Algorithmica最新文献

We consider reallocation problems in settings where the initial endowment of each agent consists of a subset of the resources. The private information of the players is their value for every possible subset of the resources. The goal is to redistribute resources among agents to maximize efficiency. Monetary transfers are allowed, but participation is voluntary. We develop incentive-compatible, individually-rational and budget-balanced mechanisms for two settings in which agents have complex multi-parameter valuations, both settings include double auctions as a special case. The first setting is combinatorial exchanges, where we provide a mechanism that achieves a logarithmic approximation to the optimal efficiency when valuations are subadditive. The second setting is Arrow–Debreu markets for a single divisible good, where we present a constant approximation mechanism. The first result is given for a Bayesian setting, where the latter result is for prior-free environments.

Consider a set of jobs connected to a directed acyclic task graph with a fixed source and sink. The edges of this graph model precedence constraints and the jobs have to be scheduled with respect to those. We introduce the server cloud scheduling problem, in which the jobs have to be processed either on a single local machine or on one of infinitely many cloud machines. For each job, processing times both on the server and in the cloud are given. Furthermore, for each edge in the task graph, a communication delay is included in the input and has to be taken into account if one of the two jobs is scheduled on the server and the other in the cloud. The server processes jobs sequentially, whereas the cloud can serve as many as needed in parallel, but induces costs. We consider both makespan and cost minimization. The main results are an FPTAS for the makespan objective for graphs with a constant source and sink dividing cut and strong hardness for the case with unit processing times and delays.

The problem of finding a maximum size matching in a graph (known as the maximum matching problem) is one of the most classical problems in computer science. Despite a significant body of work dedicated to the study of this problem in the data stream model, the state-of-the-art single-pass semi-streaming algorithm for it is still a simple greedy algorithm that computes a maximal matching, and this way obtains ({1}/{2})-approximation. Some previous works described two/three-pass algorithms that improve over this approximation ratio by using their second and third passes to improve the above mentioned maximal matching. One contribution of this paper continues this line of work by presenting new three-pass semi-streaming algorithms that work along these lines and obtain improved approximation ratios of 0.6111 and 0.5694 for triangle-free and general graphs, respectively. Unfortunately, a recent work Konrad and Naidu (Approximation, randomization, and combinatorial optimization. Algorithms and techniques, APPROX/RANDOM 2021, August 16–18, 2021. LIPIcs, vol 207, pp 19:1–19:18, 2021. https://doi.org/10.4230/LIPIcs.APPROX/RANDOM.2021.19) shows that the strategy of constructing a maximal matching in the first pass and then improving it in further passes has limitations. Additionally, this technique is unlikely to get us closer to single-pass semi-streaming algorithms obtaining a better than ({1}/{2})-approximation. Therefore, it is interesting to come up with algorithms that do something else with their first pass (we term such algorithms non-maximal-matching-first algorithms). No such algorithms were previously known, and the main contribution of this paper is describing such algorithms that obtain approximation ratios of 0.5384 and 0.5555 in two and three passes, respectively, for general graphs. The main significance of our results is not in the numerical improvements, but in demonstrating the potential of non-maximal-matching-first algorithms.

In this paper we study the computational complexity issues of the problem of secondary domination (known also as (1, 2)-domination) in several graph classes. We also study the computational complexity of the problem of determining whether the domination and secondary domination numbers are equal. In particular, we study the influence of triangles and vertices of degree 1 on these numbers. Also, an optimal algorithm for finding a minimum secondary dominating set in trees is presented.

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 (Fsubseteq E) ((Hsubseteq 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=emptyset ) 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 (frac{27}{4})-approximation algorithm. This algorithm achieves an approximation ratio of (frac{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.

We study the problem of Regularized Unconstrained Submodular Maximization (RegularizedUSM) as defined by Bodek and Feldman (Maximizing sums of non-monotone submodular and linear functions: understanding the unconstrained case, arXiv:2204.03412, 2022): given query access to a non-negative submodular function (f:2^{{mathcal {N}}}rightarrow {mathbb {R}}_{ge 0}) and a linear function (ell :2^{{mathcal {N}}}rightarrow {mathbb {R}}) over the same ground set ({mathcal {N}}), output a set (Tsubseteq {mathcal {N}}) approximately maximizing the sum (f(T)+ell (T)). An algorithm is said to provide an ((alpha ,beta ))-approximation for RegularizedUSM if it outputs a set T such that ({mathbb {E}}[f(T)+ell (T)]ge max _{Ssubseteq {mathcal {N}}}[alpha cdot f(S)+beta cdot ell (S)]). We also consider the setting where S and T are constrained to be independent in a given matroid, which we refer to as Regularized Constrained Submodular Maximization (RegularizedCSM). The special case of RegularizedCSM with monotone f has been extensively studied (Sviridenko et al. in Math Oper Res 42(4):1197–1218, 2017; Feldman in Algorithmica 83(3):853–878, 2021; Harshaw et al., in: International conference on machine learning, PMLR, 2634–2643, 2019), whereas we are aware of only one prior work that studies RegularizedCSM with non-monotone f (Lu et al. in Optimization 1–27, 2023), and that work constrains (ell ) to be non-positive. In this work, we provide improved ((alpha ,beta ))-approximation algorithms for both RegularizedUSM and RegularizedCSM with non-monotone f. Specifically, we are the first to provide nontrivial ((alpha ,beta ))-approximations for RegularizedCSM where the sign of (ell ) is unconstrained, and the (alpha ) we obtain for RegularizedUSM improves over (Bodek and Feldman in Maximizing sums of non-monotone submodular and linear functions: understanding the unconstrained case, arXiv:2204.03412, 2022) for all (beta in (0,1)). We also prove new inapproximability results for RegularizedUSM and RegularizedCSM, as well as 0.478-inapproximability for maximizing a submodular function where S and T are subject to a cardinality constraint, improving a 0.491-inapproximability result due to Oveis Gharan and Vondrak (in: Proceedings of the twenty-second annual ACM-SIAM symposium on discrete algorithms, SIAM, pp 1098–1116, 2011).

