Claudson F. Bornstein

Clique Graphs of Chordal and Path Graphs

Clique graphs of chordal and path graphs are characterized. A special class of graphs named expanded trees is discussed. It consists of a subclass of disk-Helly graphs. It is shown that the clique graph of every chordal (hence path) graph is an expanded tree. In addition, every expanded tree is the clique graph of some path (hence chordal) graph. Different characterizations of expanded trees are described, leading to a polynomial time algorithm for recognizing them.

On clique convergent graphs

A graphG isconvergent when there is some finite integern ≥ 0, such that then-th iterated clique graphKn(G) has only one vertex. The smallest suchn is theindex ofG. TheHelly defect of a convergent graph is the smallestn such thatKn(G) is clique Helly, that is, its maximal cliques satisfy the Helly property. Bandelt and Prisner proved that the Helly defect of a chordal graph is at most one and asked whether there is a graph whose Helly defect exceeds the difference of its index and diameter by more than one. In the present paper an affirmative answer to the question is given. For any arbitrary finite integern, a graph is exhibited, the Helly defect of which exceeds byn the difference of its index and diameter.

Flow metrics

We introduce flow metrics as a relaxation of path metrics (i.e. linear orderings). They are defined by polynomial-sized linear programs and have interesting properties including spreading. We use them to obtain relaxations for several NP-hard linear ordering problems such as minimum linear arrangement and minimum pathwidth. Our approach has the advantage of achieving the best-known approximation guarantees for these problems using the same relaxation and essentially the same rounding algorithm for all the problems while varying only the objective function from problem to problem. This is in contrast to the current state of the literature where each problem either has a new relaxation or a new rounding or both. We also characterize a natural projection of the flow polyhedron.

The pair completion algorithm for the homogeneous set sandwich problem

A homogeneous set is a non-trivial module of a graph, i.e., a non-empty, non-unitary, proper vertex subset such that all its elements present the same outer neighborhood. Given two graphsG1 (V, E1) and G2(V, E2), the Homogeneous Set Sandwich Problem (HSSP) asks whether there exists a graph GS(V, ES), E1 ⊆ ES ⊆ E2, which has a homogeneous set. This paper presents an algorithm that uses the concept of bias graph [S. Tang, F. Yeh, Y. Wang, An efficient algorithm for solving the homogeneous set sandwich problem, Inform. Process. Lett. 77 (2001) 17-22] to solve the problem in O(n min{|E1|,|E2|} log n) time, thus outperforming the other known HSSP deterministic algorithms for inputs where max{|E1|, |E2|} = Ω (n log n).

Data reshuffling in support of fast I/O for distributed-memory machines

Achieving high-speed network I/O on distributed memory systems is a hard problem because their architectures are, in general, ill-suited for communication with the external world One of the problems is that messages are distributed over the private memories of the distributed memory system. This can result in poor performance since communication includes a complex scatter/gather operation. This paper presents a strategy in which the task of creating large contiguous messages is performed on the distributed-memory system, thus minimizing the overhead on the network interface. The performance results for an implementation of this strategy for an iWarp system with a HIPPI interface board are presented.<<ETX>>

