On f- and h- vectors of relative simplicial complexes

A relative simplicial complex is a collection of sets of the form $\Delta \setminus \Gamma$, where $\Gamma \subset \Delta$ are simplicial complexes. Relative complexes played key roles in recent advances in algebraic, geometric, and topological combinatorics but, in contrast to simplicial complexes, little is known about their general combinatorial structure. In this paper, we address a basic question in this direction and give a characterization of $f$-vectors of relative (multi)complexes on a ground set of fixed size. On the algebraic side, this yields a characterization of Hilbert functions of quotients of homogeneous ideals over polynomial rings with a fixed number of indeterminates. Moreover, we characterize $h$-vectors of fully Cohen--Macaulay relative complexes as well as $h$-vectors of Cohen--Macaulay relative complexes with minimal faces of given dimensions. The latter resolves a question of Bj\"orner.Comment: accepted for publication in Algebraic Combinatoric

Market Equilibrium with Transaction Costs

Identical products being sold at different prices in different locations is a common phenomenon. Price differences might occur due to various reasons such as shipping costs, trade restrictions and price discrimination. To model such scenarios, we supplement the classical Fisher model of a market by introducing {\em transaction costs}. For every buyer $i$ and every good $j$, there is a transaction cost of \cij; if the price of good $j$ is $p_j$, then the cost to the buyer $i$ {\em per unit} of $j$ is p_j + \cij. This allows the same good to be sold at different (effective) prices to different buyers. We provide a combinatorial algorithm that computes $\epsilon$-approximate equilibrium prices and allocations in $O\left(\frac{1}{\epsilon}(n+\log{m})mn\log(B/\epsilon)\right)$ operations - where $m$ is the number goods, $n$ is the number of buyers and $B$ is the sum of the budgets of all the buyers

Graph Isomorphism and the Lasserre Hierarchy

In this paper we show lower bounds for a certain large class of algorithms solving the Graph Isomorphism problem, even on expander graph instances. Spielman [25] shows an algorithm for isomorphism of strongly regular expander graphs that runs in time exp(O(n^(1/3)) (this bound was recently improved to expf O(n^(1/5) [5]). It has since been an open question to remove the requirement that the graph be strongly regular. Recent algorithmic results show that for many problems the Lasserre hierarchy works surprisingly well when the underlying graph has expansion properties. Moreover, recent work of Atserias and Maneva [3] shows that k rounds of the Lasserre hierarchy is a generalization of the k-dimensional Weisfeiler-Lehman algorithm for Graph Isomorphism. These two facts combined make the Lasserre hierarchy a good candidate for solving graph isomorphism on expander graphs. Our main result rules out this promising direction by showing that even Omega(n) rounds of the Lasserre semidefinite program hierarchy fail to solve the Graph Isomorphism problem even on expander graphs.Comment: 22 pages, 3 figures, submitted to CC

Computation of the Lovasz Theta Function for Circulant Graphs

The Lovasz theta function (G) of a graph G has attracted a lot of attention for its connection with diverse issues, such as communicating without errors and computing large cliques in graphs. Indeed this function enjoys the remarkable property of being computable in polynomial time, despite being sandwitched between clique and chromatic number, two well known hard to compute quantities. In this paper we deal with the computation of the Lovasz function of certain circulant graphs, i.e., graphs whose adjacency matrix is circulant. Such graphs are important for both theoretical and practical reasons, and indeed arise in many dierent contests. The simplest circulant graph is the cycle; for the cycle, Lovasz showed a simple formula expressing the value of the theta function. We consider the theta function of circulant graphs which can be viewed as the super-position of two cycles, i.e., circulant graphs of degree 4. We invertigate the possibility to take advantage of the specis structure of the circulants in oreder to achieve higher eciency. For a circulant graph Cn;j on n vertices and with a chord length j, 2 j bn=2c, we propose an O(j) time algorithm to compute (Cn;j ) if j is odd and an O(n=j) time algorithm if j is even. This is a signicant improvement over the best known algorithms for the theta function computation for general graphs which take O(n4) time. We also derive conditions under which (Cn;j ) can be computed in O(1) time

Ubicrawler: a scalable fully distributed web crawler

We present the design and implementation of UbiCrawler, a scalable distributed web crawler, and we analyze its performance. The main features of UbiCrawler are platform independence, fault tolerance, a very effective assignment function for partitioning the domain to crawl, and more in general the complete decentralization of every task

Market Equilibrium in Exchange Economies with Some Families of Concave Utility Functions

We present explicit convex programs which characterize the equilibrium for certain additively separable utility functions and CES functions. These include some CES utility functions that do not satisfy weak gross substitutability.Exchange economy, computation of equilibria, convex feasibility problem

Computing Equilibrium in Matching Markets

Market equilibria of matching markets offer an intuitive and fair solution for matching problems without money with agents who have preferences over the items. Such a matching market can be viewed as a variation of Fisher market, albeit with rather peculiar preferences of agents. These preferences can be described by piece-wise linear concave (PLC) functions, which however, are not separable (due to each agent only asking for one item), are not monotone, and do not satisfy the gross substitute property-- increase in price of an item can result in increased demand for the item. Devanur and Kannan in FOCS 08 showed that market clearing prices can be found in polynomial time in markets with fixed number of items and general PLC preferences. They also consider Fischer markets with fixed number of agents (instead of fixed number of items), and give a polynomial time algorithm for this case if preferences are separable functions of the items, in addition to being PLC functions. Our main result is a polynomial time algorithm for finding market clearing prices in matching markets with fixed number of different agent preferences, despite that the utility corresponding to matching markets is not separable. We also give a simpler algorithm for the case of matching markets with fixed number of different items

Small Worlds

In this tutorial we present some basic ideas behind the notion of Small World. We review the state-of-the-art in the field, and put emphasis on some recent developments, in connection with analyzing the structure of the Web.-