DOCSLIB.ORG

Decomposition, Approximation, and Coloring of Odd-Minor-Free Graphs

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Decomposition, Approximation, and Coloring of Odd-Minor-Free Graphs Decomposition, Approximation, and Coloring of Odd-Minor-Free Graphs Erik D. Demaine∗ MohammadTaghi Hajiaghayiy Ken-ichi Kawarabayashiz Abstract between the pieces. For example, many optimization prob- We prove two structural decomposition theorems about graphs ex- lems can be solved exactly on graphs of bounded treewidth; cluding a fixed odd minor H, and show how these theorems can what graphs can be partitioned into a small number k of be used to obtain approximation algorithms for several algorithmic bounded-treewidth pieces? In many cases, each piece gives problems in such graphs. Our decomposition results provide new a lower/upper bound on the optimal solution for the entire structural insights into odd-H-minor-free graphs, on the one hand graph, so solving the problem exactly in each piece gives a generalizing the central structural result from Graph Minor The- k-approximation to the problem. Such a decomposition into ory, and on the other hand providing an algorithmic decomposition bounded-treewidth graphs would also be practical, as many into two bounded-treewidth graphs, generalizing a similar result for NP-hard optimization problems are now solved in practice minors. As one example of how these structural results conquer dif- using dynamic programming on low-treewidth graphs; see, ficult problems, we obtain a polynomial-time 2-approximation for e.g., [Bod05, Ami01, Tho98]. Recently, this decomposi- vertex coloring in odd-H-minor-free graphs, improving on the pre- tion approach has been successfully used for graph coloring, 1−" vious O(jV (H)j)-approximation for such graphs and generalizing which is inapproximable within n for any " > 0 unless the previous 2-approximation for H-minor-free graphs. The class P = NP [Zuc07], yet has a 2-approximation in any minor- of odd-H-minor-free graphs is a vast generalization of the well- closed graph family by this approach [DHK05]. Refinements studied H-minor-free graph families and includes, for example, all of this approach have led to PTASs for many graph problems bipartite graphs plus a bounded number of apices. Odd-H-minor- [DHK05, Bak94, Epp00]. free graphs are particularly interesting from a structural graph the- Decomposition of graphs into computationally simpler ory perspective because they break away from the sparsity of H- pieces goes back to the work of Nash-Williams in 1964 minor-free graphs, permitting a quadratic number of edges. [NW64] and of Chartrand et al. [CGH71]. In particular, Chartrand et al. [CGH71] conjectured in 1971 that any 1 Introduction planar graph can have its edges partitioned into two pieces, each inducing an outerplanar graph. This conjecture was Decomposition or partitioning of graphs into smaller pieces only just proved by Goncalves [Gon05], together with a is a fundamental way to design graph algorithms. One linear-time algorithm for computing the decomposition. In of the most famous such decompositions is the divide- particular, this result establishes that any planar graph can be and-conquer separator decomposition for planar graphs of decomposed into two graphs of treewidth 2, a result proved Lipton and Tarjan [LT80], which has been generalized to earlier [Ked96]. Ding et al. [DOSV00] proved that every arbitrary graphs via sparsest cut [LR99, ARV04]. These bounded-genus graph can be decomposed into two graphs of decompositions are based on finding relatively small cuts in bounded treewidth (where the bound depends on the genus). the graph to minimize the interaction between the pieces. To Thomas [Tho95] conjectured that every graph excluding a make the pieces relatively small, the decompositions cut the fixed minor has such a decomposition. This conjecture has graph into many pieces. been proved in two papers [DDO+04, DHK05]; the latter A different kind of decomposition that has received proof is both simpler and algorithmic. A recent extension of more attention recently is to partition the graph into a this result [DHM07] allows the bounded-treewidth pieces to small number of computationally simpler (but not neces- be formed by contractions instead of deletions/subgraphs. sarily small) pieces, without much regard to the interaction In this paper, we generalize the decomposition result to odd-minor-free graphs: every graph excluding a fixed ∗MIT Computer Science and Artificial Intelligence Laboratory, 32 Vas- sar Street, Cambridge, MA 02139, USA. [email protected] odd minor can be decomposed into two bounded-treewidth yAT&T Labs — Research, 180 Park Ave., Florham Park, NJ 07932, graphs. The family of odd-H-minor-free graphs is strictly USA, [email protected]. This work was performed while the more general than H-minor-free graphs for any graph H; for