## TechTalks from event: FOCS 2011

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## 7A

• How to Play Unique Games Against a Semi-Random Adversary Authors: Alexandra Kolla and Konstantin Makarychev and Yury Makarychev
In this paper, we study the average case complexity of the Unique Games problem. We propose a natural semi-random model, in which a unique game instance is generated in several steps. First an adversary selects a completely satisfiable instance of Unique Games, then she chooses an epsilon fraction of all edges, and finally replaces ("corrupts") the constraints corresponding to these edges with new constraints. If all steps are adversarial, the adversary can obtain any (1-epsilon) satisfiable instance, so then the problem is as hard as in the worst case. We show that known algorithms for unique games (in particular, all algorithms that use the standard SDP relaxation) fail to solve semi-random instances of Unique Games. We present an algorithm that with high probability finds a solution satisfying a (1-delta) fraction of all constraints in semi-random instances (we require that the average degree of the graph is Omega(log k)). To this end, we consider a new non-standard SDP program for Unique Games, which is not a relaxation for the problem, and show how to analyze it. We present a new rounding scheme that simultaneously uses SDP and LP solutions, which we believe is of independent interest. Finally, we study semi-random instances of Unique Games that are at most (1-epsilon) satisfiable. We present an algorithm that distinguishes between the case when the instance is a semi-random instance and the case when the instance is an (arbitrary) (1-delta)-satisfiable instance if epsilon > c delta (for some absolute constant c).
• The Grothendieck constant is strictly smaller than Krivine's bound Authors: Mark Braverman, Konstantin Makarychev, Yury Makarychev, Assaf Naor
We prove that $K_G<\frac{\pi}{2\log\left(1+\sqrt{2}\right)}$, where $K_G$ is the Grothendieck constant.
• A Parallel Approximation Algorithm for Positive Semidefinite Programming Authors: Rahul Jain and Penghui Yao
Positive semidefinite programs are an important subclass of semidefinite programs in which all matrices involved in the specification of the problem are positive semidefinite and all scalars involved are non-negative. We present a parallel algorithm, which given an instance of a positive semidefinite program of size N and an approximation factor eps > 0, runs in (parallel) time poly(1/eps) polylog(N), using poly(N) processors, and outputs a value which is within multiplicative factor of (1+eps) to the optimal. Our result generalizes analogous result of Luby and Nisan [1993] for positive linear programs and our algorithm is inspired by their algorithm.
• Rounding Semidefinite Programming Hierarchies via Global Correlation Authors: Boaz Barak and Prasad Raghavendra and David Steurer
We show a new way to round vector solutions of semidefinite programming (SDP) hierarchies into integral solutions, based on a connection between these hierarchies and the spectrum of the input graph. We demonstrate the utility of our method by providing a new SDP-hierarchy based algorithm for Unique Games. Our algorithm matches the performance of the recent algorithm of Arora, Barak and Steurer (FOCS 2010) in the worst-case, but is shown to run in polynomial time on a richer family of instances, thus ruling out more possibilities for hard instances for the Unique Games Conjecture. Specifically, we give a rounding algorithm for $O(r)$ levels of the Lasserre hierarchy that finds a good integral solution as long as, very roughly speaking, the average correlation between vectors in the SDP solution is at least $1/r$. In the case of Unique Games, the latter condition is implied by having at most $r$ large eigenvalues in the constraint graph. This improves upon prior works that required the potentially stronger condition of a bound on the number of eigenvalues in the \emph{label extended graph}. Our algorithm actually requires less than the $n^{O(r)}$ constraints specified by the $r^{th}$ level of the Lasserre hierarchy, and in particular $r$ rounds of our program can be evaluated in time $2^{O(r)}\mathrm{poly}(n)$.
