Two-agent scheduling problems on a single-machine to minimize the total weighted late work

In this paper, two-agent scheduling problems are presented. The different agents share a common processing resource, and each agent wants to minimize a cost function depending on its jobs only. The objective functions we consider are the total weighted late work and the maximum cost. The problem is to find a schedule that minimizes the objective function of agent A, while keeping the objective function of agent B cannot exceed a given bound U. Some different scenarios are presented by depending on the objective function of each agent. We address the complexity of those problems, and present the optimal polynomial time algorithms or pseudo-polynomial time algorithm to solve the scheduling problems, respectively.     

Safe sets in graphs: Graph classes and structural parameters

A safe set of a graph \(G=(V,E)\) is a non-empty subset S of V such that for every component A of G[S] and every component B of \(G[V {\setminus } S]\), we have \(|A| \ge |B|\) whenever there exists an edge of G between A and B. In this paper, we show that a minimum safe set can be found in polynomial time for trees. We then further extend the result and present polynomial-time algorithms for graphs of bounded treewidth, and also for interval graphs. We also study the parameterized complexity. We show that the problem is fixed-parameter tractable when parameterized by the solution size. Furthermore, we show that this parameter lies between the tree-depth and the vertex cover number. We then conclude the paper by showing hardness for split graphs and planar graphs.     

A coordination mechanism for a scheduling game with parallel-batching machines

In this paper we consider the scheduling problem with parallel-batching machines from a game theoretic perspective. There are m parallel-batching machines each of which can handle up to b jobs simultaneously as a batch. The processing time of a batch is the time required for processing the longest job in the batch, and all the jobs in a batch start and complete at the same time. There are n jobs. Each job is owned by a rational and selfish agent and its individual cost is the completion time of its job. The social cost is the largest completion time over all jobs, the makespan. We design a coordination mechanism for the scheduling game problem. We discuss the existence of pure Nash Equilibria and offer upper and lower bounds on the price of anarchy of the coordination mechanism. We show that the mechanism has a price of anarchy no more than \(2-\frac{2}{3b}-\frac{1}{3\max \{m,b\}}\).     

An optimal semi-online algorithm for 2-machine scheduling with an availability constraint

This paper considers a problem of semi-online scheduling jobs on two identical parallel machines with objective to minimize the makespan. We assume there is an unavailable period [B,F] on one machine and the largest job processing time P _max? is known in advance. After comparing B with P _max? we consider three cases, and we show a lower bound of the problem are 3/2, 4/3 and \((\sqrt{5}+1)/2\), respectively. We further present an optimal algorithm and prove its competitive ratio reaches the lower bound.     

Algorithms for connected <Emphasis Type="Italic">p</Emphasis>-centdian problem on block graphs

We consider the facility location problem of locating a set \(X_p\) of p facilities (resources) on a network (or a graph) such that the subnetwork (or subgraph) induced by the selected set \(X_p\) is connected. Two problems on a block graph G are proposed: one problem is to minimizes the sum of its weighted distances from all vertices of G to \(X_p\), another problem is to minimize the maximum distance from each vertex that is not in \(X_p\) to \(X_p\) and, at the same time, to minimize the sum of its distances from all vertices of G to \(X_p\). We prove that the first problem is linearly solvable on block graphs with unit edge length. For the second problem, it is shown that the set of Pareto-optimal solutions of the two criteria has cardinality not greater than n, and can be obtained in \(O(n^2)\) time, where n is the number of vertices of the block graph G.     

Covering directed graphs by in-trees

Given a directed graph D=(V,A) with a set of d specified vertices S={s ₁,…,s _d }?V and a function f : S→? where ? denotes the set of positive integers, we consider the problem which asks whether there exist ∑ _i=1 ^d f(s _i ) in-trees denoted by \(T_{i,1},T_{i,2},\ldots,T_{i,f(s_{i})}\) for every i=1,…,d such that \(T_{i,1},\ldots,T_{i,f(s_{i})}\) are rooted at s _i , each T _i,j spans vertices from which s _i is reachable and the union of all arc sets of T _i,j for i=1,…,d and j=1,…,f(s _i ) covers A. In this paper, we prove that such set of in-trees covering A can be found by using an algorithm for the weighted matroid intersection problem in time bounded by a polynomial in ∑ _i=1 ^d f(s _i ) and the size of D. Furthermore, for the case where D is acyclic, we present another characterization of the existence of in-trees covering A, and then we prove that in-trees covering A can be computed more efficiently than the general case by finding maximum matchings in a series of bipartite graphs.     

