ACM Transactions on Algorithms最新文献

We study polynomial-time approximation schemes (PTASes) for constraint satisfaction problems (CSPs) such as Maximum Independent Set or Minimum Vertex Cover on sparse graph classes.

Baker’s approach gives a PTAS on planar graphs, excluded-minor classes, and beyond. For Max-CSPs, and even more generally, maximisation finite-valued CSPs (where constraints are arbitrary non-negative functions), Romero, Wrochna, and Živný [SODA’21] showed that the Sherali-Adams LP relaxation gives a simple PTAS for all fractionally-treewidth-fragile classes, which is the most general “sparsity” condition for which a PTAS is known. We extend these results to general-valued CSPs, which include “crisp” (or “strict”) constraints that have to be satisfied by every feasible assignment. The only condition on the crisp constraints is that their domain contains an element that is at least as feasible as all the others (but possibly less valuable).

For minimisation general-valued CSPs with crisp constraints, we present a PTAS for all Baker graph classes—a definition by Dvořák [SODA’20] that encompasses all classes where Baker’s technique is known to work, except for fractionally-treewidth-fragile classes. While this is standard for problems satisfying a certain monotonicity condition on crisp constraints, we show this can be relaxed to diagonalisability—a property of relational structures connected to logics, statistical physics, and random CSPs.

In this article, we present Approximation Schemes for Capacitated Vehicle Routing Problem (CVRP) on several classes of graphs. In CVRP, introduced by Dantzig and Ramser in 1959 [14], we are given a graph G=(V,E) with metric edges costs, a depot r ∈ V, and a vehicle of bounded capacity Q. The goal is to find a minimum cost collection of tours for the vehicle that returns to the depot, each visiting at most Q nodes, such that they cover all the nodes. This generalizes classic TSP and has been studied extensively. In the more general setting, each node v has a demand d_v and the total demand of each tour must be no more than Q. Either the demand of each node must be served by one tour (unsplittable) or can be served by multiple tours (splittable). The best-known approximation algorithm for general graphs has ratio α +2(1-ε) (for the unsplittable) and α +1-ε (for the splittable) for some fixed (ε gt frac{1}{3000}), where α is the best approximation for TSP. Even for the case of trees, the best approximation ratio is 4/3 [5] and it has been an open question if there is an approximation scheme for this simple class of graphs. Das and Mathieu [15] presented an approximation scheme with time n^logO(1/ε)n for Euclidean plane ℝ². No other approximation scheme is known for any other class of metrics (without further restrictions on Q). In this article, we make significant progress on this classic problem by presenting Quasi-Polynomial Time Approximation Schemes (QPTAS) for graphs of bounded treewidth, graphs of bounded highway dimensions, and graphs of bounded doubling dimensions. For comparison, our result implies an approximation scheme for the Euclidean plane with run time n^O(log6n/ε5).

In 2015, Driemel, Krivošija, and Sohler introduced the k,ℓ-median clustering problem for polygonal curves under the Fréchet distance. Given a set of input curves, the problem asks to find k median curves of at most ℓ vertices each that minimize the sum of Fréchet distances over all input curves to their closest median curve. A major shortcoming of their algorithm is that the input curves are restricted to lie on the real line. In this article, we present a randomized bicriteria-approximation algorithm that works for polygonal curves in ℝ^d and achieves approximation factor (1+ɛ) with respect to the clustering costs. The algorithm has worst-case running time linear in the number of curves, polynomial in the maximum number of vertices per curve (i.e., their complexity), and exponential in d, ℓ, 1/ɛ and 1/δ (i.e., the failure probability). We achieve this result through a shortcutting lemma, which guarantees the existence of a polygonal curve with similar cost as an optimal median curve of complexity ℓ, but of complexity at most 2ℓ -2, and whose vertices can be computed efficiently. We combine this lemma with the superset sampling technique by Kumar et al. to derive our clustering result. In doing so, we describe and analyze a generalization of the algorithm by Ackermann et al., which may be of independent interest.

