Coverage with k-transmitters in the presence of obstacles

For a fixed integer k≥0, a k-transmitter is an omnidirectional wireless transmitter with an infinite broadcast range that is able to penetrate up to k “walls”, represented as line segments in the plane. We develop lower and upper bounds for the number of k-transmitters that are necessary and sufficient to cover a given collection of line segments, polygonal chains and polygons.     

Bin packing under linear constraints

In this paper, we study a bin packing problem in which the sizes of items are determined by k linear constraints, and the goal is to decide the sizes of items and pack them into a minimal number of unit sized bins. We first provide two scenarios that motivate this research. We show that this problem is NP-hard in general, and propose several algorithms in terms of the number of constraints. If the number of constraints is arbitrary, we propose a 2-approximation algorithm, which is based on the analysis of the Next Fit algorithm for the bin packing problem. If the number of linear constraints is a fixed constant, then we obtain a PTAS by combining linear programming and enumeration techniques, and show that it is an optimal algorithm in polynomial time when the number of constraints is one or two. It is well known that the bin packing problem is strongly NP-hard and cannot be approximated within a factor 3 / 2 unless P = NP. This result implies that the bin packing problem with a fixed number of constraints may be simper than the original bin packing problem. Finally, we discuss the case when the sizes of items are bounded.     

Tight bounds for NF-based bounded-space online bin packing algorithms

In Zheng et al. (J Comb Optim 30(2):360–369, 2015) modelled a surgery problem by the one-dimensional bin packing, and developed a semi-online algorithm to give an efficient feasible solution. In their algorithm they used a buffer to temporarily store items, having a possibility to lookahead in the list. Because of the considered practical problem they investigated the 2-parametric case, when the size of the items is at most 1 / 2. Using an NF-based online algorithm the authors proved an ACR of \(13/9 = 1.44\ldots \) for any given buffer size not less than 1. They also gave a lower bound of \(4/3=1.33\ldots \) for the bounded-space algorithms that use NF-based rules. Later, in Zhang et al. (J Comb Optim 33(2):530–542, 2017) an algorithm was given with an ACR of 1.4243,  and the authors improved the lower bound to 1.4230. In this paper we present a tight lower bound of \(h_\infty (r)\) for the r-parametric problem when the buffer capacity is 3. Since \(h_\infty (2) = 1.42312\ldots \), our result—as a special case—gives a tight bound for the algorithm-class given in 2017. To prove that the lower bound is tight, we present an NF-based online algorithm that considers the r-parametric problem, and uses a buffer with capacity of 3. We prove that this algorithm has an ACR that is equal to the lower bounds for arbitrary r.     

The k-Canadian Travelers Problem with communication

This paper studies a variation of the online k-Canadian Traveler Problem (k-CTP), in which there are multiple travelers who can communicate with each other, to share real-time blockage information of the edges. We study two different communication levels for the problem, complete communication (where all travelers can receive and send blockage information with each other) and limited communication (where only some travelers can both receive and send information while the others can only receive information). The objective is that at least one traveler finds a feasible route from the origin to the destination with as small cost as possible. We give lower bounds on the competitive ratio for both the two communication levels. Considering the urban traffic environment, we propose the Retrace-Alternating strategy and Greedy strategy for both the two communication levels, and prove that increasing the number of travelers with complete communication ability may not always improve the competitive ratio of online strategies.     

New algorithms for online rectangle filling with <Emphasis Type="BoldItalic">k</Emphasis>-lookahead

We study the online rectangle filling problem which arises in channel aware scheduling of wireless networks, and present deterministic and randomized results for algorithms that are allowed a k-lookahead for k≥2. Our main result is a deterministic min {1.848,1+2/(k−1)}-competitive online algorithm. This is the first algorithm for this problem with a competitive ratio approaching 1 as k approaches +∞. The previous best-known solution for this problem has a competitive ratio of 2 for any k≥2. We also present a randomized online algorithm with a competitive ratio of 1+1/(k+1). Our final result is a closely matching lower bound (also proved in this paper) of $1+1/(\sqrt{k+2}+\sqrt{k+1})^{2}>1+1/(4(k+2))$1+1/(\sqrt{k+2}+\sqrt{k+1})^{2}>1+1/(4(k+2)) on the competitive ratio of any randomized online algorithm against an oblivious adversary. These are the first known results for randomized algorithms for this problem.     