author was at Carnegie Mellon University. example, it includes all complete bipartite graphs K . z n=2;n=2 National Institute for Informatics, 2-1-2 Hitotsubashi, Chiyoda-ku, In interesting contrast, the Lipton-Tarjan separator decom- Tokyo 101-8430, Japan. k [email protected] position (and thus its algorithmic consequences) cannot be ized to odd-H-minor-free graphs. Our results provide the generalized to odd-minor-free graphs, because small sep- first step in this direction by providing new algorithmic arators do not exist. Another contrast between odd-H- structural tools for odd-minor-free graphs. In particular, minor-free graphs and H-minor-free graphs is that the for- the minor-free analogs of our decomposition results have mer can have a quadratic number of edges, while the latter found many applications in just the past few years—see, e.g., are always sparse. Odd-minor-free graphs have been con- [BBCH07, APS07, ASS07, Kle05, DFHT05, DHM]—and sidered extensively in the graph theory literature (see, e.g., we analogously expect our new tools to enable several new [Gue01, GGG+04, Gue05, JT95]) and recently in theoreti- algorithmic results for odd-minor-free graphs. cal computer science [KM06]. Our immediate goal is not to develop practical algo- We prove our result by generalizing a structural result rithms: it is not clear where odd-minor-free graphs would that is the heart of Graph Minor Theory [RS03], which arise in practice, and even most algorithms for H-minor- has many algorithmic applications [Gro03, DFHT05, DH05, free graphs are currently impractical because of huge bounds DHK05], to the case of odd-minor-free graphs. Specifically, from Graph Minors. Rather, the goal of this paper is to im- we prove that every odd-minor-free graph can be decom- prove our algorithmic structural understanding, in particu- posed into a clique-sum of “almost-bipartite” graphs and lar powerful decompositions enabling efficient approxima- graphs that are “almost-embeddable” into bounded-genus tion algorithms, beyond standard graph families. surfaces. (In the original Graph Minor version, only graphs of the second type exist.) Our primary challenge in this result 1.1 Odd-Minor-Free Graphs and Their Importance. is that odd-minor-free graphs can be dense, and this density Recall that a graph H is a minor of G if H can be obtained may be equally spread throughout the graph, so we cannot by contracting and deleting edges in G. Equivalently, H is a hope to find small separations to split the graph into pieces as minor of G precisely if there are jV (H)j vertex-disjoint trees in previous decomposition theorems. Nonetheless, we show in G, one tree Tv for each vertex v of H, such that for every how to decompose odd-minor-free graphs, by showing a con- edge e = fv; wg in H there is an edge e^ in G connecting the nection to the existence of clique minors; this algorithmic two corresponding trees Tv and Tw. Now H is an odd minor technique may be useful for future structural algorithms on of G if, in addition, all the vertices of the trees can be two- dense graphs. colored in such a way that (1) the edges within each tree Tv in All of our results are algorithmic: the decompositions G are bichromatic, while (2) the edge e^ connecting trees Tv can be computed in polynomial time. Our decompositions and Tw in G corresponding to each edge e = fv; wg of H is can also be used to obtain a variety of approximation algo- monochromatic. In particular, the class of odd-H-minor-free rithms for problems on odd-minor-free graphs. In particular, graphs (excluding a fixed graph H as an odd minor) is more our decomposition into two bounded-treewidth graphs (The- general than the class of H-minor-free graphs (excluding a orem 1.1 below) immediately leads to a 2-approximation for fixed graph H as a minor). many NP-complete problems in odd-minor-free graphs. For Indeed, the class of odd-H-minor-free graphs is strictly example, our 2-approximation for graph coloring in odd-Kk- more general: the complete bipartite graph Kn=2;n=2 cer- minor-free graphs improves the recent O(k)-approximation tainly contains a Kk minor for k ≤ n=2, but on the other algorithm [KM06] and generalizes the 2-approximation for hand, it does not contain Kk as an oddp minor for k ≥ 3. In Kk-minor-free graphs [DHK05]. We also believe that com- fact, any Kk-minor-free graph G is O(k log k)-degenerate, bining our clique-sum decomposition (Theorem 1.2 below) i.e, everyp induced subgraph has a vertex of degree at most with Grohe’s technique [Gro03] leads to PTASs for maxi- O(k log k); see [Kos84,p Tho01]. Thus, any Kk-minor- mum independent set, maximum clique, and minimum ver- free graph G has O(k log kn) edges. On the other hand, tex cover. Grohe’s technique for minor-free graphs uses the some odd-Kk-minor-free graphs such as Kn=2;n=2 may have property that removing small sets of vertices in such graphs Θ(n2) edges.