Separating NE from some nonuniform nondeterministic complexity classes

We investigate the question whether NE can be separated from the reduction closures of tally sets, sparse sets and NP. We show that (1) \(\mathrm{NE}\not\subseteq R^{\mathrm{NP}}_{n^{o(1)}-T}(\mathrm{TALLY})\); (2) \(\mathrm{NE}\not\subseteq R^{SN}_{m}(\mathrm{SPARSE})\); (3) \(\mathrm{NEXP}\not\subseteq \mathrm{P}^{\mathrm{NP}}_{n^{k}-T}/n^{k}\) for all k≥1; and (4) \(\mathrm{NE}\not\subseteq \mathrm{P}_{btt}(\mathrm{NP}\oplus\mathrm{SPARSE})\). Result (3) extends a previous result by Mocas to nonuniform reductions. We also investigate how different an NE-hard set is from an NP-set. We show that for any NP subset A of a many-one-hard set H for NE, there exists another NP subset A′ of H such that A′? A and A′?A is not of sub-exponential density.     

Approximation algorithms for the bus evacuation problem

We consider the bus evacuation problem. Given a positive integer B, a bipartite graph G with parts S and \(T \cup \{r\}\) in a metric space and functions \(l_i :S \rightarrow {\mathbb {Z}}_+\) and \({u_j :T \rightarrow \mathbb {Z}_+ \cup \{\infty \}}\), one wishes to find a set of B walks in G. Every walk in B should start at r and finish in T and r must be visited only once. Also, among all walks, each vertex i of S must be visited at least \(l_i\) times and each vertex j of T must be visited at most \(u_j\) times. The objective is to find a solution that minimizes the length of the longest walk. This problem arises in emergency planning situations where the walks correspond to the routes of B buses that must transport each group of people in S to a shelter in T, and the objective is to evacuate the entire population in the minimum amount of time. In this paper, we prove that approximating this problem by less than a constant is \(\text{ NP }\)-hard and present a 10.2-approximation algorithm. Further, for the uncapacitated BEP, in which \(u_j\) is infinity for each j, we give a 4.2-approximation algorithm.     

Efficient reassembling of graphs,part 1: the linear case

The reassembling of a simple connected graph \(G = (V,E)\) is an abstraction of a problem arising in earlier studies of network analysis. Its simplest formulation is in two steps:

1. (1)
   We cut every edge of G into two halves, thus obtaining a collection of \(n = |\,V\,|\) one-vertex components, such that for every \(v\in V\) the one-vertex component \(\{ v \}\) has \({{degree}}_{}(v)\) half edges attached to it.
    
2. (2)
   We splice the two halves of every edge together, not of all the edges at once, but in some ordering \(\Theta \) of the edges that minimizes two measures that depend on the edge-boundary degrees of assembled components.
    

A component A is a subset of V and its edge-boundary degree is the number of edges in G with one endpoint in A and one endpoint in \(V-A\) (which is the same as the number of half edges attached to A after all edges with both endpoints in A have been spliced together). The maximum edge-boundary degree encountered during the reassembling process is what we call the \(\varvec{\alpha }\) -measure of the reassembling, and the sum of all edge-boundary degrees is its \(\varvec{\beta }\) -measure. The \(\alpha \)-optimization (resp. \(\beta \)-optimization) of the reassembling of G is to determine an order \(\Theta \) for splicing the edges that minimizes its \(\alpha \)-measure (resp. \(\beta \)-measure). There are different forms of reassembling, depending on restrictions and variations on the ordering \(\Theta \) of the edges. We consider only cases satisfying the condition that if an edge between disjoint components A and B is spliced, then all the edges between A and B are spliced at the same time. In this report, we examine the particular case of linear reassembling, which requires that the next edge to be spliced must be adjacent to an already spliced edge. We delay other forms of reassembling to follow-up reports. We prove that \(\alpha \)-optimization of linear reassembling and minimum-cutwidth linear arrangment (\(\mathrm{CutWidth}\)) are polynomially reducible to each other, and that \(\beta \)-optimization of linear reassembling and minimum-cost linear arrangement (\(\mathrm{MinArr}\)) are polynomially reducible to each other. The known NP-hardness of \(\mathrm{CutWidth}\) and \(\mathrm{MinArr}\) imply the NP-hardness of \(\alpha \)-optimization and \(\beta \)-optimization.