A refined algorithm for maximum independent set in degree-4 graphs

The maximum independent set problem is one of the most important problems in theoretical analysis on time and space complexities of exact algorithms. Theoretical improvement on upper bounds on time complexity to solve this problem in low-degree graphs can lead to an improvement on that to the problem in general graphs. In this paper, we derive an upper bound \(O^*(1.1376^n)\) on the time complexity of a polynomial-space algorithm that solves the maximum independent set problem in an n-vertex graph with degree bounded by 4, improving all previous upper bounds on the time complexity of exact algorithms to this problem. Our algorithm is a branch-and-reduce algorithm and analyzed by using the measure-and-conquer method. To make an amortized analysis of the running time bound, we use an idea of “shift” to save some decrease of the measure from good branches to bad branches. Our algorithm first deals with small vertex cuts and vertices of degree \({\ge }5\), which may be created in our algorithm even if the input graph has maximum degree 4, then eliminates cycles of length 3 and 4 containing degree-4 vertices, and finally branches on degree-4 vertices. We invoke an exact algorithm for this problem in graphs with maximum degree 3 directly when the graph has no vertices of degree \({\ge }4\). Branching on degree-4 vertices on special local structures will be the bottleneck case, and we carefully design rules of choosing degree-4 vertices to branch on so that the resulting instances after branching decrease the measure effectively in the next step.     

Constraining the number of positive responses in adaptive,non-adaptive,and two-stage group testing

Group testing is a well known search problem that consists in detecting the defective members of a set of objects O by performing tests on properly chosen subsets (pools) of the given set O. In classical group testing the goal is to find all defectives by using as few tests as possible. We consider a variant of classical group testing in which one is concerned not only with minimizing the total number of tests but aims also at reducing the number of tests involving defective elements. The rationale behind this search model is that in many practical applications the devices used for the tests are subject to deterioration due to exposure to or interaction with the defective elements. In this paper we consider adaptive, non-adaptive and two-stage group testing. For all three considered scenarios, we derive upper and lower bounds on the number of “yes” responses that must be admitted by any strategy performing at most a certain number t of tests. In particular, for the adaptive case we provide an algorithm that uses a number of “yes” responses that exceeds the given lower bound by a small constant. Interestingly, this bound can be asymptotically attained also by our two-stage algorithm, which is a phenomenon analogous to the one occurring in classical group testing. For the non-adaptive scenario we give almost matching upper and lower bounds on the number of “yes” responses. In particular, we give two constructions both achieving the same asymptotic bound. An interesting feature of one of these constructions is that it is an explicit construction. The bounds for the non-adaptive and the two-stage cases follow from the bounds on the optimal sizes of new variants of d-cover free families and (p, d)-cover free families introduced in this paper, which we believe may be of interest also in other contexts.     

Property testing for cyclic groups and beyond

This paper studies the problem of testing if an input (Γ,°), where Γ is a finite set of unknown size and ° is a binary operation over Γ given as an oracle, is close to a specified class of groups. Friedl et al. (Proc. of STOC, 2005) have constructed an efficient tester using poly(log|Γ|) queries for the class of abelian groups. We focus in this paper on subclasses of abelian groups, and show that these problems are much harder: Ω(|Γ|^1/6) queries are necessary to test if the input is close to a cyclic group, and Ω(|Γ| ^c ) queries for some constant c are necessary to test more generally if the input is close to an abelian group generated by k elements, for any fixed integer k≥1. We also show that knowledge of the size of the ground set Γ helps only for k=1, in which case we construct an efficient tester using poly(log|Γ|) queries; for any other value k≥2 the query complexity remains Ω(|Γ| ^c ). All our upper and lower bounds hold for both the edit distance and the Hamming distance. These are, to the best of our knowledge, the first nontrivial lower bounds for such group-theoretic problems in the property testing model and, in particular, they imply the first exponential separations between the classical and quantum query complexities of testing closeness to classes of groups.     

