## On Chromatic Sums and Distributed Resource Allocation

### Cached

### Download Links

Citations: | 64 - 12 self |

### BibTeX

@MISC{Bar-Noy_onchromatic,

author = {Amotz Bar-Noy and Mihir Bellare and Hadas Shachnai and Tami Tamir and et al.},

title = {On Chromatic Sums and Distributed Resource Allocation},

year = {}

}

### Years of Citing Articles

### OpenURL

### Abstract

This paper studies an optimization problem that arises in the context of distributed resource allocation: Given a conflict graph that represents the competition of processors over resources, we seek an allocation under which no two jobs with conflicting requirements are executed simultaneously. Our objective is to minimize the average response time of the system. In alternative formulation this is known as the Minimum Color Sum (MCS) problem [24]. We show, that the algorithm based on finding iteratively a maximum independent set (MaxIS) is a 4-approximation to the MCS. This bound is tight to within a factor of 2. We give improved ratios for the classes of bipartite, bounded-degree, and line graphs. The bound generalizes to a 4ae-approximation of MCS for classes of graphs for which the maximum independent set problem can be approximated within a factor of ae. On the other hand, we show that an n1 \Gamma ffl-approximation is NP-hard, for some ffl? 0. For some instances of the resource allocation problem, such as the Dining Philosophers, an efficient solution requires edge coloring of the conflict graph. We introduce the Minimum Edge Color Sum (MECS) problem which is shown to be NP-hard. We show that a 2-approximation to MECS(G) can be obtained distributively using compact coloring within O(log² n) communication rounds.

### Citations

11366 |
Computers and Intractability : A Guide to the Theory of NP-completeness
- Garey, Johnson
- 1979
(Show Context)
Citation Context ...ependent set with the first color, and then two-colors the remaining vertices. Note that a maximum independent set of a bipartite graph can be found in polynomial time by computing a maximum matching =-=[15]-=-. Theorem 3.7 The above algorithm achieves a ratio of 9 8 to the MCS for any bipartite graph. Proof: Let α be the size of the maximum independent set of the graph. The cost of our former coloring is a... |

1227 |
Graph Theory
- Harary
- 1994
(Show Context)
Citation Context ...ing edges in G are. We say that G is a line graph, if there exists some graph G ′ , such that G = L(G ′ ). The following property of line graphs is used in the proof of the next theorem: Property 4.4 =-=[19]-=- If G = (V, E) is a line graph, then E can be partitioned into cliques, such that each vertex belongs to at most two cliques. Theorem 4.5 If G is a line graph, then any compact coloring of G is a 2-ap... |

726 | Proof verification and hardness of approximation problems - Arora, Lund, et al. - 1998 |

665 |
An Introduction to Parallel Algorithms
- Jájá
- 1992
(Show Context)
Citation Context ...l degree ∆, the MaxIS can be implemented distributively within O(∆ · log 2 n) communication rounds, by using iteratively a randomized distributed algorithm for finding a maximum matching in G (see in =-=[21]-=-). We show that compact edge coloring, that can be implemented distributively in O(log 2 n) communication rounds, yields a 2-approximation to MECS(G). Related Work The minimum color sum problem was in... |

390 | On the hardness of approximating minimization problems - Lund, Yannakakis - 1994 |

181 | Zero knowledge and the chromatic number
- Feige, Kilian
- 1998
(Show Context)
Citation Context ... i=0 Thus, we obtain a performance ratio of O(f(n)). 1 (2c f(n) ≤ O(kf(n)). ) i Feige and Kilian have recently shown that Graph Coloring (of general graphs) is hard to approximate within n 1−ϵ factor =-=[14]-=-. We thus obtain the same hardness bound for MCS. Corollary 2.2 MCS cannot be approximated within n 1−ϵ , for any ϵ > 0, unless NP = ZP P . 3 The MaxIS Algorithm 3.1 Upper Bound A natural approach for... |

146 | The Drinking Philosophers Problem
- Chandy, Misra
- 1984
(Show Context)
Citation Context ...ons Our main application is the problem of resource allocation with constraints imposed by conflicting resource requirements. In a common representation of the distributed resource allocation problem =-=[11, 27]-=-, the constraints are given by a conflict graph G, in which the nodes represent processors, and the edges indicate competition on resources, i.e., two nodes are adjacent if the corresponding processor... |