Load more

Recommended publications
  • On Treewidth and Graph Minors
    On Treewidth and Graph Minors Daniel John Harvey Submitted in total fulfilment of the requirements of the degree of Doctor of Philosophy February 2014 Department of Mathematics and Statistics The University of Melbourne Produced on archival quality paper ii Abstract Both treewidth and the Hadwiger number are key graph parameters in structural and al- gorithmic graph theory, especially in the theory of graph minors. For example, treewidth demarcates the two major cases of the Robertson and Seymour proof of Wagner's Con- jecture. Also, the Hadwiger number is the key measure of the structural complexity of a graph. In this thesis, we shall investigate these parameters on some interesting classes of graphs. The treewidth of a graph defines, in some sense, how \tree-like" the graph is. Treewidth is a key parameter in the algorithmic field of fixed-parameter tractability. In particular, on classes of bounded treewidth, certain NP-Hard problems can be solved in polynomial time. In structural graph theory, treewidth is of key interest due to its part in the stronger form of Robertson and Seymour's Graph Minor Structure Theorem. A key fact is that the treewidth of a graph is tied to the size of its largest grid minor. In fact, treewidth is tied to a large number of other graph structural parameters, which this thesis thoroughly investigates. In doing so, some of the tying functions between these results are improved. This thesis also determines exactly the treewidth of the line graph of a complete graph. This is a critical example in a recent paper of Marx, and improves on a recent result by Grohe and Marx.
    [Show full text]
  • Average Degree Conditions Forcing a Minor∗
    Average degree conditions forcing a minor∗ Daniel J. Harvey David R. Wood School of Mathematical Sciences Monash University Melbourne, Australia [email protected], [email protected] Submitted: Jun 11, 2015; Accepted: Feb 22, 2016; Published: Mar 4, 2016 Mathematics Subject Classifications: 05C83 Abstract Mader first proved that high average degree forces a given graph as a minor. Often motivated by Hadwiger's Conjecture, much research has focused on the av- erage degree required to force a complete graph as a minor. Subsequently, various authors have considered the average degree required to force an arbitrary graph H as a minor. Here, we strengthen (under certain conditions) a recent result by Reed and Wood, giving better bounds on the average degree required to force an H-minor when H is a sparse graph with many high degree vertices. This solves an open problem of Reed and Wood, and also generalises (to within a constant factor) known results when H is an unbalanced complete bipartite graph. 1 Introduction Mader [13, 14] first proved that high average degree forces a given graph as a minor1. In particular, Mader [13, 14] proved that the following function is well-defined, where d(G) denotes the average degree of a graph G: f(H) := inffD 2 R : every graph G with d(G) > D contains an H-minorg: Often motivated by Hadwiger's Conjecture (see [18]), much research has focused on f(Kt) where Kt is the complete graph on t vertices. For 3 6 t 6 9, exact bounds on the number of edges due to Mader [14], Jørgensen [6], and Song and Thomas [19] implyp that f(Kt) = 2t−4.
