An optimal online algorithm for the parallel-batch scheduling with job processing time compatibilities

We consider the online (over time) scheduling on a single unbounded parallel-batch machine with job processing time compatibilities to minimize makespan. In the problem, a constant \(\alpha >0\) is given in advance. Each job \(J_{j}\) has a normal processing time \(p_j\). Two jobs \(J_i\) and \(J_j\) are compatible if \(\max \{p_i, p_j\} \le (1+\alpha )\cdot \min \{p_i, p_j\}\). In the problem, mutually compatible jobs can form a batch being processed on the machine. The processing time of a batch is equal to the maximum normal processing time of the jobs in this batch. For this problem, we provide an optimal online algorithm with a competitive ratio of \(1+\beta _\alpha \), where \(\beta _\alpha \) is the positive root of the equation \((1+\alpha )x^{2}+\alpha x=1+\alpha \).     

Total completion time minimization in online hierarchical scheduling of unit-size jobs

This paper investigates an online hierarchical scheduling problem on m parallel identical machines. Our goal is to minimize the total completion time of all jobs. Each job has a unit processing time and a hierarchy. The job with a lower hierarchy can only be processed on the first machine and the job with a higher hierarchy can be processed on any one of m machines. We first show that the lower bound of this problem is at least \(1+\min \{\frac{1}{m}, \max \{\frac{2}{\lceil x\rceil +\frac{x}{\lceil x\rceil }+3}, \frac{2}{\lfloor x\rfloor +\frac{x}{\lfloor x\rfloor }+3}\}\), where \(x=\sqrt{2m+4}\). We then present a greedy algorithm with tight competitive ratio of \(1+\frac{2(m-1)}{m(\sqrt{4m-3}+1)}\). The competitive ratio is obtained in a way of analyzing the structure of the instance in the worst case, which is different from the most common method of competitive analysis. In particular, when \(m=2\), we propose an optimal online algorithm with competitive ratio of \(16\) \(/\) \(13\), which complements the previous result which provided an asymptotically optimal algorithm with competitive ratio of 1.1573 for the case where the number of jobs n is infinite, i.e., \(n\rightarrow \infty \).     

On the performance of mildly greedy players in cut games

We continue the study of the performance of mildly greedy players in cut games initiated by Christodoulou et al. (Theoret Comput Sci 438:13–27, 2012), where a mildly greedy player is a selfish agent who is willing to deviate from a certain strategy profile only if her payoff improves by a factor of more than \(1+\epsilon \), for some given \(\epsilon \ge 0\). Hence, in presence of mildly greedy players, the classical concepts of pure Nash equilibria and best-responses generalize to those of \((1+\epsilon )\)-approximate pure Nash equilibria and \((1+\epsilon )\)-approximate best-responses, respectively. We first show that the \(\epsilon \)-approximate price of anarchy, that is the price of anarchy of \((1+\epsilon )\)-approximate pure Nash equilibria, is at least \(\frac{1}{2+\epsilon }\) and that this bound is tight for any \(\epsilon \ge 0\). Then, we evaluate the approximation ratio of the solutions achieved after a \((1+\epsilon )\)-approximate one-round walk starting from any initial strategy profile, where a \((1+\epsilon )\)-approximate one-round walk is a sequence of \((1+\epsilon )\)-approximate best-responses, one for each player. We improve the currently known lower bound on this ratio from \(\min \left\{ \frac{1}{4+2\epsilon },\frac{\epsilon }{4+2\epsilon }\right\} \) up to \(\min \left\{ \frac{1}{2+\epsilon },\frac{2\epsilon }{(1+\epsilon )(2+\epsilon )}\right\} \) and show that this is again tight for any \(\epsilon \ge 0\). An interesting and quite surprising consequence of our results is that the worst-case performance guarantee of the very simple solutions generated after a \((1+\epsilon )\)-approximate one-round walk is the same as that of \((1+\epsilon )\)-approximate pure Nash equilibria when \(\epsilon \ge 1\) and of that of subgame perfect equilibria (i.e., Nash equilibria for greedy players with farsighted, rather than myopic, rationality) when \(\epsilon =1\).     

Upper bounds for the total rainbow connection of graphs

A two-agent scheduling problem on parallel machines is considered. Our objective is to minimize the makespan for agent A, subject to an upper bound on the makespan for agent B. When the number of machines, denoted by \(m\), is chosen arbitrarily, we provide an \(O(n)\) algorithm with performance ratio \(2-\frac{1}{m}\), i.e., the makespan for agent A given by the algorithm is no more than \(2-\frac{1}{m}\) times the optimal value, while the makespan for agent B is no more than \(2-\frac{1}{m}\) times the threshold value. This ratio is proved to be tight. Moreover, when \(m=2\), we present an \(O(nlogn)\) algorithm with performance ratio \(\frac{1+\sqrt{17}}{4}\approx 1.28\) which is smaller than \(\frac{3}{2}\). The ratio is weakly tight.     