Flows with unit path capacities and related packing and covering problems

Since the seminal work of Ford and Fulkerson in the 1950s, network flow theory is one of the most important and most active areas of research in combinatorial optimization. Coming from the classical maximum flow problem, we introduce and study an apparently basic but new flow problem that features a couple of interesting peculiarities. We derive several results on the complexity and approximability of the new problem. On the way we also discover two closely related basic covering and packing problems that are of independent interest. Starting from an LP formulation of the maximum s-t-flow problem in path variables, we introduce unit upper bounds on the amount of flow being sent along each path. The resulting (fractional) flow problem is NP-hard; its integral version is strongly NP-hard already on very simple classes of graphs. For the fractional problem we present an FPTAS that is based on solving the k shortest paths problem iteratively. We show that the integral problem is hard to approximate and give an interesting O(log?m)-approximation algorithm, where m is the number of arcs in the considered graph. For the multicommodity version of the problem there is an $O(\sqrt{m})Since the seminal work of Ford and Fulkerson in the 1950s, network flow theory is one of the most important and most active areas of research in combinatorial optimization. Coming from the classical maximum flow problem, we introduce and study an apparently basic but new flow problem that features a couple of interesting peculiarities. We derive several results on the complexity and approximability of the new problem. On the way we also discover two closely related basic covering and packing problems that are of independent interest. Starting from an LP formulation of the maximum s-t-flow problem in path variables, we introduce unit upper bounds on the amount of flow being sent along each path. The resulting (fractional) flow problem is NP-hard; its integral version is strongly NP-hard already on very simple classes of graphs. For the fractional problem we present an FPTAS that is based on solving the k shortest paths problem iteratively. We show that the integral problem is hard to approximate and give an interesting O(log m)-approximation algorithm, where m is the number of arcs in the considered graph. For the multicommodity version of the problem there is an O(?m)O(\sqrt{m}) -approximation algorithm. We argue that this performance guarantee is best possible, unless P=NP.     

Incremental Facility Location Problem and Its Competitive Algorithms

We consider the incremental version of the k-Facility Location Problem, which is a common generalization of the facility location and the k-median problems. The objective is to produce an incremental sequence of facility sets F ₁?F ₂?????F _n , where each F _k contains at most k facilities. An incremental facility sequence or an algorithm producing such a sequence is called c -competitive if the cost of each F _k is at most c times the optimum cost of corresponding k-facility location problem, where c is called competitive ratio. In this paper we present two competitive algorithms for this problem. The first algorithm produces competitive ratio 8α, where α is the approximation ratio of k-facility location problem. By recently result (Zhang, Theor. Comput. Sci. 384:126–135, 2007), we obtain the competitive ratio \(16+8\sqrt{3}+\epsilon\). The second algorithm has the competitive ratio Δ+1, where Δ is the ratio between the maximum and minimum nonzero interpoint distances. The latter result has its self interest, specially for the small metric space with Δ≤8α?1.     

Sharp bounds of the Zagreb indices of k-trees

For a graph G, the first Zagreb index M ₁ is equal to the sum of squares of the vertex degrees, and the second Zagreb index M ₂ is equal to the sum of the products of degrees of pairs of adjacent vertices. The Zagreb indices have been the focus of considerable research in computational chemistry dating back to Gutman and Trinajsti? in 1972. In 2004, Das and Gutman determined sharp upper and lower bounds for M ₁ and M ₂ values for trees along with the unique trees that obtain the minimum and maximum M ₁ and M ₂ values respectively. In this paper, we generalize the results of Das and Gutman to the generalized tree, the k-tree, where the results of Das and Gutman are for k=1. Also by showing that maximal outerplanar graphs are 2-trees, we also extend a result of Hou, Li, Song, and Wei who determined sharp upper and lower bounds for M ₁ and M ₂ values for maximal outerplanar graphs.     