    [Show full text]
  • The Hadwiger Number, Chordal Graphs and Ab-Perfection Arxiv
    The Hadwiger number, chordal graphs and ab-perfection∗ Christian Rubio-Montiel [email protected] Instituto de Matem´aticas, Universidad Nacional Aut´onomade M´exico, 04510, Mexico City, Mexico Department of Algebra, Comenius University, 84248, Bratislava, Slovakia October 2, 2018 Abstract A graph is chordal if every induced cycle has three vertices. The Hadwiger number is the order of the largest complete minor of a graph. We characterize the chordal graphs in terms of the Hadwiger number and we also characterize the families of graphs such that for each induced subgraph H, (1) the Hadwiger number of H is equal to the maximum clique order of H, (2) the Hadwiger number of H is equal to the achromatic number of H, (3) the b-chromatic number is equal to the pseudoachromatic number, (4) the pseudo-b-chromatic number is equal to the pseudoachromatic number, (5) the arXiv:1701.08417v1 [math.CO] 29 Jan 2017 Hadwiger number of H is equal to the Grundy number of H, and (6) the b-chromatic number is equal to the pseudo-Grundy number. Keywords: Complete colorings, perfect graphs, forbidden graphs characterization. 2010 Mathematics Subject Classification: 05C17; 05C15; 05C83. ∗Research partially supported by CONACyT-Mexico, Grants 178395, 166306; PAPIIT-Mexico, Grant IN104915; a Postdoctoral fellowship of CONACyT-Mexico; and the National scholarship programme of the Slovak republic. 1 1 Introduction Let G be a finite graph. A k-coloring of G is a surjective function & that assigns a number from the set [k] := 1; : : : ; k to each vertex of G.A k-coloring & of G is called proper if any two adjacent verticesf haveg different colors, and & is called complete if for each pair of different colors i; j [k] there exists an edge xy E(G) such that x &−1(i) and y &−1(j).
    [Show full text]
  • Some Remarks on Even-Hole-Free Graphs
    Some remarks on even-hole-free graphs Zi-Xia Song∗ Department of Mathematics, University of Central Florida, Orlando, FL 32816, USA Abstract A vertex of a graph is bisimplicial if the set of its neighbors is the union of two cliques; a graph is quasi-line if every vertex is bisimplicial. A recent result of Chudnovsky and Seymour asserts that every non-empty even-hole-free graph has a bisimplicial vertex. Both Hadwiger’s conjecture and the Erd˝os-Lov´asz Tihany conjecture have been shown to be true for quasi-line graphs, but are open for even-hole-free graphs. In this note, we prove that for all k ≥ 7, every even-hole-free graph with no Kk minor is (2k − 5)-colorable; every even-hole-free graph G with ω(G) < χ(G) = s + t − 1 satisfies the Erd˝os-Lov´asz Tihany conjecture provided that t ≥ s>χ(G)/3. Furthermore, we prove that every 9-chromatic graph G with ω(G) ≤ 8 has a K4 ∪ K6 minor. Our proofs rely heavily on the structural result of Chudnovsky and Seymour on even-hole-free graphs. 1 Introduction All graphs in this paper are finite and simple. For a graph G, we use V (G) to denote the vertex set, E(G) the edge set, |G| the number of vertices, e(G) the number of edges, δ(G) the minimum degree, ∆(G) the maximum degree, α(G) the independence number, ω(G) the clique number and χ(G) the chromatic number. A graph H is a minor of a graph G if H can be obtained from a subgraph of G by contracting edges.