Machine-learned predictors, although achieving very good results for inputs resembling training data, cannot possibly provide perfect predictions in all situations. Still, decision-making systems that are based on such predictors need not only benefit from good predictions, but should also achieve a decent performance when the predictions are inadequate. In this paper, we propose a prediction setup for arbitrary metrical task systems (MTS)(e.g., caching, k-server and convex body chasing) and online matching on the line. We utilize results from the theory of online algorithms to show how to make the setup robust. Specifically for caching, we present an algorithm whose performance, as a function of the prediction error, is exponentially better than what is achievable for general MTS. Finally, we present an empirical evaluation of our methods on real world datasets, which suggests practicality.

We consider a fundamental online scheduling problem in which jobs with processing times and deadlines arrive online over time at their release dates. The task is to determine a feasible preemptive schedule on a single or multiple possibly unrelated machines that maximizes the number of jobs that complete before their deadline. Due to strong impossibility results for competitive analysis on a single machine, we require that jobs contain some slack ɛ > 0, which means that the feasible time window for scheduling a job is at least 1+ɛ times its processing time on each eligible machine. Our contribution is two-fold: (i) We give the first non-trivial online algorithms for throughput maximization on unrelated machines, and (ii), this is the main focus of our paper, we answer the question on how to handle commitment requirements which enforce that a scheduler has to guarantee at a certain point in time the completion of admitted jobs. This is very relevant, e.g., in providing cloud-computing services, and disallows last-minute rejections of critical tasks. We present an algorithm for unrelated machines that is (Theta (frac{1}{varepsilon }))-competitive when the scheduler must commit upon starting a job. Somewhat surprisingly, this is the same optimal performance bound (up to constants) as for scheduling without commitment on a single machine. If commitment decisions must be made before a job’s slack becomes less than a δ-fraction of its size, we prove a competitive ratio of (mathcal {O}(frac{1}{varepsilon - delta })) for 0 < δ < ɛ. This result nicely interpolates between commitment upon starting a job and commitment upon arrival. For the latter commitment model, it is known that no (randomized) online algorithm admits any bounded competitive ratio. While we mainly focus on scheduling without migration, our results also hold when comparing against a migratory optimal solution in case of identical machines.

The generalized k-server problem is a far-reaching extension of the k-server problem with several applications. Here, each server s_i lies in its own metric space M_i. A request is a k-tuple r = (r₁,r₂,… ,r_k, which is served by moving some server s_i to the point r_i ∈ M_i, and the goal is to minimize the total distance traveled by the servers. Despite much work, no f(k)-competitive algorithm is known for the problem for k > 2 servers, even for special cases such as uniform metrics and lines.

Here, we consider the problem in uniform metrics and give the first f(k)-competitive algorithms for general k. In particular, we obtain deterministic and randomized algorithms with competitive ratio k · 2^k and O(k³ log k), respectively. Our deterministic bound is based on a novel application of the polynomial method to online algorithms, and essentially matches the long-known lower bound of 2^k-1. We also give a 2^2O(k)-competitive deterministic algorithm for weighted uniform metrics, which also essentially matches the recent doubly exponential lower bound for the problem.

We initiate the study of k-edge-connected orientations of undirected graphs through edge flips for k ≥ 2. We prove that in every orientation of an undirected 2k-edge-connected graph, there exists a sequence of edges such that flipping their directions one by one does not decrease the edge connectivity, and the final orientation is k-edge connected. This yields an “edge-flip based” new proof of Nash-Williams’ theorem: A undirected graph G has a k-edge-connected orientation if and only if G is 2k-edge connected. As another consequence of the theorem, we prove that the edge-flip graph of k-edge-connected orientations of an undirected graph G is connected if G is (2k+2)-edge connected. This has been known to be true only when k=1.