     

Estimating hybrid frequency moments of data streams

We consider the problem of estimating hybrid frequency moments of two dimensional data streams. In this model, data is viewed to be organized in a matrix form (A _i,j )_1≤i,j,≤n . The entries A _i,j are updated coordinate-wise, in arbitrary order and possibly multiple times. The updates include both increments and decrements to the current value of A _i,j . The hybrid frequency moment F _p,q (A) is defined as \(\sum_{j=1}^{n}(\sum_{i=1}^{n}{A_{i,j}}^{p})^{q}\) and is a generalization of the frequency moment of one-dimensional data streams.We present the first \(\tilde{O}(1)\) space algorithm for the problem of estimating F _p,q for p∈[0,2] and q∈[0,1] to within an approximation factor of 1±ε. The \(\tilde{O}\) notation hides poly-logarithmic factors in the size of the stream m, the matrix size n and polynomial factors of ε ^?1. We also present the first \(\tilde{O}(n^{1-1/q})\) space algorithm for estimating F _p,q for p∈[0,2] and q∈(1,2].     

Optimal on-line algorithms for one batch machine with grouped processing times

In this paper, we study on-line scheduling problems on a batch machine with the assumption that all jobs have their processing times in [p, (1+φ)p], where p>0 and \(\phi=(\sqrt{5}-1)/2\). Jobs arrive over time. First, we deal with the on-line problem on a bounded batch machine with the objective to minimize makespan. A class of algorithms with competitive ratio \((\sqrt{5}+1)/2\) are given. Then we consider the scheduling on an unbounded batch machine to minimize the time by which all jobs have been delivered, and provide a class of on-line algorithms with competitive ratio \((\sqrt{5}+1)/2\). The two class of algorithms are optimal for the problems studied here.     

Online scheduling with chain precedence constraints of equal-length jobs on parallel machines to minimize makespan

We study the online scheduling problem on m identical parallel machines to minimize makespan, i.e., the maximum completion time of the jobs, where m is given in advance and the jobs arrive online over time. We assume that the jobs, which arrive at some nonnegative real times, are of equal-length and are restricted by chain precedence constraints. Moreover, the jobs arriving at distinct times are independent, and so, only the jobs arriving at a common time are restricted by the chain precedence constraints. In the literature, a best possible online algorithm of a competitive ratio 1.3028 is given for the case \(m=2\). But the problem is unaddressed for \(m\ge 3\). In this paper, we present a best possible online algorithm for the problem with \(m\ge 3\), where the algorithm has a competitive ratio of 1.3028 for \(3\le m\le 5\) and 1.3146 for \(m\ge 6\).     

Algebraic and combinatorial properties of ideals and algebras of uniform clutters of TDI systems

Let \(\mathcal{C}\) be a uniform clutter and let A be the incidence matrix of \(\mathcal{C}\). We denote the column vectors of A by v ₁,…,v _q . Under certain conditions we prove that \(\mathcal{C}\) is vertex critical. If \(\mathcal{C}\) satisfies the max-flow min-cut property, we prove that A diagonalizes over ? to an identity matrix and that v ₁,…,v _q form a Hilbert basis. We also prove that if \(\mathcal{C}\) has a perfect matching such that \(\mathcal{C}\) has the packing property and its vertex covering number is equal to 2, then A diagonalizes over ? to an identity matrix. If A is a balanced matrix we prove that any regular triangulation of the cone generated by v ₁,…,v _q is unimodular. Some examples are presented to show that our results only hold for uniform clutters. These results are closely related to certain algebraic properties, such as the normality or torsion-freeness, of blowup algebras of edge ideals and to finitely generated abelian groups. They are also related to the theory of Gröbner bases of toric ideals and to Ehrhart rings.     

ILP formulation of the degree-constrained minimum spanning hierarchy problem

Given a connected edge-weighted graph G and a positive integer B, the degree-constrained minimum spanning tree problem (DCMST) consists in finding a minimum cost spanning tree of G such that the degree of each vertex in the tree is less than or equal to B. This problem, which has been extensively studied over the last few decades, has several practical applications, mainly in networks. However, some applications do not especially impose a subgraph as a solution. For this purpose, a more flexible so-called hierarchy structure has been proposed. Hierarchy, which can be seen as a generalization of trees, is defined as a homomorphism of a tree in a graph. In this paper, we discuss the degree-constrained minimum spanning hierarchy (DCMSH) problem which is NP-hard. An integer linear program (ILP) formulation of this new problem is given. Properties of the solution are analysed, which allows us to add valid inequalities to the ILP. To evaluate the difference of cost between trees and hierarchies, the exact solution of DCMST and z problems are compared. It appears that, in sparse random graphs, the average percentage of improvement of the cost varies from 20 to 36% when the maximal authorized degree of vertices B is equal to 2, and from 11 to 31% when B is equal to 3. The improvement increases as the graph size increases.     