• Lasserre Hierarchy, Higher Eigenvalues, and Approximation Schemes for Graph Partitioning and Quadratic Integer Programming with PSD Objectives Authors: Venkatesan Guruswami and Ali Kemal Sinop
We present an approximation scheme for optimizing certain Quadratic Integer Programming problems with positive semidefinite objective functions and global linear constraints. This framework includes well known graph problems such as Uniform sparsest cut, Minimum graph bisection, and Small Set expansion, as well as the Unique Games problem. These problems are notorious for the existence of huge gaps between the known algorithmic results and NP-hardness results. Our algorithm is based on rounding semidefinite programs from the Lasserre hierarchy, and the analysis uses bounds for low-rank approximations of a matrix in Frobenius norm using columns of the matrix.For all the above graph problems, we give an algorithm running in time $n^{O(r/\eps^2)}$ with approximation ratio $\frac{1+\eps}{\min\{1,\lambda_r\}}$, where $\lambda_r$ is the $r$'th smallest eigenvalue of the normalized graph Laplacian $\Lnorm$. In the case of graph bisection and small set expansion, the number of vertices in the cut is within lower-order terms of the stipulated bound. Our results imply $(1+O(\eps))$ factor approximation in time $n^{O(r^\ast)}$ where $r^\ast$ is the number of eigenvalues of $\Lnorm$ smaller $1-\eps$. This perhaps gives some indication as to why even showing mere APX-hardness for these problems has been elusive, since the reduction must produce graphs with a slowly growing spectrum (and classes like planar graphs which are known to have such a spectral property often admit good algorithms owing to their nice structure). For Unique Games, we give a factor $(1+\frac{2+\eps}{\lambda_r})$ approximation for minimizing the number of unsatisfied constraints in $n^{O(r/\eps)}$ time. This improves an earlier bound for solving Unique Games on expanders, and also shows that Lasserre SDPs are powerful enough to solve well-known integrality gap instances for the basic SDP. We also give an algorithm for independent sets in graphs that performs well when the Laplacian does not have too many eigenvalues bigger than $1+o(1)$.

## 7B

• Markov Layout Authors: Flavio Chierichetti and Ravi Kumar and Prabhakar Raghavan
Consider the following problem of laying out a set of $n$ images that match a query onto the nodes of a $\sqrt{n}\times\sqrt{n}$ grid. We are given a score for each image, as well as the distribution of patterns by which a user's eye scans the nodes of the grid and we wish to maximize the expected total score of images selected by the user. This is a special case of the \emph{Markov layout problem}, in which we are given a Markov chain $M$ together with a set o f objects to be placed at the states of the Markov chain. Each object has a utility to the user if viewed, as well as a stopping probability with which the user ceases to look further at objects. We point out that this layout problem is prototypical in a number of applications in web search and advertising, particularly in the emerging genre of search results pages from major engines. In a different class of applications, the states of the Markov chain are web pages at a publishers website and the objects are advertisements. In this paper we study the approximability of the Markov layout problem. Our main result is an $O(\log n)$ approximation algorithm for the most general version of the problem. The core idea behind the algorithm is to transform an optimization problem over partial permutations into an optimization problem over sets by losing a logarithmic factor in approximation; the latter problem is then shown to be submodular with two matroid constraints, which admits a constant-factor approximation. In contrast, we also show the problem is APX-hard via a reduction from {\sc Cubic Max-Bisection}. We then study harder variants of the problem in which no \emph{gaps} --- states of $M$ with no object placed on them --- are allowed. By exploiting the geometry, we obtain an $O(\log^{3/2} n)$ approximation algorithm when the digraph underlying $M$ is a grid and an $O(\log n)$ approximation algorithm when it is a tree. These special cases are especially appropriate for our applications.
• Limitations of Randomized Mechanisms for Combinatorial Auctions Authors: Shaddin Dughmi and Jan Vondrak
The design of computationally efficient and incentive compatible mechanisms that solve or approximate fundamental resource allocation problems is the main goal of algorithmic mechanism design. A central example in both theory and practice is welfare-maximization in combinatorial auctions. Recently, a randomized mechanism has been discovered for combinatorial auctions that is truthful in expectation and guarantees a $(1-1/e)$-approximation to the optimal social welfare when players have coverage valuations \cite{DRY11}. This approximation ratio is the best possible even for non-truthful algorithms, assuming $P \neq NP$ \cite{KLMM05}. Given the recent sequence of negative results for combinatorial auctions under more restrictive notions of incentive compatibility \cite{DN07,BDFKMPSSU10,Dobzin11}, this development raises a natural question: Are truthful-in-expectation mechanisms compatible with polynomial-time approximation in a way that deterministic or universally truthful mechanisms are not? In particular, can polynomial-time truthful-in-expectation mechanisms guarantee a near-optimal approximation ratio for more general variants of combinatorial auctions? We prove that this is not the case. Specifically, the result of \cite{DRY11} cannot be extended to combinatorial auctions with submodular valuations in the value oracle model. (Absent strategic considerations, a $(1-1/e)$-approximation is still achievable in this setting \cite{V08}.) More precisely, we prove that there is a constant $\gamma>0$ such that there is no randomized mechanism that is truthful-in-expectation --- or even approximately truthful-in-expectation --- and guarantees an $m^{-\gamma}$-approximation to the optimal social welfare for combinatorial auctions with submodular valuations in the value oracle model. We also prove an analogous result for the flexible combinatorial public projects (CPP) problem, where a truthful-in-expectation $(1-1/e)$-approximation for coverage valuations has been recently developed \cite{Dughmi11}. We show that there is no truthful-in-expectation --- or even approximately truthful-in-expectation --- mechanism that achieves an $m^{-\gamma}$-approximation to the optimal social welfare for combinatorial public projects with submodular valuations in the value oracle model. Both our results present an unexpected separation between coverage functions and submodular functions, which does not occur for these problems without strategic considerations.
