Title: Graph partitioning into isolated, high conductance clusters: Theory, computation and applications to preconditioning.

Yiannis Koutis, CMU

Abstract:

We study the problem of decomposing a weighted graph with $n$ vertices into a collection $P$ of vertex disjoint clusters such that, for all clusters $C$ in $P$, the graph induced by the vertices in $C$ and the edges leaving $C$, has conductance bounded below by a constant $\phi$. We show that for constant average degree graphs we can compute a decomposition $P$ such that $|P| < n/a$, where $a$ is a constant, in $O(\log n)$ parallel time with $O(n)$ work. We show how these decompositions can be used in the first known linear work parallel and quite practical construction of provably good preconditioners for the important class of fixed degree graph Laplacians. On a more theoretical note, we present upper bounds on the Euclidean distance of eigenvectors of the normalized Laplacian from the space of vectors which consists of the cluster-wise constant vectors.

Abstract:

We study the problem of deciding whether an n-variate polynomial, presented as an arithmetic circuit G, contains a degree k square-free term with an odd coefficient. We show that if G can be evaluated over the integers modulo 2^(k+1) in time t and space s, the problem can be decided with constant probability in O((kn+t)2^k) time and O(kn+s) space. Based on this, we present new and faster algorithms for several parameterized problems, among which: (i) an O(2^(mk)) algorithm for the m-set k-packing problem and (ii) an O(2^(3k/2)) algorithm for the simple k-path problem, or an O(2^k) algorithm if the graph has an induced k-subgraph with an odd number of Hamiltonian paths.

But if you have tried this route, you may notice that TeX has “optimized away” your effort by subtly padding the pages with more vertical spaces, thereby keeping the page count constant…

Long story short, you want to use \raggedbottom. Put it in the preamble and recompile. With this, LaTeX will keep the breaks at the same places, but it will not pad and hence the bottom of the pages will be ragged (duh!). Now you can usually see why your effort did not produce the desired effect: the space that you just freed up does not allow the current page/column to absorb enough material from the next.

With the ability to find a short page/column by inspection, now you can identify the earliest place where rewording is more likely help. Repeat the reword-recompile cycle a couple times, and the page count will go down. Just be sure to take out \raggedbottom when you are done!

P.S. The two-page mode in your previewer can make the inspection more effective.

Abstract:

Recently, Kalai et. al gave a variant of the “Low-Degree Algorithm” for agnostic learning (learning with arbitrary classification noise) under the uniform distribution on {0,1}^n. One result of their work is an agnostic learning algorithm with respect to the class of linear threshold functions under certain restricted instance distributions, including the uniform distribution on {0,1}^n.

In this talk, we extend these ideas to product distributions on instance spaces X_1 x … X_n. We develop a variant of the “Low-Degree Algorithm” for these distributions, and we show that our algorithm agnostically learns with respect to the class of threshold functions under these distributions. We prove this by extending the “noise sensitivity method” to arbitrary product spaces, showing that threshold functions over arbitrary product spaces are no more noise sensitive than their Boolean counterparts.

[…] a panelist at IPDPS (in Miami, a couple of weeks ago) assert that parallel-processing topics have been disappearing from CS curricula in recent years. As anecdotal evidence, he pointed out the topic’s removal in the 2nd edition of Introduction to Algorithms […]

The article goes on to give several other examples to prove his point. In the end, Michael wrote

[…] multicore computing platforms are now the norm. Recognizing that reality, let’s make the adjustment time short.

Are we expecting a surge of algorithm and data structure textbooks with an emphasis in multicore?