    [Show full text]
  • Prizes and Awards Session
    PRIZES AND AWARDS SESSION Wednesday, July 12, 2021 9:00 AM EDT 2021 SIAM Annual Meeting July 19 – 23, 2021 Held in Virtual Format 1 Table of Contents AWM-SIAM Sonia Kovalevsky Lecture ................................................................................................... 3 George B. Dantzig Prize ............................................................................................................................. 5 George Pólya Prize for Mathematical Exposition .................................................................................... 7 George Pólya Prize in Applied Combinatorics ......................................................................................... 8 I.E. Block Community Lecture .................................................................................................................. 9 John von Neumann Prize ......................................................................................................................... 11 Lagrange Prize in Continuous Optimization .......................................................................................... 13 Ralph E. Kleinman Prize .......................................................................................................................... 15 SIAM Prize for Distinguished Service to the Profession ....................................................................... 17 SIAM Student Paper Prizes ....................................................................................................................
    [Show full text]
  • Fractional Colouring and Hadwiger's Conjecture
    Journal of Combinatorial Theory, Series B 74, 147152 (1998) Article No. TB981835 Fractional Colouring and Hadwiger's Conjecture Bruce Reed Equipe Combinatoire, Case 189, Universite de Paris VI, 4 Place Jussieu, 75252 Paris Cedex 05, France and Paul Seymour* Department of Mathematics, Princeton University, Princeton, New Jersey 08544 View metadata, citation and similar papersReceived at Octobercore.ac.uk 28, 1997 brought to you by CORE provided by Elsevier - Publisher Connector Let G be a loopless graph with no Kp+1 minor. We prove that the ``fractional chromatic number'' of G is at most 2p; that is, it is possible to assign a rational q(S)0 to every stable set SV(G) so that S % v q(S)=1 for every vertex v, and S q(S)2p. 1998 Academic Press 1. INTRODUCTION In this paper, all graphs are finite and without loops or multiple edges; Kp+1 denotes the complete graph with p+1 vertices; and H is a minor of G if H can be obtained from a subgraph of G by contracting edges. A famous conjecture of Hadwiger [4], dating from 1943, asserts (1.1) (Conjecture) For all p1, every graph with no Kp+1 minor is p-colorable. This is easy for p3, true but difficult for p=4 and 5 (in both cases [9, 6] it is equivalent to the four-colour theorem [1, 2, 7]), and open for p6. It is therefore of interest to examine weakenings of (1.1). If G has no Kp+1 minor, the best upper bound known on its chromatic number is O( p(log p)1Â2).
    [Show full text]
  • On the Odd-Minor Variant of Hadwiger's Conjecture
    On the odd-minor variant of Hadwiger’s conjecture Jim Geelen∗, Bert Gerards†, Bruce Reed‡, Paul Seymour§ Adrian Vetta¶ ???; revised April 18, 2011 ∗Department of Combinatorics and Optimization, University of Waterloo. Email: [email protected] †Centrum voor Wiskunde en Informatica, and Eindhoven University of Technology. Email: [email protected] ‡School of Computer Science, McGill University. Email: [email protected] §Department of Mathematics, Princeton University. Email: [email protected] ¶Department of Mathematics and Statistics, and School of Computer Science, McGill University. Email: [email protected] Abstract A Kl-expansion consists of l vertex-disjoint trees, every two of which are joined by an edge. We call such an expansion odd if its vertices can be two-coloured so that the edges of the trees are bichromatic but the edges between trees are monochromatic. We show that, for every l, if a graph contains no odd Kl-expansion then its chromatic number is O(l√log l). In doing so, we obtain a characterization of graphs which contain no odd Kl-expansion which is of independent interest. We also prove that given a graph and a subset S of its vertex set, either there are k vertex-disjoint odd paths with endpoints in S, or there is a set X of at most 2k 2 vertices such that every odd path − with both ends in S contains a vertex in X. Finally, we discuss the algorithmic implications of these results. 1 The Main Results All graphs in this paper are finite, and have no loops or parallel edges.