Semi-online scheduling with “end of sequence” information

We study a variant of classical scheduling, which is called scheduling with “end of sequence” information. It is known in advance that the last job has the longest processing time. Moreover, the last job is marked, and thus it is known for every new job whether it is the final job of the sequence. We explore this model on two uniformly related machines, that is, two machines with possibly different speeds. Two objectives are considered, maximizing the minimum completion time and minimizing the maximum completion time (makespan). Let s be the speed ratio between the two machines, we consider the competitive ratios which are possible to achieve for the two problems as functions of s. We present algorithms for different values of s and lower bounds on the competitive ratio. The proposed algorithms are best possible for a wide range of values of s. For the overall competitive ratio, we show tight bounds of ϕ + 1 ≈ 2.618 for the first problem, and upper and lower bounds of 1.5 and 1.46557 for the second problem. The authors would like to dedicate this paper to the memory of our colleague and friend Yong He who passed away in August 2005 after struggling with illness. D. Ye: Research was supported in part by NSFC (10601048).     

k-tuple total domination in cross products of graphs

For k??1 an integer, a set S of vertices in a graph G with minimum degree at least?k is a k-tuple total dominating set of G if every vertex of G is adjacent to at least k vertices in S. The minimum cardinality of a k-tuple total dominating set of G is the k-tuple total domination number of G. When k=1, the k-tuple total domination number is the well-studied total domination number. In this paper, we establish upper and lower bounds on the k-tuple total domination number of the cross product graph G×H for any two graphs G and H with minimum degree at least?k. In particular, we determine the exact value of the k-tuple total domination number of the cross product of two complete graphs.     

Finding longest increasing and common subsequences in streaming data

We present algorithms and lower bounds for the Longest Increasing Subsequence (LIS) and Longest Common Subsequence (LCS) problems in the data-streaming model. To decide if the LIS of a given stream of elements drawn from an alphabet αbet has length at least k, we discuss a one-pass algorithm using O(k log αbetsize) space, with update time either O(log k) or O(log log αbetsize); for αbetsize = O(1), we can achieve O(log k) space and constant-time updates. We also prove a lower bound of Ω(k) on the space requirement for this problem for general alphabets αbet, even when the input stream is a permutation of αbet. For finding the actual LIS, we give a ⌈log (1 + 1/ɛ)-pass algorithm using O(k^1+ɛlog αbetsize) space, for any ɛ > 0. For LCS, there is a trivial Θ(1)-approximate O(log n)-space streaming algorithm when αbetsize = O(1). For general alphabets αbet, the problem is much harder. We prove several lower bounds on the LCS problem, of which the strongest is the following: it is necessary to use Ω(n/ρ²) space to approximate the LCS of two n-element streams to within a factor of ρ, even if the streams are permutations of each other. A preliminary version of this paper appears in the Proceedings of the 11th International Computing and Combinatorics Conference (COCOON'05), August 2005, pp. 263–272.     

Standard directed search strategies and their applications

Given a digraph that contains an intruder hiding on vertices or along edges, a directed searching problem is to find the minimum number of searchers required to capture the intruder. In this paper, we propose the standard directed search strategies, which can simplify arguments in proving search numbers, lower bounds, and relationships between search models. Using these strategies, we characterize k-directed-searchable graphs when k≤3. We also use standard directed search strategies to investigate the searching problems on oriented grids. We give tight lower and upper bounds for the directed search number of general oriented grids. Finally, we show how to compute the directed search number of a Manhattan grid. Yang’s research was supported in part by NSERC and MITACS.     

Almost optimal solutions for bin coloring problems

In this paper we study two interesting bin coloring problems: Minimum Bin Coloring Problem (MinBC) and Online Maximum Bin Coloring Problem (OMaxBC), motivated from several applications in networking. For the MinBC problem, we present two near linear time approximation algorithms to achieve almost optimal solutions, i.e., no more than OPT+2 and OPT+1 respectively, where OPT is the optimal solution. For the OMaxBC problem, we first introduce a deterministic 2-competitive greedy algorithm, and then give lower bounds for any deterministic and randomized (against adaptive offline adversary) online algorithms. The lower bounds show that our deterministic algorithm achieves the best possible competitive ratio. The research of this paper was partially supported by an NSF CAREER award CCF-0546509.     

Online scheduling on uniform machines with two hierarchies

In this paper we study online scheduling problem on m parallel uniform machines with two hierarchies. The objective is to minimize the maximum completion time (makespan). Machines are provided with different capability. The machines with speed s can schedule all jobs, while the other machines with speed 1 can only process partial jobs. Online algorithms for any 0<s<∞ are provided in the paper. For the case of k=1 and m=2, and the case of some values of s, k=1 and m=3, the algorithms are the best possible, where k is the number of machines with hierarchy 1, and m is the number of machines. Lower bounds for some special cases are also presented.     

