Approximation algorithms for combinatorial problems

In this paper we introduce the minimum-order approach to frequency assignment and present a theory which relates this approach to the traditional one. This new approach is potentially more desirable than the traditional one. We model assignment problems as both frequency-distance constrained and frequency constrained optimization problems. The frequency constrained approach should be avoided if distance separation is employed to mitigate interference. A restricted class of graphs, called disk graphs, plays a central role in frequency-distance constrained problems. We introduce two generalizations of chromatic number and show that many frequency assignment problems are equivalent to generalized graph coloring problems. Using these equivalences and recent results concerning the complexity of graph coloring, we classify many frequency assignment problems according to the "execution time efficiency" of algorithms that may be devised for their solution. We discuss applications to important real world problems and identify areas for further work.

Frequency assignment: Theory and applications

We consider the problem of coloring k-colorable graphs with the fewest possible colors. We present a randomized polynomial time algorithm that colors a 3-colorable graph on n vertices with min{O(D1/3 log1/2 D log n), O(n1/4 log1/2n)} colors where D is the maximum degree of any vertex. Besides giving the best known approximation ratio in terms of n, this marks the first nontrivial approximation result as a function of the maximum degree D. This result can be generalized to k-colorable graphs to obtain a coloring using min{O(D1-2/k log1/2 D log n), O(n1−3/(k+1) log1/2n)} colors. Our results are inspired by the recent work of Goemans and Williamson who used an algorithm for semidefinite optimization problems, which generalize linear programs, to obtain improved approximations for the MAX CUT and MAX 2-SAT problems. An intriguing outcome of our work is a duality relationship established between the value of the optimum solution to our semidefinite program and the Lovasz t-function. We show lower bounds on the gap between the optimum solution of our semidefinite program and the actual chromatic number; by duality this also demonstrates interesting new facts about the t-function.

/pdf/approximate-graph-coloring-by-semidefinite-programming-463z414g31.pdf

Approximate graph coloring by semidefinite programming

Graph coloring problems, in which one would like to color the vertices of a given graph with a small number of colors so that no two adjacent vertices receive the same color, arise in many applications, including various scheduling and partitioning problems. In this paper the complexity and performance of algorithms which construct such colorings are investigated. For a graph G, let k(G) denote the minimum possible number of colors required to color G and, for any graph coloring algorithm A, let A(G) denote the number of colors used by A when applied to G. Since the graph coloring problem is known to be “NP-complete,” it is considered unlikely that any efficient algorithm can guarantee A(G) = k(G) for all input graphs. In this paper it is proved that even coming close to khgr;(G) with a fast algorithm is hard. Specifically, it is shown that if for some constant r d there exists a polynomial-time algorithm A which guarantees A(G) ≤ r·k(G) + d, then there also exists a polynomial-time algorithm A which guarantees A(G) = k(G).

The Complexity of Near-Optimal Graph Coloring

A language and a program for stating and solving combinatorial problems

A technique for colouring a graph applicable to large scale timetabling problems

Managers of computer facilities frequently need to be able to measure the throughput of their installations. Throughput is based on the amount of work a system can perform in a given time in a particular environment, and is usually measured as the reciprocal of the running time of selected jobs. The throughput of a computer system is a function of the hardware configuration, the operating system software and the workload characteristics. Once the characteristics of the workload are identified, the configuration and operating system can be optimized in terms of throughput for that environment. Improvements resulting from system changes can be evaluated using throughput measurements.

Throughput measurement using a synthetic job stream

■ A panel appointed by the U.S. National Institute of Standards and Technology has recommended that the U.S. Government Open Systems interconnection Protile (GOSIP) policy, which mandates the acquisition of Open Systems Interconnection (0S1) protocols, be broadened to include protocols from the Internet Protocol Suite. This article reviews the major networking standards policy issues considered by the panel and draws on their discussions and conclusions, as well as on views raised by the many individuals and organizations who offered comments on the panel’s draft report. o meet requirements for data internetworking, the federal government in 1988 adopted the Government Open Systems Interconnection (0S1) Profile (GOSIP), based on the 0S1 international standards. The objective of GOSIP is to achieve interconnection and interoperability of computers and systems that are acquired from different manufacturers in an open systems environment. Beginning in 1990, GOSIP has been required by federal agencies when acquiring computer networking products and services and communications systems, or else to acquire services that provide equivalent functionality to the protocols defined in GOSIP. Because GOSIP standards were a result of the international standards process and were expected to be implemented worldwide, they were expected to displace the Transmission Control Protocol/Internet Protocol suite (generally known as TCP/IP) and proprietary protocols. In practice, some GOSIP prcduets have been much slower in coming to market than expected and have not been widely deployed to date, while TCP/IP standards have become commodity products that are widely used in LANs and private networks. More importantly, a substantial infrastmcture, the worldwide Internet, which supports the TCP/IP standards has continued to develop, while comparable infrastructure has not been developed for GOSIP. As a result of promoting GOSIP through procurement mandate, some agencies have acquired GOSIP-compliant hardware and software that is often not used or installed. In the meantime, agencies have continued to use proprietary solutions or have moved to adopt portions of the TCP/IP suite. As a consequence of this situation, the Federal Intemetworking Requirements Panel (FIRP) was established by the National Institute of Standards and Technology (NIST), at the request of the Office of Management and Budget, to reassess federal requirements for open systems networks. The panel was chartered to recommend policy on the government’s use of networking standards, particularly the TCP/IP protocol suite (also referred to as the Internet Protocol Suite or IPS) and the 0S1 protocol suite and,

Federal networking standards: policy issues

■ A workshop on Information and Communication Technology Standardization Policy was held by the European Commission in November 1994. The European Union has a policy requiring European and international standards in public procurements. There is growing realization in the Commission that many formal standards are not successful in the marketplace, and that de facto standards from consortia often provide a cheaper alternative for open systems. The workshop resulted in a series of recommendations that can be expected to lead to changes in European policy on standardization over the next year. There were many parallels between the workshop and the U.S. Federal Internetworking Requirements Panel (FIRP), which in 1994 recommended changes to U.S. Government Open System Interconnection Profile (GOSIP) policy. he workshop was motivated primarily by recommendations made to the European Council in June of 1994 on “Europe and the Global Information Society.” The recommendations are contained in the “Bangemann Report,” which was prepared by a group of prominent persons and chaired by Commissioner Martin Bangemann [Bangemann 1994]. The report sets out a vision for Europe in the Global Information Infrastructure, and in many ways is analogous to the Clinton/Gore February 1993 paper on technology for America’s economic growth [Clinton and Gore 1993] and the National Information Infrastructure (NII) Agenda for Action of September 1993 [IITF 1993]. The report, which does not favor more public money, urges the European Union to put its faith in market mechanisms, to deregulate, and to encourage a competitive market for information services. It also proposes an action plan involving ten key applications. The report makes some critical observations concerning standards: “Standards institutes have an honorable record in producing European standards, but the standardization process as it stands today raises a number of concerns about fitness for purpose, lack of interoperability, and priority setting that is not sufficiently market driven.” In line with its overall vision and on the basis of its assessment of the current organization and output, “the Bangemann Group recommends a review of the European standardization process in order to increase its speed and responsiveness to markets.”

David C. Wood

Papers

A technique for colouring a graph applicable to large scale timetabling problems

Throughput measurement using a synthetic job stream

Throughput measurement using a synthetic job stream

Federal networking standards: policy issues

European standardization policy