An approximation algorithm for <Emphasis Type="Italic">k</Emphasis>-facility location problem with linear penalties using local search scheme

In this paper, we consider an extension of the classical facility location problem, namely k-facility location problem with linear penalties. In contrast to the classical facility location problem, this problem opens no more than k facilities and pays a penalty cost for any non-served client. We present a local search algorithm for this problem with a similar but more technical analysis due to the extra penalty cost, compared to that in Zhang (Theoretical Computer Science 384:126–135, 2007). We show that the approximation ratio of the local search algorithm is \(2 + 1/p + \sqrt{3+ 2/p+ 1/p^2} + \epsilon \), where \(p \in {\mathbb {Z}}_+\) is a parameter of the algorithm and \(\epsilon >0\) is a positive number.     

Semi-online scheduling with combined information on two identical machines in parallel

This paper is concerned with a semi-online scheduling problem with combined information on two identical parallel machines to minimize the makespan, where all the jobs have processing times in the interval \([1,\,t]\)  \((t\ge 1)\) and the jobs arrive in non-increasing order of their processing times. The objective is to minimize the makespan. For \(t\ge 1\), we obtain a lower bound \(\max _{N=1,2,3,\ldots }\left\{ \min \{\frac{4N+3}{4N+2}\,,\frac{Nt+N+1}{2N+1}\}\right\} \) and show that the competitive ratio of the \(LS\) algorithm achieves the lower bound.     

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 \(m\ge 6\).     

Online tradeoff scheduling on a single machine to minimize makespan and maximum lateness

In this paper, we consider the following single machine online tradeoff scheduling problem. A set of n independent jobs arrive online over time. Each job \(J_{j}\) has a release date \(r_{j}\), a processing time \(p_{j}\) and a delivery time \(q_{j}\). The characteristics of a job are unknown until it arrives. The goal is to find a schedule that minimizes the makespan \(C_{\max } = \max _{1 \le j \le n} C_{j}\) and the maximum lateness \(L_{\max } = \max _{1 \le j \le n} L_{j}\), where \(L_{j} = C_{j} + q_{j}\). For the problem, we present a nondominated \(( \rho , 1 + \displaystyle \frac{1}{\rho } )\)-competitive online algorithm for each \(\rho \) with \( 1 \le \rho \le \displaystyle \frac{\sqrt{5} + 1}{2}\).     

Scheduling dedicated jobs with variative processing times

We consider the following optimization problem. There is a set of \(n\) dedicated jobs that are to be processed on \(m\) parallel machines. The job set is partitioned into subsets and jobs of each subset have a common due date. Processing times of jobs are interconnected and they are the subject of the decision making. The goal is to choose a processing time for each job in a feasible way and to construct a schedule that minimizes the maximum lateness. We show that the problem is NP-hard even if \(m=1\) and that it is NP-hard in the strong sense if \(m\) is a variable. We prove that there is no approximate polynomial algorithm with guaranteed approximation ratio less than 2. We propose an integer linear formulation for the problem and perform experiments. The experiments show that the solutions obtained with CPLEX within the limit of 5 min are on average about 5 % from the optimum value for instances with up to 150 jobs, 16 subsets and 11 machines. Most instances were solved to optimality and the average CPLEX running time was 32 s for these instances.     

2-Edge connected dominating sets and 2-Connected dominating sets of a graph

A \(k\)-connected (resp. \(k\)-edge connected) dominating set \(D\) of a connected graph \(G\) is a subset of \(V(G)\) such that \(G[D]\) is \(k\)-connected (resp. \(k\)-edge connected) and each \(v\in V(G)\backslash D\) has at least one neighbor in \(D\). The \(k\) -connected domination number (resp. \(k\) -edge connected domination number) of a graph \(G\) is the minimum size of a \(k\)-connected (resp. \(k\)-edge connected) dominating set of \(G\), and denoted by \(\gamma _k(G)\) (resp. \(\gamma '_k(G)\)). In this paper, we investigate the relation of independence number and 2-connected (resp. 2-edge-connected) domination number, and prove that for a graph \(G\), if it is \(2\)-edge connected, then \(\gamma '_2(G)\le 4\alpha (G)-1\), and it is \(2\)-connected, then \(\gamma _2(G)\le 6\alpha (G)-3\), where \(\alpha (G)\) is the independent number of \(G\).     