137 | Approximating maximum independent sets by excluding subgraphs
- Boppana, Halld6rsson
- 1992
(Show Context)
Citation Context ... MaxIS algorithm is a 4ρ-approximation to the MCS. This immediately gives us a fairly good characterization of the approximability of MCS on various classes of graphs: O(n/ log 2 n) on general graphs =-=[10]-=-, O(∆ log log ∆/ log ∆) on graphs of maximum degree ∆ [35], O(n .2134 ) on 3-colorable graphs [8], and at most 4 on all perfect graphs and partial k-trees, among others. We show below, that the bound ... |

118 | L.: What can be computed locally
- Naor, Stockmeyer
- 1995
(Show Context)
Citation Context ...A. Bar-Noy et al., On Chromatic Sums.. 6 For some resource allocation problems, such as the classic Dining Philosophers, efficient solution requires an edge coloring of the conflict graph (see, e.g., =-=[27, 28, 34]-=-). The measure used for these problems is the maximal waiting chain, which is the number of colors needed to edge color the conflict graph G. For these problems, minimizing the average response time c... |

75 | Complexity bounds for multiprocessor scheduling with resource constraints
- Garey, Johnson
- 1975
(Show Context)
Citation Context ...ted with a vector of requirement for resources, and jobs cannot be scheduled simultaneously, if the sum of their requirements for a specific resource exceeds the total amount of that resource (see in =-=[16, 33]-=-). Outline of the Paper The rest of this paper is organized as follows: In Section 2 we give some definitions and prove a hardness result. In Section 3 we define the MaxIS algorithm and show that MaxI... |

38 |
A Heurlstlc of Scheduling Parallel Tasks and its Analysis
- Wang, Cheng
- 1992
(Show Context)
Citation Context ... two successive schedules of that task. Other works related to the present context address the more general problem of scheduling under constraints. Typical examples are a predetermined partial order =-=[13, 36]-=- or resource constrained scheduling. In the latter case, each of the jobs is associated with a vector of requirement for resources, and jobs cannot be scheduled simultaneously, if the sum of their req... |

36 | Improved distributed algorithms for coloring and network decomposition problems - Panconesi, Srinivasan - 1992 |

34 |
the Chromatic Sum
- Kubicka, Schwenk
- 1989
(Show Context)
Citation Context ...licting requirements are executed simultaneously. Our objective is to minimize the average response time of the system. In alternative formulation this is known as the Minimum Color Sum (MCS) problem =-=[25]-=-. We show, that the algorithm based on finding iteratively a maximum independent set (MaxIS) is a 4-approximation to the MCS. This bound is tight to within a factor of 2. We give improved ratios for t... |

28 | Efficient fault-tolerance algorithms for distributed resource allocation
- Choy, Singh
- 1995
(Show Context)
Citation Context ... pj at round k + 1. Messages may be of arbitrary length and local computation is instantaneous and unlimited. We assume that processors have unique numerical id’s. Some resource allocation algorithms =-=[12]-=- use a preprocessing which results in a legal coloring ofA. Bar-Noy et al., On Chromatic Sums.. 21 the communication graph. The color of a processor indicates the maximal length of a waiting chain fo... |

28 |
Upper bounds for static resource allocation in a distributed system
- Lynch
- 1981
(Show Context)
Citation Context ...ons Our main application is the problem of resource allocation with constraints imposed by conflicting resource requirements. In a common representation of the distributed resource allocation problem =-=[11, 27]-=-, the constraints are given by a conflict graph G, in which the nodes represent processors, and the edges indicate competition on resources, i.e., two nodes are adjacent if the corresponding processor... |

26 |
A dining philosophers algorithm with polynomial response time
- Awerbuch, Saks
- 1990
(Show Context)
Citation Context ..., studies algorithms that minimize the maximal response time per processor, or alternatively – the maximal waiting chain in the system, in solutions for the Dining Philosophers version of the problem =-=[2, 5, 11, 27]-=-. In this context, the term of one shot resource allocation problem was coined by Rhee [32]. The one shot problem is used in his work to show that it is NP-hard to minimize the maximal response time f... |

26 | Scheduling with conflicts and applications to traffic signal control - Irani, Leung - 1996 |