We consider the classic problem of computing the Longest Common Subsequence (LCS) of two strings of length n. The 40-year-old quadratic-time dynamic programming algorithm has recently been shown to be near-optimal by Abboud, Backurs, and Vassilevska Williams [FOCS’15] and Bringmann and Künnemann [FOCS’15] assuming the Strong Exponential Time Hypothesis. This has led the community to look for subquadratic approximation algorithms for the problem.

Yet, unlike the edit distance problem for which a constant-factor approximation in almost-linear time is known, very little progress has been made on LCS, making it a notoriously difficult problem also in the realm of approximation. For the general setting, only a naive O(n^ɛ/2-approximation algorithm with running time OŠ(n^2-ɛ has been known, for any constant 0 < ɛ ≤ 1. Recently, a breakthrough result by Hajiaghayi, Seddighin, Seddighin, and Sun [SODA’19] provided a linear-time algorithm that yields a O(n^0.497956-approximation in expectation; improving upon the naive (O(sqrt {n}))-approximation for the first time.

In this paper, we provide an algorithm that in time O(n_2-ɛ) computes an OŠ(n^2ɛ/5-approximation with high probability, for any 0 < ɛ ≤ 1. Our result (1) gives an OŠ(n^0.4-approximation in linear time, improving upon the bound of Hajiaghayi, Seddighin, Seddighin, and Sun, (2) provides an algorithm whose approximation scales with any subquadratic running time O(n^2-ɛ), improving upon the naive bound of O(n^ɛ/2) for any ɛ, and (3) instead of only in expectation, succeeds with high probability.

Since counting subgraphs in general graphs is, by and large, a computationally demanding problem, it is natural to try and design fast algorithms for restricted families of graphs. One such family that has been extensively studied is that of graphs of bounded degeneracy (e.g., planar graphs). This line of work, which started in the early 80’s, culminated in a recent work of Gishboliner et al., which highlighted the importance of the task of counting homomorphic copies of cycles (i.e., cyclic walks) in graphs of bounded degeneracy.

Our main result in this paper is a surprisingly tight relation between the above task and the well-studied problem of detecting (standard) copies of directed cycles in general directed graphs. More precisely, we prove the following:

• One can compute the number of homomorphic copies of C_2k and C_2k+1 in n-vertex graphs of bounded degeneracy in time Õ(n^d_k), where the fastest known algorithm for detecting directed copies of C_k in general m-edge digraphs runs in time Õ(m^d_k).

• Conversely, one can transform any O(n^b_k) algorithm for computing the number of homomorphic copies of C_2k or of C_2k+1 in n-vertex graphs of bounded degeneracy, into an Õ(m^b_k) time algorithm for detecting directed copies of C_k in general m-edge digraphs.

We emphasize that our first result does not use a black-box reduction (as opposed to the second result which does). Instead, we design an algorithm for computing the number of C_k-homomorphisms in degenerate graphs and show that one part of its analysis can be reduced to the analysis of the fastest known algorithm for detecting directed cycles in general digraphs, which was carried out in a recent breakthrough of Dalirrooyfard, Vuong and Vassilevska Williams. As a by-product of our algorithm, we obtain a new algorithm for detecting k-cycles in directed and undirected graphs of bounded degeneracy that is faster than all previously known algorithms for 7 ≤ k ≤ 11, and faster for all k ≥ 7 if the matrix multiplication exponent is 2.

Given a graph G and an integer k, the Interval Vertex Deletion (IVD) problem asks whether there exists a subset S⊆V(G) of size at most k such that G − S is an interval graph. This problem is known to be NP-complete [Yannakakis, STOC’78]. Originally in 2012, Cao and Marx showed that IVD is fixed parameter tractable: they exhibited an algorithm with running time (10^k n^{mathcal {O}(1)} ) [Cao and Marx, SODA’14]. The existence of a polynomial kernel for IVD remained a well-known open problem in Parameterized Complexity. In this paper, we settle this problem in the affirmative.