Online scheduling with rejection and reordering: exact algorithms for unit size jobs

We study an online scheduling problem with rejection on \(m\ge 2\) identical machines, in which we deal with unit size jobs. Each arriving job has a rejection value (a rejection cost or penalty for minimization problems, and a rejection profit for maximization problems) associated with it. A buffer of size \(K\) is available to store \(K\) jobs. A job which is not stored in the buffer must be either assigned to a machine or rejected. Upon the arrival of a new job, the job can be stored in the buffer if there is a free slot (possibly created by evicting other jobs and assigning or rejecting every evicted job). At termination, the buffer must be emptied. We study four variants of the problem, as follows. We study the makespan minimization problem, where the goal is to minimize the sum of the makespan and the penalty of rejected jobs, and the \(\ell _p\) norm minimization problem, where the goal is to minimize the sum of the \(\ell _p\) norm of the vector of machine completion times and the penalty of rejected jobs. We also study two maximization problems, where the goal in the first version is to maximize the sum of the minimum machine load (the cover value of the machines) and the total rejection profit, and in the second version the goal is to maximize a function of the machine completion times (which measures the balance of machine loads) and the total rejection profit. We show that an optimal solution (an exact solution for the offline problem) can always be obtained in this environment, and determine the required buffer size. Specifically, for all four variants we present optimal algorithms with \(K=m-1\) and prove that in each case, using a buffer of size at most \(m-2\) does not allow the design of an optimal algorithm, which makes our algorithms optimal in this respect as well. The lower bounds hold even for the special case where the rejection value is equal for all input jobs.     

Strong minimum energy hierarchical topology in wireless sensor networks

Given a set of \(n\) sensors, the strong minimum energy topology (SMET) problem in a wireless sensor network is to assign transmit powers to all sensors such that (i) the graph induced only using the bi-directional links is connected, that is, there is a path between every pair of sensors, and (ii) the sum of the transmit powers of all the sensors is minimum. This problem is known to be NP-hard. In this paper, we study a special case of the SMET problem, namely , the \(k\)-strong minimum energy hierarchical topology (\(k\)-SMEHT) problem. Given a set of \(n\) sensors and an integer \(k\), the \(k\)-SMEHT problem is to assign transmission powers to all sensors such that (i) the graph induced using only bi-directional links is connected, (ii) at most \(k\) nodes of the graph induced using only bi-directional links have two or more neighbors, that is they are non-pendant nodes, and (iii) the sum of the transmit powers of all the sensors in \(G\) is minimum. We show that \(k\)-SMEHT problem is NP-hard for arbitrary \(k\). However, we propose a \(\frac{k+1}{2}\)-approximation algorithm for \(k\)-SMEHT problem, when \(k\) is a fixed constant. Finally, we propose a polynomial time algorithm for the \(k\)-SMEHT problem for \(k=2\).     

Disjunctive total domination in graphs

Let \(G\) be a graph with no isolated vertex. In this paper, we study a parameter that is a relaxation of arguably the most important domination parameter, namely the total domination number, \(\gamma _t(G)\). A set \(S\) of vertices in \(G\) is a disjunctive total dominating set of \(G\) if every vertex is adjacent to a vertex of \(S\) or has at least two vertices in \(S\) at distance \(2\) from it. The disjunctive total domination number, \(\gamma ^d_t(G)\), is the minimum cardinality of such a set. We observe that \(\gamma ^d_t(G) \le \gamma _t(G)\). We prove that if \(G\) is a connected graph of order \(n \ge 8\), then \(\gamma ^d_t(G) \le 2(n-1)/3\) and we characterize the extremal graphs. It is known that if \(G\) is a connected claw-free graph of order \(n\), then \(\gamma _t(G) \le 2n/3\) and this upper bound is tight for arbitrarily large \(n\). We show this upper bound can be improved significantly for the disjunctive total domination number. We show that if \(G\) is a connected claw-free graph of order \(n > 14\), then \(\gamma ^d_t(G) \le 4n/7\) and we characterize the graphs achieving equality in this bound.     