Abstract:

We present some results on packing graphs in dense multipartite graphs. This is a question very similar to the Hajnal-Szemeredi theorem, which gives sufficient minimum-degree conditions for an $n$-vertex graph to have a subgraph consisting of $\lfloor n/r\rfloor$ vertex-disjoint copies of $K_r$. This is a packing, or tiling, of the graph by copies of $K_r$. The Hajnal-Szemeredi theorem has been generalized to finding minimum-degree conditions that guarantee packings of non-complete graphs, notably by Alon and Yuster and by Kuhn and Osthus. We consider a multipartite version of this problem. That is, given an $r$-partite graph with $N$ vertices in each partition, what is the minimum-degree required of the bipartite graph induced by each pair of color-classes so that the graph contains $N$ vertex-disjoint copies of $K_r$? The question has been answered for $r=3,4$, provided $r$ is sufficiently large. When $r=3$ and $N$ is sufficiently large, a degree condition of $(2/3)N$ is sufficient with the exception of a single tripartite graph when $N$ is an odd multiple of $3$. When $r=4$ and $N$ is sufficiently large, a degree condition of $(3/4)N$ is sufficient and there is no exceptional graph. There are also bounds on the degree condition for higher $r$ by Csaba and Mydlarz. This question has also been generalized to finding minimum-degree conditions for packings of some arbitrary $r$-colorable graph in an $r$-partite. The case $r=2$ is highly nontrivial for packing arbitrary bipartite graphs and was answered very precisely by Zhao. The case $r=3$ is even more complex and we provide some tight bounds on the required degree condition. This talk includes joint work with Cs. Magyar, with E. Szemeredi and with Y. Zhao.

Abstract:

We address a question in graphs called the stable paths problem, which is an abstraction of a network routing problem concerning the Border Gateway Protocol (BGP). The main tool we use is Scarf’s Lemma. This talk will describe Scarf’s Lemma and how it is related to other results more familiar to combinatorialists, and then will explain its implications for the stable paths problem.

Nash Bargaining via Flexible Budget Markets

Vijay V. Vazirani, Georgia Tech

In his seminal 1950 paper, John Nash defined the bargaining problem; the ensuing theory of bargaining lies today at the heart of game theory. In this work, we initiate an algorithmic study of Nash bargaining problems.

We consider a class of Nash bargaining problems whose solution can be stated as a convex program. For these problems, we show that there corresponds a market whose equilibrium allocations yield the solution to the convex program and hence the bargaining problem. For several of these markets, we give combinatorial, polynomial time algorithms, using the primal-dual paradigm.

Unlike the traditional Fisher market model, in which buyers spend a fixed amount of money, in these markets, each buyer declares a lower bound on the amount of utility she wishes to derive. The amount of money she actually spends is a specific function of this bound and the announced prices of goods.

Over the years, a fascinating theory has started forming around a convex program given by Eisenberg and Gale in 1959. Besides market equilibria, this theory touches on such disparate topics as TCP congestion control and efficient solvability of nonlinear programs by combinatorial means. Our work shows that the Nash bargaining problem fits harmoniously in this collage of ideas.

Mohit Singh

Wednesday, April 30, 2008, 3:30 pm, 384 Posner

Abstract:

Linear programming has been a successful tool in combinatorial optimization to achieve polynomial time algorithms for problems in P and also to achieve good approximation algorithms for problems which are NP-hard. We demonstrate that iterative methods give a general framework to analyze linear programming formulations of polynomial time solvable problems as well as NP-hard problems.

In this thesis, we focus on degree bounded network design problems. The most well-studied problem in this class is the Minimum Bounded Degree Spanning Tree problem defined as follows. Given a weighted undirected graph with degree bound B, the task is to find a spanning tree of minimum cost that satisfies the degree bound. We present a polynomial time algorithm that returns a spanning tree of optimal cost and maximum degree B+1. This generalizes a result of Furer and Raghavachari to weighted graphs, and thus settles a 15-year-old conjecture of Goemans affirmatively. This is also the best possible result for the problem in polynomial time unless P=NP.

We also study degree bounded versions of general network design problems including the minimum bounded degree Steiner tree problem, the minimum bounded degree Steiner forest problem, minimum bounded degree k-edge connected subgraph problem and the minimum bounded degree arborescence problem. We show that iterative methods give bi-criteria approximation algorithms that return a solution whose cost is within a small constant factor of the optimal solution and the degree bounds are violated by an additive factor in undirected graphs and a small multiplicative factor in directed graphs. These results also imply first additive approximation algorithms for various degree constrained network design problems in undirected graphs.