26 |
Improved Algorithms for Distributed Resource Allocation
- Steyer, Peterson
- 1988
(Show Context)
Citation Context ...A. Bar-Noy et al., On Chromatic Sums.. 6 For some resource allocation problems, such as the classic Dining Philosophers, efficient solution requires an edge coloring of the conflict graph (see, e.g., =-=[27, 28, 34]-=-). The measure used for these problems is the maximal waiting chain, which is the number of colors needed to edge color the conflict graph G. For these problems, minimizing the average response time c... |

25 | Guaranteeing Fair Service to Persistent Dependent Tasks
- Bar-Noy, Mayer, et al.
- 1998
(Show Context)
Citation Context ...tion problem was coined by Rhee [32]. The one shot problem is used in his work to show that it is NP-hard to minimize the maximal response time for a static conflict graph. Bar-Noy et al. consider in =-=[6]-=- the problem of scheduling persistent tasks with conflicting resource requirement. Since the tasks are scheduled repeatedly, the response time for a given task is the maximal time that elapses between... |

21 | An Introduction to Parallel Algorithms (Addison-Wesley - JaJa - 1992 |

20 |
The optimum cost chromatic partition problem
- Jansen
- 1997
(Show Context)
Citation Context ...problem was introduced by Kubicka in [23]. In [25] it is shown that computing the MCS of a given graph is NP-hard. A polynomial time algorithm is given for the case where G is a tree. Jansen shows in =-=[22]-=- that the MCS is solvable in polynomial time for partial k-trees. In [24] it is shown that approximating MCS within an additive constant factor is NP-hard, and that a first-fit algorithm yields a d 2 ... |

19 |
Optimal separations between concurrent-write parallel machines
- Boppana
- 1989
(Show Context)
Citation Context ...cessing resource requests as they arrive. The following general upper bound on the chromatic sum has been observed several times in the past. Let m denote the number of edges in the graph. Lemma 4.2 (=-=[9, 24]-=-) The sum of any compact coloring is at most m + n. This bound is tight for disjoint collection of cliques. It can be attained by a parallel algorithm [17].A. Bar-Noy et al., On Chromatic Sums.. 17 T... |

19 | Já Já , An Introduction to Parallel Algorithms - unknown authors - 1992 |

17 |
The chromatic sum of a graph
- Kubicka
- 1989
(Show Context)
Citation Context ... edge coloring, that can be implemented distributively in O(log 2 n) communication rounds, yields a 2-approximation to MECS(G). Related Work The minimum color sum problem was introduced by Kubicka in =-=[23]-=-. In [25] it is shown that computing the MCS of a given graph is NP-hard. A polynomial time algorithm is given for the case where G is a tree. Jansen shows in [22] that the MCS is solvable in polynomi... |

16 | E cient asynchronous distributed symmetry breaking - Awerbuch, Cowen, et al. - 1994 |

14 |
Distributed Resource Allocation Algorithms
- Bar-Ilan, Peleg
- 1992
(Show Context)
Citation Context ..., studies algorithms that minimize the maximal response time per processor, or alternatively – the maximal waiting chain in the system, in solutions for the Dining Philosophers version of the problem =-=[2, 5, 11, 27]-=-. In this context, the term of one shot resource allocation problem was coined by Rhee [32]. The one shot problem is used in his work to show that it is NP-hard to minimize the maximal response time f... |

10 | An efficient parallel algorithm that finds independent sets of guaranteed size
- Goldberg, Spencer
- 1990
(Show Context)
Citation Context ... number of edges in the graph. Lemma 4.2 ([9, 24]) The sum of any compact coloring is at most m + n. This bound is tight for disjoint collection of cliques. It can be attained by a parallel algorithm =-=[17]-=-.A. Bar-Noy et al., On Chromatic Sums.. 17 Theorem 4.3 Any compact coloring of a graph G = (V, E) provides a ∆+2 3 -approximation to MCS(G), and that is tight. Proof: All edges have at least one endp... |

9 | Tight Approximations for Resource Constrained Scheduling Problems
- Srivastav, Stangier
- 1994
(Show Context)
Citation Context ...ted with a vector of requirement for resources, and jobs cannot be scheduled simultaneously, if the sum of their requirements for a specific resource exceeds the total amount of that resource (see in =-=[16, 33]-=-). Outline of the Paper The rest of this paper is organized as follows: In Section 2 we give some definitions and prove a hardness result. In Section 3 we define the MaxIS algorithm and show that MaxI... |