OFDP: a distributed algorithm for finding disjoint paths with minimum total length in wireless sensor networks

This paper investigates the MINimum-length-\(k\)-Disjoint-Paths (MIN-\(k\)-DP) problem: in a sensor network, given two nodes \(s\) and \(t\), a positive integer \(k\), finding \(k\) (node) disjoint paths connecting \(s\) and \(t\) with minimum total length. An efficient distributed algorithm named Optimally-Finding-Disjoint-Paths (OFDP) is proposed for this problem. OFDP guarantees correctness and optimality, i.e., (1) it will find \(k\) disjoint paths if there exist \(k\) disjoint paths in the network or the maximum number of disjoint paths otherwise; (2) the disjoint paths it outputs do have minimum total length. To the best of our knowledge, OFDP is the first distributed algorithm that can solve the MIN-\(k\)-DP problem with correctness and optimality guarantee. Compared with the existing centralized algorithms which also guarantee correctness and optimality, OFDP is shown to be much more efficient by simulation results.     

Total coloring of planar graphs without short cycles

The total chromatic number of a graph \(G\), denoted by \(\chi ''(G)\), is the minimum number of colors needed to color the vertices and edges of \(G\) such that no two adjacent or incident elements get the same color. It is known that if a planar graph \(G\) has maximum degree \(\Delta (G)\ge 9\), then \(\chi ''(G)=\Delta (G)+1\). In this paper, it is proved that if \(G\) is a planar graph with \(\Delta (G)\ge 7\), and for each vertex \(v\), there is an integer \(k_v\in \{3,4,5,6,7,8\}\) such that there is no \(k_v\)-cycle which contains \(v\), then \(\chi ''(G)=\Delta (G)+1\).     

Special Issue for FAW 2014

For a graph \(G=(V,E)\), a dominating set is a set \(D\subseteq V\) such that every vertex \(v\in V\setminus D\) has a neighbor in \(D\). The minimum outer-connected dominating set (Min-Outer-Connected-Dom-Set) problem for a graph \(G\) is to find a dominating set \(D\) of \(G\) such that \(G[V\setminus D]\), the induced subgraph by \(G\) on \(V\setminus D\), is connected and the cardinality of \(D\) is minimized. In this paper, we consider the complexity of the Min-Outer-Connected-Dom-Set problem. In particular, we show that the decision version of the Min-Outer-Connected-Dom-Set problem is NP-complete for split graphs, a well known subclass of chordal graphs. We also consider the approximability of the Min-Outer-Connected-Dom-Set problem. We show that the Min-Outer-Connected-Dom-Set problem cannot be approximated within a factor of \((1-\varepsilon ) \ln |V|\) for any \(\varepsilon >0\), unless NP \(\subseteq \) DTIME(\(|V|^{\log \log |V|}\)). For sufficiently large values of \(\varDelta \), we show that for graphs with maximum degree \(\varDelta \), the Min-Outer-Connected-Dom-Set problem cannot be approximated within a factor of \(\ln \varDelta -C \ln \ln \varDelta \) for some constant \(C\), unless P \(=\) NP. On the positive side, we present a \(\ln (\varDelta +1)+1\)-factor approximation algorithm for the Min-Outer-Connected-Dom-Set problem for general graphs. We show that the Min-Outer-Connected-Dom-Set problem is APX-complete for graphs of maximum degree 4.     

An FPTAS for generalized absolute 1-center problem in vertex-weighted graphs

Given a vertex-weighted undirected connected graph \(G = (V, E, \ell , \rho )\), where each edge \(e \in E\) has a length \(\ell (e) > 0\) and each vertex \(v \in V\) has a weight \(\rho (v) > 0\), a subset \(T \subseteq V\) of vertices and a set S containing all the points on edges in a subset \(E' \subseteq E\) of edges, the generalized absolute 1-center problem (GA1CP), an extension of the classic vertex-weighted absolute 1-center problem (A1CP), asks to find a point from S such that the longest weighted shortest path distance in G from it to T is minimized. This paper presents a simple FPTAS for GA1CP by traversing the edges in \(E'\) using a positive real number as step size. The FPTAS takes \(O( |E| |V| + |V|^2 \log \log |V| + \frac{1}{\epsilon } |E'| |T| {\mathcal {R}})\) time, where \({\mathcal {R}}\) is an input parameter size of the problem instance, for any given \(\epsilon > 0\). For instances with a small input parameter size \({\mathcal {R}}\), applying the FPTAS with \(\epsilon = \Theta (1)\) to the classic vertex-weighted A1CP can produce a \((1 + \Theta (1))\)-approximation in at most O(|E| |V|) time when the distance matrix is known and \(O(|E| |V| + |V|^2 \log \log |V|)\) time when the distance matrix is unknown, which are smaller than Kariv and Hakimi’s \(O(|E| |V| \log |V|)\)-time algorithm and \(O(|E| |V| \log |V| + |V|^3)\)-time algorithm, respectively.     