We also show the generality of the iterative methods and apply it to the degree constrained matroid problem, multi-criteria spanning tree problem, multi-criteria matroid basis problem and the generalized assignment problem achieving or matching best known approximation algorithms for them.

Thesis Committee:
Prof. R. Ravi, Carnegie Mellon University (Chair)
Prof. Gerard Cornuejols, Carnegie Mellon University
Prof. Alan Frieze, Carnegie Mellon University
Prof. Michel Goemans, Massachusetts Institute of Technology
Prof. Anupam Gupta, Carnegie Mellon University

Abstract:

We consider the online single-server job scheduling problem. It is known that to minimize the average response time of jobs in this setting, at all times the job with the shortest remaining service time must be scheduled. This requires that the server knows about the sizes of all the jobs. However, in the scenario where the server does not know the sizes of the jobs whereas the jobs know their own sizes, the server can not rely on the jobs to truthfully reveal their sizes since a job may reduce its own response time by misreporting. While there are mechanisms in the literature that achieve truthful revelation, such mechanisms are based on imposing a tax and hence involve “real” money - which is not always desirable.

In this work, we propose a novel token based scheduling game. We prove that while playing the above scheduling game, all the jobs trying to minimize their own response time will end up implementing the shortest remaining service time first scheduling policy themselves.

Andrew Binstock and Donald Knuth converse on the success of open source, the problem with multicore architecture, the disappointing lack of interest in literate programming, the menace of reusable code, and that urban legend about winning a programming contest with a single compilation.

Among other things, he explained why the numbering of Volume 4 fascicles starts at 0 and not 1. (You may recall that there is no Volume 0 and so zero-based counting cannot be exactly the reason. Well, I mean not exactly.)

Pall Melsted, CMU
April 25, 2008, 3:30PM, Wean 7220

Abstract:

We present a linear expected time algorithm for finding maximum cardinality matchings in sparse random graphs. This is optimal and improves on previous results by a logarithmic factor.

This is joint work with Prasad Chebolu and Alan Frieze.

Abstract:

Our focus in this talk will be the size of linear programming formulations of combinatorial optimization problems. We may view this parameter as akin to traditional measures of complexity, such as computational time and space. We will focus on problems in P, in particular the minimum cut problem. For a graph $(V,E)$, existing linear formulations for the minimum cut problem require $\Theta(|V||E|)$ variables and constraints. These formulations can be interpreted as a composition of $|V|-1$ polyhedra for minimum $s$-$t$ cuts paralleling early algorithmic approaches to finding globally minimum cuts, which relied on $|V|-1$ calls to a minimum $s$-$t$ cut algorithm. We present the first formulation to beat this bound, one that uses $O(|V|^2)$ variables and $O(|V|^3)$ constraints. Our formulation directly implies a smaller compact linear relaxation for the Traveling Salesman Problem that is equivalent in strength to the standard subtour relaxation.

Abstract: I will describe some latest results in predicate encryption. The crypto construction allows a user to store her personal files on a remote untrusted server, and make expressive search queries to retrieve certain documents. The remote untrusted server learns no unintended information.

Viswanath Nagarajan, Thesis Proposal for Ph.D. in Algorithms, Combinatorics and Optimization.
10:30am Wednesday 23-April
Room 384, 3rd floor Posner Hall (Tepper School of Business)

Broadly speaking, any scheduling problem can be characterized as serving a set of requests using a limited set of resources, subject to constraints detailing how the resources may serve requests. Due to the complicating nature of constraints in typical scheduling problems, most of them are NP-complete and hence we do not expect efficient (i.e. polynomial time) exact algorithms. The two main approaches to practical solutions of such problems are (i) exact algorithms that compute the optimal solution but take exponential time in the worst case, and (ii) heuristic algorithms that run in polynomial time but find near-optimal solutions. An approximation algorithm is an efficient heuristic along with a worst-case guarantee on the quality of the near-optimal solutions found by it. The goal of this thesis is to design approximation algorithms for some scheduling problems, with an emphasis on Vehicle Routing Problems.