8 |
Approximation algorithms for the chromatic sum
- Kubicka, Kubicki, et al.
- 1989
(Show Context)
Citation Context ... numbers (colors). The Minimum Color problem is to find a vertex coloring which uses the minimum number of colors. In this paper we consider a related problem known as Minimum Color Sum (MCS) problem =-=[24, 25]-=-. Given a graph G = (V, E), find a vertex coloring Ψ : V → N for G such that ∑ v∈V Ψ(v) is minimized. We note that the problems are not equivalent. For instance, bipartite graphs can be colored with t... |

7 | Efficiency of Partial Synchrony, and Resource Allocation in Distributed Systems
- Rhee
- 1994
(Show Context)
Citation Context ...nd the edges indicate competition on resources, i.e., two nodes are adjacent if the corresponding processors cannot run their jobs simultaneously. We focus on the one shot resource allocation problem =-=[32, 4]-=-, in which we have to allocate resources to one batch of requests. The allocation of resources should satisfy the two following conditions: • Mutual exclusion: No two conflicting jobs are executed sim... |

5 | Local labeling and resource allocation using preprocessing
- Attiya, Shachnai, et al.
- 1999
(Show Context)
Citation Context ...nd the edges indicate competition on resources, i.e., two nodes are adjacent if the corresponding processors cannot run their jobs simultaneously. We focus on the one shot resource allocation problem =-=[32, 4]-=-, in which we have to allocate resources to one batch of requests. The allocation of resources should satisfy the two following conditions: • Mutual exclusion: No two conflicting jobs are executed sim... |

5 |
An O(n 0.2143 ) coloring for 3-colorable graphs
- Blum, and, et al.
(Show Context)
Citation Context ...erization of the approximability of MCS on various classes of graphs: O(n/ log 2 n) on general graphs [10], O(∆ log log ∆/ log ∆) on graphs of maximum degree ∆ [35], O(n .2134 ) on 3-colorable graphs =-=[8]-=-, and at most 4 on all perfect graphs and partial k-trees, among others. We show below, that the bound in Theorem 3.1 applies also to the MWCS problem. In that case, the weighted MaxIS algorithm (W Ma... |

3 | On the Sum Coloring Problem on Interval Graphs. Istituto di Analisi dei Sistemi ed Informatica (IASI-CNR), R - Nicoloso, Sarrafzadeh, et al. - 1994 |

2 |
Private communication
- Vishwanathan
- 1998
(Show Context)
Citation Context ...mediately gives us a fairly good characterization of the approximability of MCS on various classes of graphs: O(n/ log 2 n) on general graphs [10], O(∆ log log ∆/ log ∆) on graphs of maximum degree ∆ =-=[35]-=-, O(n .2134 ) on 3-colorable graphs [8], and at most 4 on all perfect graphs and partial k-trees, among others. We show below, that the bound in Theorem 3.1 applies also to the MWCS problem. In that c... |

1 | Approximating the Chromatic Sum - orsson, M, et al. - 1993 |

1 | Graph Theory. Addison-Wesley. Bar-Noy et al., On Chromatic Sums - Harary - 1969 |

1 | Proof verio/cation and hardness of approximation problems - Arora, Lund, et al. - 1992 |

1 | EOEcient Fault Tolerant Algorithms in Distributed Systems - Choy, Singh - 1992 |

1 | An eOEcient parallel algorithm that o/nds independent sets of guaranteed size - Goldberg, Spencer - 1990 |

1 |
Approximating the Chromatic Sum
- Halldórsson, Radhakrishnan
- 1993
(Show Context)
Citation Context ...nimum weighted color sum of a graph G, denoted by MWCS(G), is the minimum WCS(G, Ψ) over all the legal colorings Ψ. v∈V 1 Remark: The current paper is a merger and extension of the two papers [7] and =-=[18]-=-. 2 Throughout the paper we use also the term chromatic sum when referring to the minimum color sum of a graph.A. Bar-Noy et al., On Chromatic Sums.. 9 2.2 Hardness of the MCS In this section we give... |

1 | Graph Theory. Addison-Wesley. [20 - Harary - 1969 |

1 | Private communication. [35 - Vishwanathan - 1995 |