Polynomial Time Approximation Research Articles

Multiagent MST Cover: Pleasing All Optimally via a Simple Voting Rule

Given a connected graph on whose edges we can build roads to connect the nodes, a number of agents hold possibly different perspectives on which edges should be selected by assigning different edge weights. Our task is to build a minimum number of roads so that every agent has a spanning tree in the built subgraph whose weight is the same as a minimum spanning tree in the original graph. We first show that this problem is NP-hard and does not admit better than ((1-o(1)) ln k)-approximation polynomial-time algorithms unless P = NP, where k is the number of agents. We then give a simple voting algorithm with an optimal approximation ratio. Moreover, our algorithm only needs to access the agents' rankings on the edges. Finally, we extend our problem to submodular objective functions and Matroid rank constraints.

Complexity and Approximation for Discriminating and Identifying Code Problems in Geometric Setups

We study geometric variations of the discriminating code problem. In the discrete version of the problem, a finite set of points P and a finite set of objects S are given in \(\mathbb {R}^d\). The objective is to choose a subset \(S^* \subseteq S\) of minimum cardinality such that for each point \(p_i \in P\), the subset \(S_i^* \subseteq S^*\) covering \(p_i\) satisfies \(S_i^*\ne \emptyset \), and each pair \(p_i,p_j \in P\), \(i \ne j\), we have \(S_i^* \ne S_j^*\). In the continuous version of the problem, the solution set \(S^*\) can be chosen freely among a (potentially infinite) class of allowed geometric objects. In the 1-dimensional case (\(d=1\)), the points in P are placed on a horizontal line L, and the objects in S are finite-length line segments aligned with L (called intervals). We show that the discrete version of this problem is NP-complete. This is somewhat surprising as the continuous version is known to be polynomial-time solvable. This is also in contrast with most geometric covering problems, which are usually polynomial-time solvable in one dimension. Still for the 1-dimensional discrete version, we design a polynomial-time 2-approximation algorithm. We also design a PTAS for both discrete and continuous versions in one dimension, for the restriction where the intervals are all required to have the same length. We then study the 2-dimensional case (\(d=2\)) for axis-parallel unit square objects. We show that both continuous and discrete versions are NP-complete, and design polynomial-time approximation algorithms that produce \((16\cdot OPT+1)\)-approximate and \((64\cdot OPT+1)\)-approximate solutions respectively, using rounding of suitably defined integer linear programming problems. Finally, we apply our techniques to a related variant of the discrete problem, where instead of points and geometric objects we just have a set S of objects. The goal is to select a small subset \(S^*\) of objects so that all objects of S are discriminated by their intersection with the objects of \(S^*\). This problem can be viewed as a graph problem by stating it in terms of the vertices of the geometric intersection graph of S. Under this graph-theoretical form, it is known as the identifying code problem. We show that the identifying code problem for axis-parallel unit square intersection graphs (in \(d=2\)) can be solved in the same manner as for the discrete version of the discriminating code problem for unit square objects described above, and all our positive approximation results still hold in this setting.

Fast Restoring the Controllability of Networked Systems With Symmetric Weights

Structural controllability theory enables us to attain controllability of networked systems independent of edge weights. However, the theory is restricted to a strong assumption that all edge weights are independent of each other, limiting the practical application in most real life scenarios. In this paper, we develop a new theory to restore the controllability of networked systems with coupled edge weights, i.e., symmetric edge weights. Firstly, we show that an uncontrollable symmetric networked system can be transformed into a controllable temporal network by the sym-cactus-based segmentation method and provide the rigorous theoretical proof. Then we consider the least number of segmentation to restore the controllability as quickly as possible and reduce the problem to the classical set coverage problem. Finally, an approximate polynomial-time algorithm and a simulation example are provided to demonstrate the validity of our method.

On computational capabilities of Ising machines based on nonlinear oscillators

Dynamical Ising machines are actively investigated from the perspective of finding efficient heuristics for NP-hard optimization problems. However, the existing data demonstrate super-polynomial scaling of the running time with the system size, which is incompatible with large NP-hard problems. We show that oscillator networks implementing the Kuramoto model of synchronization are capable of demonstrating polynomial scaling. The dynamics of these networks is related to the semidefinite programming relaxation of the Ising model ground state problem. Consequently, such networks, as we numerically demonstrate, are capable of producing the best possible approximation in polynomial time. To reach such performance, however, the reconstruction of the binary Ising state (rounding) must be specially addressed. We demonstrate that commonly implemented forced collapse to a close-to-Ising state may diminish the computational capabilities up to their complete invalidation. Therefore, consistent treatment of rounding may cardinally improve various operation metrics of already existing and upcoming dynamical Ising machines.