Two uniform machines with nearly equal speeds: unified approach to known sum and known optimum in semi on-line scheduling

We consider semi on-line scheduling on two uniform machines. The speed of the slow machine is normalized to 1 while the speed of the fast machine is assumed to be s≥1. Jobs of size J ₁,J ₂,… arrive one at a time, and each J _i (i≥1) has to be assigned to one of the machines before J _i+1 arrives. The assignment cannot be changed later. The processing time of the ith job is J _i on the slow machine and J _i /s on the fast one. The objective is to minimize the makespan. We study both the case where the only information known in advance is the total size ∑_i≥1 J _i of the jobs and the case where the only information known in advance is the optimum makespan. For each of these two cases, we almost completely determine the best possible competitive ratio of semi on-line algorithms compared to the off-line optimum, as a function of s in the range \(1\le s<\frac{1+\sqrt{17}}{4}\approx1.2808\), except for a very short subinterval around s=1.08. We also prove that the best competitive ratio achievable for known optimum is at least as good as the one for known sum, even for any number of uniform machines of any speeds.     

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.     

Scheduling with interjob communication on parallel processors

Consider a scheduling problem in which a set of tasks needs to be scheduled on m parallel processors. Each task \(T_i\) consists of a set of jobs with interjob communication demands, represented by a weighted, undirected graph \(G_i\). The processors are assumed to be interconnected by a shared communication channel, which can be used by jobs to communicate among each other while being processed in parallel. In each time step, the scheduler assigns jobs to the processors and allows any processed job to use a certain capacity of the channel in order to satisfy (parts of) its communication demands to adjacent jobs processed in the same step. The goal is to find a schedule with minimum length in which the communication demands of all jobs are satisfied. We show that this problem is NP-hard in the strong sense even if the number of processors is constant and the underlying graph is a single path or a forest with arbitrary constant maximum degree. Consequently, we design and analyze approximation algorithms with asymptotic approximation ratio \(\min \{1.8, 1.5 \frac{m}{m-1}\}+1\) if the underlying graph G, the union of the \(G_i\), is a forest. For general graphs it is \(\min \left\{ 1.8, \frac{1.5m}{m-1}\right\} \cdot \left( \text {arb}(G) + \frac{5}{3}\right) \), where \(\text {arb}(G)\) denotes the arboricity of G.     

A practical greedy approximation for the directed Steiner tree problem

The directed Steiner tree (DST) NP-hard problem asks, considering a directed weighted graph with n nodes and m arcs, a node r called root and a set of k nodes X called terminals, for a minimum cost directed tree rooted at r spanning X. The best known polynomial approximation ratio for DST is a \(O(k^\varepsilon )\)-approximation greedy algorithm. However, a much faster k-approximation, returning the shortest paths from r to X, is generally used in practice. We give two new algorithms : a fast k-approximation called Greedy\(_\text {FLAC}\) running in \(O(m \log (n)k + \min (m, nk)nk^2)\) and a \(O(\sqrt{k})\)-approximation called Greedy\(_\text {FLAC}^\triangleright \) running in \(O(nm + n^2 \log (n)k +n^2 k^3)\). We provide computational results to show that, Greedy\(_\text {FLAC}\) rivals in practice with the running time of the fast k-approximation and returns solution with smaller cost in practice.     

Resource-sharing systems and hypergraph colorings

A resource-sharing system is modeled by a hypergraph H in which a vertex represents a process and an edge represents a resource consisting of all vertices (processes) that have access to it. A schedule of H=(V,E) is a mapping f:?→2 ^V , where f(i) is an independent set of H which consists of processes that operate at round i. The rate of f is defined as \({\rm rate}(f)=\limsup_{n\to\infty}\sum_{i=1}^{n}|f(i)|/(n|V|)\), which is the average fraction of operating processes at each round. The purpose of this paper is to study optimal rates for various classes of schedules under different fairness conditions. In particular, we give relations between these optimal rates and fractional/circular chromatic numbers. For the special case of the hypergraph is a connected graph, a new derivation for the previous result by Yeh and Zhu is also given.     

On the minimum routing cost clustered tree problem

For an edge-weighted graph \(G=(V,E,w)\), in which the vertices are partitioned into k clusters \(\mathcal {R}=\{R_1,R_2,\ldots ,R_k\}\), a spanning tree T of G is a clustered spanning tree if T can be cut into k subtrees by removing \(k-1\) edges such that each subtree is a spanning tree for one cluster. In this paper, we show the inapproximability of finding a clustered spanning tree with minimum routing cost, where the routing cost is the total distance summed over all pairs of vertices. We present a 2-approximation for the case that the input is a complete weighted graph whose edge weights obey the triangle inequality. We also study a variant in which the objective function is the total distance summed over all pairs of vertices of different clusters. We show that the problem is polynomial-time solvable when the number of clusters k is 2 and NP-hard for \(k=3\). Finally, we propose a polynomial-time 2-approximation algorithm for the case of three clusters.