• Bayesian Combinatorial Auctions: Expanding Single Buyer Mechanisms to Many Buyers Authors: Saeed Alaei
For Bayesian combinatorial auctions, we present a general framework for reducing the mechanism design problem for many buyers to the mechanism design problem for one buyer. Our generic reduction works for any separable objective (e.g., welfare, revenue, etc) and any space of valuations (e.g. submodular, additive, etc) and any distribution of valuations as long as valuations of different buyers are distributed independently (not necessarily identically). Roughly speaking, we present two generic $n$-buyer mechanisms that use $1$-buyer mechanisms as black boxes. We show that if we have an $\alpha$-approximate $1$-buyer mechanism for each buyer\footnote{Note that we can use different $1$-buyer mechanisms for different buyers.} then our generic $n$-buyer mechanisms are $\frac{1}{2}\alpha$-approximation of the optimal $n$-buyer mechanism. Furthermore, if we have several copies of each item and no buyer ever needs more than $\frac{1}{k}$ of all copies of each item then our generic $n$-buyer mechanisms are $\gamma_k \alpha$-approximation of the optimal $n$-buyer mechanism where $\gamma_k \ge 1-\frac{1}{\sqrt{k+3}}$. Observe that $\gamma_k$ is at least $\frac{1}{2}$ and approaches $1$ as $k$ increases. Applications of our main theorem include the following improvements on results from the literature. For each of the following models we construct a $1$-buyer mechanism and then apply our generic expansion: For revenue maximization in combinatorial auctions with hard budget constraints, \cite{BGGM10} presented a $\frac{1}{4}$-approximate BIC mechanism for additive/correlated valuations and an $O(1)$-approximate\footnote{$O(1)=\frac{1}{96}$} sequential posted pricing mechanism for additive/independent valuations. We improve this to a $\gamma_k$-approximate BIC mechanism and a $\gamma_k (1-\frac{1}{e})$-approximate sequential posted pricing mechanism respectively. For revenue maximization in combinatorial auctions with unit demand buyers, \cite{CHMS10} presented a $\frac{1}{6.75}$-approximate sequential posted pricing mechanism. We improve this to a $\frac{1}{2} \gamma_k$ approximate sequential posted pricing mechanism. We also present a $\gamma_k$-approximate sequential posted pricing mechanism for unit-demand multi-unit auctions(homogeneous) with hard-budget constraints. Furthermore, our sequential posted pricing mechanisms assume no control or prior information about the order in which buyers arrive.
• Extreme-Value Theorems for Optimal Multidimensional Pricing Authors: Yang Cai and Constantinos Daskalakis
We provide a Polynomial Time Approximation Scheme for the {\em multi-dimensional unit-demand pricing problem}, when the buyer's values are independent (but not necessarily identically distributed.) For all $\epsilon>0$, we obtain a $(1+\epsilon)$-factor approximation to the optimal revenue in time polynomial, when the values are sampled from Monotone Hazard Rate (MHR) distributions, quasi-polynomial, when sampled from regular distributions, and polynomial in $n^{{\rm poly}(\log r)}$, when sampled from general distributions supported on a set $[u_{min}, r u_{min}]$. We also provide an additive PTAS for all bounded distributions.Our algorithms are based on novel extreme value theorems for MHR and regular distributions, and apply probabilistic techniques to understand the statistical properties of revenue distributions, as well as to reduce the size of the search space of the algorithm. As a byproduct of our techniques, we establish structural properties of optimal solutions. We show that, for all $\epsilon >0$, $g(1/\epsilon)$ distinct prices suffice to obtain a $(1+\epsilon)$-factor approximation to the optimal revenue for MHR distributions, where $g(1/\epsilon)$ is a quasi-linear function of $1/\epsilon$ that does not depend on the number of items. Similarly, for all $\epsilon>0$ and $n>0$, $g(1/\epsilon \cdot \log n)$ distinct prices suffice for regular distributions, where $n$ is the number of items and $g(\cdot)$ is a polynomial function. Finally, in the i.i.d. MHR case, we show that, as long as the number of items is a sufficiently large function of $1/\epsilon$, a single price suffices to achieve a $(1+\epsilon)$-factor approximation.