Clustering sequence graphs

In application domains ranging from social networks to e-commerce, it is important to cluster users with respect to both their relationships (e.g., friendship or trust) and their actions (e.g., visited locations or rated products). Motivated by these applications, we introduce here the task of clustering the nodes of a sequence graph, i.e., a graph whose nodes are labeled with strings (e.g., sequences of users’ visited locations or rated products). Both string clustering algorithms and graph clustering algorithms are inappropriate to deal with this task, as they do not consider the structure of strings and graph simultaneously. Moreover, attributed graph clustering algorithms generally construct poor solutions because they need to represent a string as a vector of attributes, which inevitably loses information and may harm clustering quality. We thus introduce the problem of clustering a sequence graph. We first propose two pairwise distance measures for sequence graphs, one based on edit distance and shortest path distance and another one based on SimRank. We then formalize the problem under each measure, showing also that it is NP-hard. In addition, we design a polynomial-time 2-approximation algorithm, as well as a heuristic for the problem. Experiments using real datasets and a case study demonstrate the effectiveness and efficiency of our methods.

Compliant Conditions for Polynomial Time Approximation of Operator Counts

In this brief abstract, we develop a computationally simpler version of the operator count heuristic for a particular class of domains. The contribution of this abstract is thus threefold, we (1) propose an efficient closed form approximation to the operator count heuristic; (2) leverage compressed sensing techniques to obtain an integer approximation in polynomial time; and (3) discuss the relationship of the proposed formulation to existing heuristics and investigate properties of domains where such approaches are useful.

Minimizing total interference in asymmetric sensor networks

The problem of computing a connected network with minimum interference is a fundamental problem in wireless sensor networks. Several models of interference have been studied in the literature. The most common one is the receiver-centric, in which the interference of a node p is defined as the number of other nodes whose transmission range covers p. In this paper, we study the problem of assigning a transmission range to each sensor, such that the resulting network is strongly connected and the total interference of the network is minimized.For the one-dimensional case, we show how to solve the problem optimally in O(n3) time. For the two-dimensional case, we show that the problem is NP-complete and give a polynomial-time 2-approximation algorithm for the problem.

Algorithms for hierarchical and semi-partitioned parallel scheduling

We propose a model for scheduling jobs in a parallel machine setting that takes into account the cost of migrations by assuming that the processing time of a job may depend on the specific set of machines among which the job is migrated. For the makespan minimization objective, the model generalizes classical scheduling problems such as unrelated parallel machine scheduling, as well as novel ones such as semi-partitioned and clustered scheduling. In the case of a hierarchical family of machines, we derive a compact integer linear programming (ILP) formulation for the job assignment subproblem, and show how to turn arbitrary ILP solutions into valid schedules. We also derive a polynomial-time 2-approximation algorithm for the problem. Extensions that incorporate memory capacity constraints are also discussed.

Safe Approximation—An Efficient Solution for a Hard Routing Problem

The Disjoint Connecting Paths problem and its capacitated generalization, called Unsplittable Flow problem, play an important role in practical applications such as communication network design and routing. These tasks are NP-hard in general, but various polynomial-time approximations are known. Nevertheless, the approximations tend to be either too loose (allowing large deviation from the optimum), or too complicated, often rendering them impractical in large, complex networks. Therefore, our goal is to present a solution that provides a relatively simple, efficient algorithm for the unsplittable flow problem in large directed graphs, where the task is NP-hard, and is known to remain NP-hard even to approximate up to a large factor. The efficiency of our algorithm is achieved by sacrificing a small part of the solution space. This also represents a novel paradigm for approximation. Rather than giving up the search for an exact solution, we restrict the solution space to a subset that is the most important for applications, and excludes only a small part that is marginal in some well-defined sense. Specifically, the sacrificed part only contains scenarios where some edges are very close to saturation. Since nearly saturated links are undesirable in practical applications, therefore, excluding near saturation is quite reasonable from the practical point of view. We refer the solutions that contain no nearly saturated edges as safe solutions, and call the approach safe approximation. We prove that this safe approximation can be carried out efficiently. That is, once we restrict ourselves to safe solutions, we can find the exact optimum by a randomized polynomial time algorithm.