Vehicle routing problems (VRPs) form a rich class of variants of the basic Traveling Salesman Problem, that are also practically motivated. In VRPs, a fleet of vehicles represents the resources used to serve a set of client-requests (such as transporting objects to the clients). Many VRPs just seek to minimize cost incurred by the vehicles while serving client requests; a goal in this thesis is to study VRPs that incorporate some additional criteria on the vehicle routes.

All VRPs are defined in relation to a metric space (i.e. set of locations with a distance function on them). Most of the work on approximation algorithms for VRPs has focussed on symmetric metrics. The corresponding problems on asymmetric metrics become considerably harder. Another goal of this thesis is to design algorithms for VRPs on asymmetric metrics.

Committee: R. Ravi (Chair), Gerard Cornuejols, Anupam Gupta, Mike Trick

Abstract:

The classical secretary problem studies how to select online an element with maximum value in a randomly ordered sequence. The problem is closely connected with online mechanism design in which agents {e} with private values v(e) for a good arrive sequentially in random order and the mechanism designer wishes to allocate the good to an agent with maximum value. The difficulty lies in the fact that an agent’s allocation must be decided irrevocably upon arrival. A mechanism for this problem is called alpha-competitive if it gets, in expectation, at least a 1/alpha fraction of the (expected) optimal offline solution. It is well-known how to design constant-competitive algorithms for the classical secretary problem and several variants. In this talk we will discuss the discounted secretary problem, in which there is a time-dependent “discount” factor d(t) and the benefit derived from assigning the good at time t to agent e is the product of d(t) and v(e). For instance, the special case when d(t) is decreasing captures the natural tension between selling early and waiting to maximize the value of the agent receiving the good. We provide nearly matching logarithmic upper and lower bounds for this problem, and show a constant-competitive algorithm when the expected optimum is known in advance.

Abstract:

A variety of modern bioinformatics and systems biology problems can be approached systematically from an optimization point of view. This talk will focus on protein side-chain prediction, protein structural alignment, structure determination from X-ray diffraction measurements, and metabolic systems analysis and design. To solve these problems, we have employed machinery from linear algebra, dynamic programming, combinatorial optimization, and mixed-integer nonlinear programming. Many of the underlying biological problems are purely continuous in nature but have, to this date, been approached mostly via combinatorial optimization algorithms that are applied to discrete approximations. Other problems naturally present a strong and difficult combinatorial component.

Abstract:

In 1970, Folkman proved that for a given integer r and a graph G of order n there exists a graph H with the same clique number as G such that every r coloring of vertices of H yields at least one monochromatic copy of G. His proof gives no good bound on the order of graph H, i.e., the order of H is bounded by an iterated power function of n. In this talk we will give an alternative proof of Folkman’s theorem with the relatively small order of H bounded from above by O(n^3 log^3 n). This is joint work with Vojtech Rodl.

TITLE: What makes a good Steiner point?

Benoit Hudson
Toyota Technological Institute at Chicago

ABSTRACT:

The mesh refinement problem is to take an input geometry (defined by a set of points, curves, and surfaces), and output a set of points that both “respects'’ the geometry and has good “quality.'’ What it means for a tetrahedral mesh to respect curved surfaces is already interesting and will take some explaining. Even knowing what the goal is, mesh refinement algorithms typically are of the form: until the output is good enough, add points. But where should we add these additional Steiner points? And how do we know that the algorithm will stop? Most prior work is very specific about where to add points, and thus needs its own very specific proof that the algorithm ends.

In this talk, I will give a set of rules for choosing Steiner points. Any algorithm that follows my rules — as most previous algorithms do — will terminate. After hearing me out, you will know how to represent curved surfaces with linear elements, and you will be able to design your very own meshing algorithm with confidence.