident to a single node. To overcome this barrier we introduce a new graph sparsification technique that deterministically computes a low-degree subgraph with additional desired properties: degrees in the subgraph are sufficiently small that nodes' neighborhoods can be stored on single machines, and solving the problem on the subgraph provides significant global progress towards solving the problem for the original input graph. Using this framework to derandomize the well-known randomized algorithm of Luby [SICOMP'86], we obtain O(log Δ+log log n)$-round deterministic MPC algorithms for solving the fundamental problems of Maximal Matching and Maximal Independent Set with O(ne) space on each machine for any constant e > 0. Based on the recent work of Ghaffari et al. [FOCS'18], this additive O(log log n) factor is conditionally essential. These algorithms can also be shown to run in O(log Δ) rounds in the closely related model of CONGESTED CLIQUE, improving upon the state-of-the-art bound of O(log2 Δ) rounds by Censor-Hillel et al. [DISC'17].

This study deals with the topology discovery for the capacitated minimum spanning tree network. The problem is composed of finding the best way to link nodes to a source node and, in graph-theoretical terms, it is to determine a minimal spanning tree with a capacity constraint. In this paper, a heuristic algorithm with two phases is presented. Computational complexity analysis and simulation confirm that our algorithm produces better results than the previous other algorithms in short running time. The algorithm can be applied to find the least cost multicast trees in the local computer network.

Introduction. Varieties of the problem of packing of rectangular parallelepipeds have a wide practical application in various fields of activity, for example, in the optimal filling of containers, in the design and layout of a wide variety of technological objects and systems, in the creation of backup copies on removable media, in the optimization of storage, protection and transportation of goods, in additive manufacturing, etc. This work continues research on this topic and is devoted to the problem of balanced sparse packing of a given set of identically oriented rectangular parallelepipeds of different sizes into a rectangular parallelepiped of minimum volume. It presents a mathematical model for this packing problem and a parallel algorithm for solving it. This algorithm is based on the reduction of the original problem to an unconditional optimization problem using penalty functions, which is solved by the multistart method, in which r-algorithm is used to find local minima from the set of generated starting points. The purpose. Construction of a mathematical model and development of an algorithm for solving the problem of balanced sparse packing of a given set of identically oriented rectangular parallelepipeds into a rectangular parallelepiped of minimum volume. Results. A mathematical model and a parallel algorithm for balanced sparse packing of identically oriented rectangular parallelepipeds into a rectangular parallelepiped of minimum volume are presented. The algorithm is based on reducing the problem with the help of penalty functions to an unconditional nondifferentiable optimization problem, for finding the solution of which multistart method is used in combination with r-algorithm for finding local minima. The results of numerical experiments are given. Conclusions. The application of the algorithm described in the work based on non-smooth optimization methods allows to improve the results of balanced sparse packing of rectangular parallelepipeds in an acceptable time. Numerical experiments showed effectiveness of the algorithm in practice. Keywords: balanced sparse packing, multistart method, r-algorithm, penalty function, "Master-Slave" procedure, numerical experiments.

Algebraic coding theory is one of the areas that routinely gives rise to computational problems involving various structured matrices, such as Hankel, Vandermonde, Cauchy matrices, and certain generalizations thereof. Their structure has often been used to derive efficient algorithms; however, the use of the structure was pattern-specific, without applying a unified technique. In contrast, in several other areas, where structured matrices are also widely encountered, the concept of displacement rank was found to be useful to derive efficient algorithms in a unified manner (i.e., not depending on a particular pattern of structure). The latter technique allows one to compress, in a unified way, different types of n × n structured matrices to only αn parameters. This typically leads to computational savings (in many applications the number α, called the displacement rank, is a small fixed constant).In this paper we demonstrate the power of the displacement structure approach by deriving in a unified way efficient algorithms for a number of decoding problems. We accelerate the Sudan's list decoding algorithm for Reed-Solomon codes, its generalization to algebraic-geometric codes by Shokrollahi and Wasserman, and the improvement of Guruswami and Sudan in the case of Reed-Solomon codes. In particular, we notice that matrices that occur in the context of list decoding have low displacement rank, and use this fact to derive algorithms that use O(n2 l) and O(n7/3 l) operations over the base field for list decoding of Reed-Solomon codes and algebraic-geometric codes from certain plane curves, respectively. Here l denotes the length of the list; assuming that l is constant, this gives algorithms of running time O(n2) and O(n7/3), which is the same as the running time of conventional decoding algorithms. We also present efficient parallel algorithms for the above tasks.To the best of our knowledge this is the first application of the concept of displacement rank to the unified derivation of several decoding algorithms; the technique can be useful in finding efficient and fast methods for solving other decoding problems.

Thorup and Zwick showed that for any integer k≥ 1, it is possible to preprocess any positively weighted undirected graph G=(V,E), with |E|=m and |V|=n, in Õ(kmn \(^{\rm 1/{\it k}}\)) expected time and construct a data structure (a (2k –1)-approximate distance oracle) of size O(kn \(^{\rm 1+1/{\it k}}\)) capable of returning in O(k) time an approximation \(\hat{\delta}(u,v)\) of the distance δ(u,v) from u to v in G that satisfies \(\delta(u,v) \leq \hat{\delta}(u,v) \leq (2k -1)\cdot \delta(u,v)\), for any two vertices u,v∈ V. They also presented a much slower Õ(kmn) time deterministic algorithm for constructing approximate distance oracle with the slightly larger size of O(kn \(^{\rm 1+1/{\it k}}\)log n). We present here a deterministic Õ(kmn \(^{\rm 1/{\it k}}\)) time algorithm for constructing oracles of size O(kn \(^{\rm 1+1/{\it k}}\)). Our deterministic algorithm is slower than the randomized one by only a logarithmic factor.Using our derandomization technique we also obtain the first deterministic linear time algorithm for constructing optimal spanners of weighted graphs. We do that by derandomizing the O(km) expected time algorithm of Baswana and Sen (ICALP’03) for constructing (2k–1)-spanners of size O(kn \(^{\rm 1+1/{\it k}}\)) of weighted undirected graphs without incurring any asymptotic loss in the running time or in the size of the spanners produced.