    [Show full text]
  • The Order of the Largest Complete Minor in a Random Graph
    THE ORDER OF THE LARGEST COMPLETE MINOR IN A RANDOM GRAPH NIKOLAOS FOUNTOULAKIS, DANIELA KUHN¨ AND DERYK OSTHUS Abstract. Let ccl(G) denote the order of the largest complete minor in a graph G (also called the contraction clique number) and let Gn,p denote a random graph on n vertices with edge probability p. Bollob´as,Catlin and Erd˝os[5] asymptotically determined ccl(Gn,p) when p is a constant.Luczak, Pittel and Wierman [10] gave bounds on ccl(Gn,p) when p is very close to 1/n, i.e. inside the phase transition. We show that for every ε > 0 there exists a constant C suchp that whenever C/n < p < 1 − ε then asymptotically almost surely ccl(G )= (1 ± ε)n/ log (np), where b := 1/(1 − p). If p = C/n for a constant C > 1, n,p b √ then asymptotically almost surely ccl(Gn,p)= Θ( n). This extends the results in [5] and answers a question of Krivelevich and Sudakov [9]. 1. Introduction 1.1. Main results. A graph H is a minor of G if for every vertex h ∈ H there is a connected subset Bh ⊆ V (G) such that all the Bh are disjoint and G contains an edge between Bh 0 and Bh0 whenever hh is an edge of H. The Bh’s are called the branch sets. We denote by ccl(G) the order of the largest complete minor in G. The study of the largest complete minor contained in a given graph has its origins in Hadwiger’s conjecture which states that if the chromatic number of a graph G is at least k, then G contains Kk minor.
    [Show full text]
  • Clique Minors in Graphs with a Forbidden Subgraph
    Clique minors in graphs with a forbidden subgraph Matija Bucić∗ Jacob Foxy Benny Sudakovz Abstract The classical Hadwiger conjecture dating back to 1940’s states that any graph of chromatic number at least r has the clique of order r as a minor. Hadwiger’s conjecture is an example of a well-studied class of problems asking how large a clique minor one can guarantee in a graph with certain restrictions. One problem of this type asks what is the largest size of a clique minor in a graph on n vertices of independence number α(G) at most r. If true Hadwiger’s conjecture would imply the existence of a clique minor of order n/α(G). Results of Kühn and Osthus and Krivelevich and Sudakov imply that if one assumes in addition that G is H-free for some bipartite graph H then one can find a polynomially larger clique minor. This has recently been extended to triangle-free graphs by Dvořák and Yepremyan, answering a question of Norin. We complete the picture and show that the same is true for arbitrary graph H, answering a question of Dvořák and Yepremyan. In particular, we show that any Ks-free 1+ 1 graph has a clique minor of order cs(n/α(G)) 10(s−2) , for some constant cs depending only on s. The 1 exponent in this result is tight up to a constant factor in front of the s−2 term. 1 Introduction A graph Γ is said to be a minor of a graph G if for every vertex v of Γ we can choose a connected subgraph Gu of G, such that subgraphs Gu are vertex disjoint and G contains an edge between Gv and Gv0 whenever v and v0 make an edge in Γ.
    [Show full text]
  • SPECIAL ISSUE in HONOR of GEOFF WHITTLE Geoff Whittle Turned 60 in 2010. to Mark
    SPECIAL ISSUE IN HONOR OF GEOFF WHITTLE DILLON MAYHEW, JAMES OXLEY, AND CHARLES SEMPLE Geoff Whittle turned 60 in 2010. To mark this occasion, this special issue (guest- edited by Dillon Mayhew and Charles Semple) aims to recognize and celebrate his mathematical contributions. Perhaps surprisingly, we can confidently say that Ge- off’s best work lies ahead of him, for some of the most significant results from his longstanding collaboration with Jim Geelen and Bert Gerards have been announced, but not yet published. This collaboration has produced matroid analogues of the well-quasi-ordering results from Robertson and Seymour’s monumental Graph Mi- nors Project. In addition, it is anticipated that Rota’s conjecture, a problem long considered to be one of the most difficult in discrete mathematics, will also be set- tled by Jim, Bert, and Geoff. It is already clear that their work deserves to be ranked alongside that of W. T. Tutte and Paul Seymour as the most significant ever done in matroid theory. In this preface, we will introduce matroids after giving some biographical details of Geoff’s life. Our main task is to give a brief description of Geoff’s contributions to the development of matroid theory. 1. Brief Biography Geoff was born in 1950 in Launceston, a city in northern Tasmania, Australia. He completed high school in Launceston in 1968, and, in the following year, he worked as a miner in Tasmania, Western Australia, and the Northern Territory before leaving to travel through Asia and Europe. In 1971, he enrolled at the University of Tasmania, graduating in 1973 with a bachelor’s degree in Philosophy and Mathematics.