Special Section on the 48th Annual ACM Symposium on Theory of Computing (STOC 2016)

Special Section on the 48th Annual ACM Symposium on Theory of Computing (STOC 2016)

Near-gathering of energy-constrained mobile agents

We study the task of gathering k energy-constrained mobile agents in an undirected edge-weighted graph. Each agent is initially placed on an arbitrary node and has a limited amount of energy, which constrains the distance it can move. Since this may render gathering at a single point impossible, we study three variants of near-gathering:The goal is to move the agents into a configuration that minimizes either (i) the radius of a ball containing all agents, (ii) the maximum distance between any two agents, or (iii) the average distance between the agents. We prove that (i) is polynomial-time solvable, (ii) has a polynomial-time 2-approximation with a matching NP-hardness lower bound, while (iii) admits a polynomial-time 2(1−1k)-approximation, but no FPTAS, unless P=NP. We extend some of our results to additive approximation.

Regimes of Classical Simulability for Noisy Gaussian Boson Sampling.

As a promising candidate for exhibiting quantum computational supremacy, Gaussian boson sampling (GBS) is designed to exploit the ease of experimental preparation of Gaussian states. However, sufficiently large and inevitable experimental noise might render GBS classically simulable. In this work, we formalize this intuition by establishing a sufficient condition for approximate polynomial-time classical simulation of noisy GBS-in the form of an inequality between the input squeezing parameter, the overall transmission rate, and the quality of photon detectors. Our result serves as a nonclassicality test that must be passed by any quantum computational supremacy demonstration based on GBS. We show that, for most linear-optical architectures, where photon loss increases exponentially with the circuit depth, noisy GBS loses its quantum advantage in the asymptotic limit. Our results thus delineate intermediate-sized regimes where GBS devices might considerably outperform classical computers for modest noise levels. Finally, we find that increasing the amount of input squeezing is helpful to evade our classical simulation algorithm, which suggests a potential route to mitigate photon loss.

Transportation and Batching Scheduling for Minimizing Total Weighted Completion Time

We consider the coordination of transportation and batching scheduling with one single vehicle for minimizing total weighted completion time. The computational complexity of the problem with batch capacity of at least 2 was posed as open in the literature. For this problem, we show the unary NP-hardness for every batch capacity at least 3 and present a polynomial-time 3-approximation algorithm when the batch capacity is at least 2.

Dynamic programming optimization in line of sight networks

Line of Sight (LoS) networks were designed to model wireless communication in settings which may contain obstacles. For fixed positive integer d, and positive integer ω, a graph G=(V,E) is a (d-dimensional) LoS network with range parameter ω if it can be embedded in a finite cube of the d-dimensional integer grid so that each pair of vertices in V are adjacent if and only if their embedding coordinates differ only in one position and such difference is less than ω.In this paper we investigate a dynamic programming (DP) approach which can be used to obtain efficient algorithmic solutions for various combinatorial problems in LoS networks. In particular DP solves the Maximum Independent Set (MIS) problem in LoS networks optimally, for any ω, on narrow LoS networks (i.e. networks which can be embedded in a k×k…×k×n region, for some fixed k independent of n). In the unrestricted case it has been shown that the (decision version of the) problem is NP-hard when 2<ω=O(n1−δ), for fixed 0<δ<1. We describe how DP can be used as a building block in the design of good approximation algorithms in this case. In particular we present a semi-online polynomial-time approximation scheme for the MIS problem in narrow d-dimensional LoS networks, as well as a polynomial-time 2-approximation algorithm and a fast polynomial time approximation scheme for the MIS problem in arbitrary d-dimensional LoS networks. Finally we comment on how the approach can be adapted to prove similar results for a number of important optimization problems in LoS networks.

A Simple Algorithm for $r$-gatherings on the Line

In this paper we study recently proposed variant of the facility location problem called the r-gathering problem. Given sets C and F of points on the plane and distance d(c, f) for each \(c\in C\) and \(f\in F\), an r-gathering of C to F is an assignment A of C to open facilities \(F^{'} \subset F\) such that r or more customers are assigned to each open facility. The cost of an r-gathering is the maximum distance d(c, f) between \(c\in C\) and \(A(c)\in F'\) among the assignment, which is \(\max _{c\in C}\{ d(c,A(c)) \}\). The r-gathering problem finds the r-gathering minimize the cost. A polynomial time 3-approximation algorithm for the r-gathering problem is known. When all C and F are on the line an \(O((|C|+|F|)\log (|C|+|F|) )\) time algorithm and an \(O(|C|+|F|\log ^2 r+|F|\log |F|)\) time algorithm to solve the r-gathering problem are known. In this paper we give a simple \(O(|C|+r^2|F|)\) time algorithm to solve the r-gathering problem. Since in typical case \(r<<|F|<<|C|\) holds our new algorithm is faster than the known algorithms.