Tradeoffs between parallelism and fill in nested dissection

In this paper we demonstrate that parallelism and fill can be traded off in orders for Gaussian elimination. While the well-known nested dissection algorithm produces very parallel elimination orders, we show that by reducing the parallelism it is possible to reduce the fill that the orders generate. In particular, we present a new “less parallel nested dissection” algorithm (LPND). We prove that, unlike standard nested dissection, when applied to a chordal graph LPND finds a zero-fill elimination order. Our implementation of LPND generates less fill than state-of-the-art implementations of the nested dissection (METIS), minimum-degree @MD), and hybrid (BEND) algorithms on a large body of test matrices, at the cost of a small reduction in the paralellism in the orders that it produces. We have also implemented a nested dissection algorithm that is different from METIS and that uses the same separator algorithm used by our implementation of LPND. This algorithm, like LPND, generates less fill than METIS, and on large graphs generates significantly less fill than AMD. The latter comparison is notable, because although it is known that, for certain classes of graphs, minimum-degree produces asymptotically more fill than nested dissection, minimumdegree is believed to produce low-fill orderings in practice. Our experiments contradict this belief. ‘Universidade Federal do Rio de Janeiro, (claudson@acd.ufrj.br). Claudson was supported in part by NSF Grant CCR-9505472. *School of Computer Science, Carnegie Mellon University, and Akamai Technologies, Inc. Bruce Maggs is supported in part by the Air Force Materiel Command (AFMC) and DARPA under Contracts F19628-93-CO193 and F19628-96-G0061,by DARPA Contract NOOO14-95-1-1246, and by an NSF National Young Investigator Award, No. CCR-94-57766, with matching funds providedby NEC Research Institute and Sun Microsystems. 3Department of Computer Science, Carnegie Mellon University, (gImiIler@cs.cmu.edu). Gary Miller issupportedin part by NSF Grants CCR-9505472 and CCR-9706572. Permission to make digital or hard copies ofall or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. To copy otherwise, to republish, to post on servers or to redistribute to lists. requires prior spccitic permission and/or a fee. SPAA ‘99 Saint Malo, France Copyright ACM 1999 l-581 13-124-O/99/06.,.

Parallelizing elimination orders with linear fill

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Intersection Graphs of Orthodox Paths in Trees

This paper presents an algorithm for finding parallel elimination orders for Gaussian elimination. Viewing a system of equations as a graph, the algorithm can be applied directly to interval graphs and chordal graphs. For general graphs, the algorithm can be used to parallelize the order produced by some other heuristic such as minimum degree. In this case, the algorithm is applied to the chordal completion that the heuristic generates from the input graph. In general, the input to the algorithm is a chordal graph G with n nodes and m edges. The algorithm produces an order with height at most O(log/sup 3/ n) times optimal, fill at most O(m), and work at most O(W*(G)), where W*(G) is the minimum possible work over all elimination orders for G. Experimental results show that when applied after some other heuristic, the increase in work and fill is usually small. In some instances the algorithm obtains an order that is actually better, in terms of work and fill, than the original one. We also present an algorithm that produces an order with a factor of log n less height, but with a factor of O(/spl radic/log n) more fill.

Forbidden induced subgraphs for bounded p -intersection number

Abstract We study the graph classes ORTH [ h , s , t ] introduced by Jamison and Mulder, and focus on the case s = 2 , which is closely related to the well-known VPT and EPT graphs. We collect general properties of the graphs in ORTH [ h , 2 , t ] , and provide a characterization in terms of tree layouts. Answering a question posed by Golumbic, Lipshteyn, and Stern, we show that ORTH [ h + 1 , 2 , t ] \ ORTH [ h , 2 , t ] is non-empty for every h ≥ 3 and t ≥ 3 . We derive decomposition properties, which lead to efficient recognition algorithms for the graphs in ORTH [ h , 2 , 2 ] for every h ≥ 3 . Finally, we show that the graphs in ORTH [ 3 , 2 , 3 ] are line graphs of planar graphs.

On behalf of the seller and society: bicriteria mechanisms for unit-demand auctions

A graph G has p -intersection number at most d if it is possible to assign to every vertex u of G , a subset S ( u ) of some ground set U with | U | = d in such a way that distinct vertices u and v of G are adjacent in G if and only if | S ( u ) ? S ( v ) | ? p . We show that every minimal forbidden induced subgraph for the hereditary class G ( d , p ) of graphs whose p -intersection number is at most d , has order at most ( 2 d + 1 ) 2 , and that the exponential dependence on d in this upper bound is necessary. For p ? { d - 1 , d - 2 } , we provide more explicit results characterizing the graphs in G ( d , p ) without isolated/universal vertices using forbidden induced subgraphs.