    [Show full text]
  • An Introduction to the Discharging Method Via Graph Coloring
    An Introduction to the Discharging Method via Graph Coloring Daniel W. Cranston∗ Douglas B. West† October 10, 2016 Abstract We provide a “how-to” guide to the use and application of the Discharging Method. Our aim is not to exhaustively survey results proved by this technique, but rather to demystify the technique and facilitate its wider use, using applications in graph coloring as examples. Along the way, we present some new proofs and new problems. 1 Introduction Arguments that can be phrased in the language of the Discharging Method have been used in graph theory for more than 100 years, though that name is much more recent. The most famous application of the method is the proof of the Four Color Theorem, stating that graphs embeddable in the plane have chromatic number at most 4. However, the method remains mysterious to many. Our aim is to explain its use and make the method more widely accessible. Although we mention many applications, including stronger versions of results proved here, cataloguing applications is not our goal. Borodin [23] presents a survey of applications of discharging to coloring of plane graphs. Discharging is most commonly used as a tool in a two-pronged approach to inductive proofs, typically for sparse graphs. In this context, it is used to prove that a global sparse- ness hypothesis guarantees the existence of some desired local structure. The method has been applied to many types of problems (including graph embeddings and decompositions, spread of infections in networks, geometric problems, etc.). Nevertheless, we present only applications in graph coloring (where it has been used most often), in order to emphasize the discharging techniques.
    [Show full text]
  • Professor Shang-Hua Teng
    Professor Shang-Hua Teng Department of Computer Science, University of Southern California 1039 25 Street 3710 McClintock, RTH 505, Los Angeles, CA 90089 Santa Monica, CA 90403 (213) 740-3667, [email protected] (617) 440-4281 EMPLOYMENT: University Professor USC (2017 – ) Seeley G. Mudd Professor (Computer Science and Mathematics) USC (2012 – ) Chair and Seeley G. Mudd Professor (Computer Science) USC (2009 – 2012) Professor (Computer Science) Boston University (2002 – 2009) Research Affiliate Professor (Mathematics) MIT (1999 – Present) Visiting Research Scientist Microsoft Research Asia and New England (2004 – Present) Senior Research Scientist Akamai (1997 – 2009) Visiting Professor (Computer Science) Tsinghua University (2004 – 2008) Professor (Computer Science)[Associate Professor before 1999] UIUC (2000 – 2002) Research Scientist IBM Almaden Research Center (1997 – 1999) Assistant Professor (Computer Science) Univ. of Minnesota (1994 – 1997) Research Scientist Intel Corporation (Summer 1994, 1995) Instructor (Mathematics) MIT (1992 – 1994) Computer Scientist NASA Ames Research Center (Summer 1993) Research Scientist Xerox Palo Alto Research Center (1991 – 1992) EDUCATION: Ph.D. (Computer Science) Carnegie Mellon University (1991) Thesis: “A Unified Geometric Approach to Graph Partitioning.” Advisor: Gary Lee Miller M.S. (Computer Science) University of Southern California (1988) Thesis: “Security, Verifiablity, and Universality in Distributed Computing”. Advised by Leonard Adleman and Ming-Deh Huang B.S. (Computer Science) & B.A. (Electrical
    [Show full text]