Approximability of the Problem of Finding a Vector Subset with the Longest Sum

We answer the question of existence of polynomial-time constant-factor approximation algorithms for the space of nonfixed dimension. We prove that, in Euclidean space the problem is solvable in polynomial time with accuracy $$\sqrt a $$ , where α = 2/π, and if P ≠ NP then there are no polynomial algorithms with better accuracy. It is shown that, in the case of the lp spaces, the problem is APX-complete if p ∈ [1, 2] and not approximable with constant accuracy if P ≠ NP and p ∈ (2,∞).

Hardness of Approximation for H -free Edge Modification Problems

The H - free E dge D eletion problem asks, for a given graph G and integer k , whether it is possible to delete at most k edges from G to make it H -free—that is, not containing H as an induced subgraph. The H - free E dge C ompletion problem is defined similarly, but we add edges instead of deleting them. The study of these two problem families has recently been the subject of intensive studies from the point of view of parameterized complexity and kernelization. In particular, it was shown that the problems do not admit polynomial kernels (under plausible complexity assumptions) for almost all graphs H , with several important exceptions occurring when the class of H -free graphs exhibits some structural properties. In this work, we complement the parameterized study of edge modification problems to H -free graphs by considering their approximability. We prove that whenever H is 3-connected and has at least two nonedges, then both H - free E dge D eletion and H - free E dge C ompletion are very hard to approximate: they do not admit poly(OPT)-approximation in polynomial time, unless P=NP, or even in time subexponential in OPT, unless the exponential time hypothesis fails. The assumption of the existence of two nonedges appears to be important: we show that whenever H is a complete graph without one edge, then H - free E dge D eletion is tightly connected to the M in H orn D eletion problem, whose approximability is still open. Finally, in an attempt to extend our hardness results beyond 3-connected graphs, we consider the cases of H being a path or a cycle, and we achieve an almost complete dichotomy there.

Multiwinner analogues of the plurality rule: axiomatic and algorithmic perspectives

We characterize the class of committee scoring rules that satisfy the fixed-majority criterion. We argue that rules in this class are multiwinner analogues of the single-winner Plurality rule, which is uniquely characterized as the only single-winner scoring rule that satisfies the simple majority criterion. We define top-k-counting committee scoring rules and show that the fixed-majority consistent rules are a subclass of the top-k-counting rules. We give necessary and sufficient conditions for a top-k-counting rule to satisfy the fixed-majority criterion. We show that, for many top-k-counting rules, the complexity of winner determination is high (formally, we show that the problem of deciding if there exists a committee with at least a given score is {mathrm {NP}}-hard), but we also show examples of rules with polynomial-time winner determination procedures. For some of the computationally hard rules, we provide either exact FPT algorithms or approximate polynomial-time algorithms.

Analysis of Iterated Greedy Heuristic for Vertex Clique Covering

The aim of the vertex clique covering problem (CCP) is to cover the vertices of a graph with as few cliques as possible. We analyse the iterated greedy (IG) algorithm for CCP, which was previously shown to provide strong empirical results for real-world networks. It is demonstrated how the techniques of analysis for randomised search heuristics can be applied to IG, and several practically relevant results are obtained. We show that for triangle-free graphs, IG solves CCP optimally in expected polynomial time. Secondly, we show that IG finds the optimum for CCP in a specific case of sparse random graphs in expected polynomial time with high probability. For Barabási-Albert model of scale-free networks, which is a canonical model explaining the growth of social, biological or computer networks, we obtain that IG obtains an asymptotically optimal approximation in polynomial time in expectation. Last but not least, we propose a slightly modified variant of IG, which guarantees expected polynomial-time convergence to the optimum for graphs with non-overlapping triangles.

Polynomial-Time Approximation Algorithm for the Problem of Cardinality-Weighted Variance-Based 2-Clustering with a Given Center

A strongly NP-hard problem of partitioning a finite set of points of Euclidean space into two clusters is considered. The solution criterion is the minimum of the sum (over both clusters) of weighted sums of squared distances from the elements of each cluster to its geometric center. The weights of the sums are equal to the cardinalities of the desired clusters. The center of one cluster is given as input, while the center of the other is unknown and is determined as the point of space equal to the mean of the cluster elements. A version of the problem is analyzed in which the cardinalities of the clusters are given as input. A polynomial-time 2-approximation algorithm for solving the problem is constructed.