Two recent lower bounds on the compressibility of repetitive sequences, (delta le gamma ), have received much attention. It has been shown that a length-n string S over an alphabet of size (sigma ) can be represented within the optimal (O(delta log tfrac{nlog sigma }{delta log n})) space, and further, that within that space one can find all the occ occurrences in S of any length-m pattern in time (O(mlog n + occ log ^epsilon n)) for any constant (epsilon >0). Instead, the near-optimal search time (O(m+({occ+1})log ^epsilon n)) has been achieved only within (O(gamma log frac{n}{gamma })) space. Both results are based on considerably different locally consistent parsing techniques. The question of whether the better search time could be supported within the (delta )-optimal space remained open. In this paper, we prove that both techniques can indeed be combined to obtain the best of both worlds: (O(m+({occ+1})log ^epsilon n)) search time within (O(delta log tfrac{nlog sigma }{delta log n})) space. Moreover, the number of occurrences can be computed in (O(m+log ^{2+epsilon }n)) time within (O(delta log tfrac{nlog sigma }{delta log n})) space. We also show that an extra sublogarithmic factor on top of this space enables optimal (O(m+occ)) search time, whereas an extra logarithmic factor enables optimal O(m) counting time.

Given two matroids on the same ground set, the matroid intersection problem asks for a common base, i.e., a subset of the ground set that is a base in both the matroids. The weighted version of the problem asks for a common base with maximum weight. In the case of linearly representable matroids, the weighted version is known to have a randomized parallel (RNC) algorithm based on the isolation lemma, when the given weights are polynomially bounded (Narayanan et al. in SIAM J Comput 23(2): 387–397, 1994). Finding a deterministic parallel (NC) algorithm, even for the unweighted decision question, has been a long-standing open question. The above RNC algorithm can be viewed as a randomized reduction from weighted search to weighted decision, which works for arbitrary matroids. We derandomize this reduction, i.e., we give an NC algorithm for weighted matroid intersection search using oracle access to its decision version.

A subset of ([n] = {1,2,ldots ,n}) is called stable if it forms an independent set in the cycle on the vertex set [n]. In 1978, Schrijver proved via a topological argument that for all integers n and k with (n ge 2k), the family of stable k-subsets of [n] cannot be covered by (n-2k+1) intersecting families. We study two total search problems whose totality relies on this result. In the first problem, denoted by (textsc {Schrijver}(n,k,m)), we are given an access to a coloring of the stable k-subsets of [n] with (m = m(n,k)) colors, where (m le n-2k+1), and the goal is to find a pair of disjoint subsets that are assigned the same color. While for (m = n-2k+1) the problem is known to be (textsf{PPA})-complete, we prove that for (m < d cdot lfloor frac{n}{2k+d-2} rfloor ), with d being any fixed constant, the problem admits an efficient algorithm. For (m = lfloor n/2 rfloor -2k+1), we prove that the problem is efficiently reducible to the (textsc {Kneser}) problem. Motivated by the relation between the problems, we investigate the family of unstable k-subsets of [n], which might be of independent interest. In the second problem, called Unfair Independent Set in Cycle, we are given (ell ) subsets (V_1, ldots , V_ell ) of [n], where (ell le n-2k+1) and (|V_i| ge 2) for all (i in [ell ]), and the goal is to find a stable k-subset S of [n] satisfying the constraints (|S cap V_i| le |V_i|/2) for (i in [ell ]). We prove that the problem is (textsf{PPA})-complete and that its restriction to instances with (n=3k) is at least as hard as the Cycle plus Triangles problem, for which no efficient algorithm is known. On the contrary, we prove that there exists a constant c for which the restriction of the problem to instances with (n ge c cdot k) can be solved in polynomial time.

Given a class of graphs ({mathcal {H}}), the problem (oplus text {{Sub}}({mathcal {H}})) is defined as follows. The input is a graph (Hin {mathcal {H}}) together with an arbitrary graph G. The problem is to compute, modulo 2, the number of subgraphs of G that are isomorphic to H. The goal of this research is to determine for which classes ({mathcal {H}}) the problem (oplus text {{Sub}}({mathcal {H}})) is fixed-parameter tractable (FPT), i.e., solvable in time (f(|H|)cdot |G|^{O(1)}). Curticapean, Dell, and Husfeldt (ESA 2021) conjectured that (oplus text {{Sub}}({mathcal {H}})) is FPT if and only if the class of allowed patterns ({mathcal {H}}) is matching splittable, which means that for some fixed B, every (H in {mathcal {H}}) can be turned into a matching (a graph in which every vertex has degree at most 1) by removing at most B vertices. Assuming the randomised Exponential Time Hypothesis, we prove their conjecture for (I) all hereditary pattern classes ({mathcal {H}}), and (II) all tree pattern classes, i.e., all classes ({mathcal {H}}) such that every (Hin {mathcal {H}}) is a tree. We also establish almost tight fine-grained upper and lower bounds for the case of hereditary patterns (I).