Clustering with or without the approximation

We study algorithms for clustering data that were recently proposed by Balcan et al. (SODA’09: 19th Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 1068–1077, 2009a) and that have already given rise to several follow-up papers. The input for the clustering problem consists of points in a metric space and a number k, specifying the desired number of clusters. The algorithms find a clustering that is provably close to a target clustering, provided that the instance has the “(1+α,ε)-property”, which means that the instance is such that all solutions to the k-median problem for which the objective value is at most (1+α) times the optimal objective value correspond to clusterings that misclassify at most an ε fraction of the points with respect to the target clustering. We investigate the theoretical and practical implications of their results. Our main contributions are as follows. First, we show that instances that have the (1+α,ε)-property and for which, additionally, the clusters in the target clustering are large, are easier than general instances: the algorithm proposed in Balcan et al. (SODA’09: 19th Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 1068–1077, 2009a) is a constant factor approximation algorithm with an approximation guarantee that is better than the known hardness of approximation for general instances. Further, we show that it is NP-hard to check if an instance satisfies the (1+α,ε)-property for a given (α,ε); the algorithms in Balcan et al. (SODA’09: 19th Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 1068–1077, 2009a) need such α and ε as input parameters, however. We propose ways to use their algorithms even if we do not know values of α and ε for which the assumption holds. Finally, we implement these methods and other popular methods, and test them on real world data sets. We find that on these data sets there are no α and ε so that the dataset has both (1+α,ε)-property and sufficiently large clusters in the target solution. For the general case where there are no assumptions about the cluster sizes, we show that on our data sets the performance guarantee proved by Balcan et a. (SODA’09: 19th Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 1068–1077, 2009a) is meaningless for the values of α,ε for which the data set has the (1+α,ε)-property. The algorithm nonetheless gives reasonable results, although it is outperformed by other methods.     

On the readability of monotone Boolean formulae

Golumbic et al. (Discrete Appl. Math. 154:1465–1477, 2006) defined the readability of a monotone Boolean function f to be the minimum integer k such that there exists an ∧−∨-formula equivalent to f in which each variable appears at most k times. They asked whether there exists a polynomial-time algorithm, which given a monotone Boolean function f, in CNF or DNF form, checks whether f is a read-k function, for a fixed k. In this paper, we partially answer this question already for k=2 by showing that it is NP-hard to decide if a given monotone formula represents a read-twice function. It follows also from our reduction that it is NP-hard to approximate the readability of a given monotone Boolean function f:{0,1} ⁿ →{0,1} within a factor of O(n)\mathcal{O}(n) . We also give tight sublinear upper bounds on the readability of a monotone Boolean function given in CNF (or DNF) form, parameterized by the number of terms in the CNF and the maximum size in each term, or more generally the maximum number of variables in the intersection of any constant number of terms. When the variables of the DNF can be ordered so that each term consists of a set of consecutive variables, we give much tighter logarithmic bounds on the readability.     

Enumerating the edge-colourings and total colourings of a regular graph

In this paper, we are interested in computing the number of edge colourings and total colourings of a connected graph. We prove that the maximum number of k-edge-colourings of a connected k-regular graph on n vertices is k?((k?1)!) ^n/2. Our proof is constructive and leads to a branching algorithm enumerating all the k-edge-colourings of a connected k-regular graph in time O ^?(((k?1)!) ^n/2) and polynomial space. In particular, we obtain a algorithm to enumerate all the 3-edge-colourings of a connected cubic graph in time O ^?(2 ^n/2)=O ^?(1.4143 ⁿ ) and polynomial space. This improves the running time of O ^?(1.5423 ⁿ ) of the algorithm due to Golovach et al. (Proceedings of WG 2010, pp. 39–50, 2010). We also show that the number of 4-total-colourings of a connected cubic graph is at most 3?2^3n/2. Again, our proof yields a branching algorithm to enumerate all the 4-total-colourings of a connected cubic graph.