An $$O^{*}(1.4366^n)$$-time exact algorithm for maximum $$P_2$$P2-packing in cubic graphs

Given a graph \(G=(V, E)\), a \(P_2\)-packing \(\mathcal {P}\) is a collection of vertex disjoint copies of \(P_2\)s in \(G\) where a \(P_2\) is a simple path with three vertices and two edges. The Maximum \(P_2\)-Packing problem is to find a \(P_2\)-packing \(\mathcal {P}\) in the input graph \(G\) of maximum cardinality. This problem is NP-hard for cubic graphs. In this paper, we give a branch-and-reduce algorithm for the Maximum \(P_2\)-Packing problem in cubic graphs. We analyze the running time of the algorithm using measure-and-conquer and show that it runs in time \(O^{*}(1.4366^n)\) which is faster than previous known exact algorithms where \(n\) is the number of vertices in the input graph.     

A local search approximation algorithm for a squared metric <Emphasis Type="Italic">k</Emphasis>-facility location problem

We consider a scheduling problem where machines need to be rented from the cloud in order to process jobs. There are two types of machines available which can be rented for machine-type dependent prices and for arbitrary durations. However, a machine-type dependent setup time is required before a machine is available for processing. Jobs arrive online over time, have deadlines and machine-type dependent sizes. The objective is to rent machines and schedule jobs so as to meet all deadlines while minimizing the rental cost. As we observe the slack of jobs to have a fundamental influence on the competitiveness, we parameterize instances by their (minimum) slack. An instance is called to have a slack of \(\beta \) if, for all jobs, the difference between the job’s release time and the latest point in time at which it needs to be started is at least \(\beta \). While for \(\beta < s\) no finite competitiveness is possible, our main result is an online algorithm for \(\beta = (1+\varepsilon )s\) with Open image in new window , where s denotes the largest setup time. Its competitiveness only depends on \(\varepsilon \) and the cost ratio of the machine types and is proven to be optimal up to a factor of Open image in new window .     

Improved approximations for buy-at-bul and shallow-light $$$$-Steiner trees and $$(,2)$$(,2)-subgraph

In this paper we give improved approximation algorithms for some network design problems. In the bounded-diameter or shallow-light \(k\)-Steiner tree problem (SL\(k\)ST), we are given an undirected graph \(G=(V,E)\) with terminals \(T\subseteq V\) containing a root \(r\in T\), a cost function \(c:E\rightarrow \mathbb {R}^+\), a length function \(\ell :E\rightarrow \mathbb {R}^+\), a bound \(L>0\) and an integer \(k\ge 1\). The goal is to find a minimum \(c\)-cost \(r\)-rooted Steiner tree containing at least \(k\) terminals whose diameter under \(\ell \) metric is at most \(L\). The input to the buy-at-bulk \(k\)-Steiner tree problem (BB\(k\)ST) is similar: graph \(G=(V,E)\), terminals \(T\subseteq V\) containing a root \(r\in T\), cost and length functions \(c,\ell :E\rightarrow \mathbb {R}^+\), and an integer \(k\ge 1\). The goal is to find a minimum total cost \(r\)-rooted Steiner tree \(H\) containing at least \(k\) terminals, where the cost of each edge \(e\) is \(c(e)+\ell (e)\cdot f(e)\) where \(f(e)\) denotes the number of terminals whose path to root in \(H\) contains edge \(e\). We present a bicriteria \((O(\log ^2 n),O(\log n))\)-approximation for SL\(k\)ST: the algorithm finds a \(k\)-Steiner tree with cost at most \(O(\log ^2 n\cdot \text{ opt }^*)\) where \(\text{ opt }^*\) is the cost of an LP relaxation of the problem and diameter at most \(O(L\cdot \log n)\). This improves on the algorithm of Hajiaghayi et al. (2009) (APPROX’06/Algorithmica’09) which had ratio \((O(\log ^4 n), O(\log ^2 n))\). Using this, we obtain an \(O(\log ^3 n)\)-approximation for BB\(k\)ST, which improves upon the \(O(\log ^4 n)\)-approximation of Hajiaghayi et al. (2009). We also consider the problem of finding a minimum cost \(2\)-edge-connected subgraph with at least \(k\) vertices, which is introduced as the \((k,2)\)-subgraph problem in Lau et al. (2009) (STOC’07/SICOMP09). This generalizes some well-studied classical problems such as the \(k\)-MST and the minimum cost \(2\)-edge-connected subgraph problems. We give an \(O(\log n)\)-approximation algorithm for this problem which improves upon the \(O(\log ^2 n)\)-approximation algorithm of Lau et al. (2009).