• Efficient computation of approximate pure Nash equilibria in congestion games Authors: Ioannis Caragiannis and Angelo Fanelli and Nick Gravin and Alexander Skopalik
Congestion games constitute an important class of games in which computing an exact or even approximate pure Nash equilibrium is in general {\sf PLS}-complete. We present a surprisingly simple polynomial-time algorithm that computes $O(1)$-approximate Nash equilibria in these games. In particular, for congestion games with linear latency functions, our algorithm computes $(2+\epsilon)$-approximate pure Nash equilibria in time polynomial in the number of players, the number of resources and $1/\epsilon$. It also applies to games with polynomial latency functions with constant maximum degree $d$; there, the approximation guarantee is $d^{O(d)}$. The algorithm essentially identifies a polynomially long sequence of best-response moves that lead to an approximate equilibrium; the existence of such short sequences is interesting in itself. These are the first positive algorithmic results for approximate equilibria in non-symmetric congestion games. We strengthen them further by proving that, for congestion games that deviate from our mild assumptions, computing $\rho$-approximate equilibria is {\sf PLS}-complete for any polynomial-time computable $\rho$.

## 8

• On Range Searching in the Group Model and Combinatorial Discrepancy Authors: Kasper Green Larsen
In this paper we establish an intimate connection between dynamic range searching in the group model and combinatorial discrepancy. Our result states that, for a broad class of range searching data structures (including all known upper bounds), it must hold that $t_u t_q = \Omega(\disc^2/\lg n)$ where $t_u$ is the worst case update time, $t_q$ the worst case query time and $\disc$ is the combinatorial discrepancy of the range searching problem in question. This relation immediately implies a whole range of exceptionally high and near-tight lower bounds for all of the basic range searching problems. We list a few of them in the following: \begin{itemize} \item For halfspace range searching in $d$-dimensional space, we get a lower bound of $t_u t_q = \Omega(n^{1-1/d}/\lg n)$. This comes within a $\lg n \lg \lg n$ factor of the best known upper bound. \item For orthogonal range searching in $d$-dimensional space, we get a lower bound of $t_u t_q = \Omega(\lg^{d-2+\mu(d)}n)$, where $\mu(d)>0$ is some small but strictly positive function of $d$. \item For ball range searching in $d$-dimensional space, we get a lower bound of $t_u t_q = \Omega(n^{1-1/d}/\lg n)$. \end{itemize} We note that the previous highest lower bound for any explicit problem, due to Patrascu [STOC'07], states that $t_q =\Omega((\lg n/\lg(\lg n+t_u))^2)$, which does however hold for a less restrictive class of data structures. Our result also has implications for the field of combinatorial discrepancy. Using textbook range searching solutions, we improve on the best known discrepancy upper bound for axis-aligned rectangles in dimensions $d \geq 3$.
• A Randomized Rounding Approach to the Traveling Salesman Problem Authors: Shayan Oveis Gharan and Amin Saberi and Mohit Singh
For some positive constant $\epsilon_0$, we give a $(\frac{3}{2}-\epsilon_0$)-approximation algorithm for the following problem: given a graph $G_0=(V,E_0)$, find the shortest tour that visits every vertex at least once. This is a special case of the metric traveling salesman problem when the underlying metric is defined by shortest path distances in $G_0$. The result improves on the $\frac{3}{2}$-approximation algorithm due to Christofides for this special case.Similar to Christofides, our algorithm finds a spanning tree whose cost is upper bounded by the optimum, then it finds the minimum cost Eulerian augmentation (or T-join) of that tree. The main difference is in the selection of the spanning tree. Except in certain cases where the solution of LP is nearly integral, we select the spanning tree randomly by sampling from a maximum entropy distribution defined by the linear programming relaxation. Despite the simplicity of the algorithm, the analysis builds on a variety of ideas such as properties of strongly Rayleigh measures from probability theory, graph theoretical results on the structure of near minimum cuts, and the integrality of the T-join polytope from polyhedral theory. Also, as a byproduct of our result, we show new properties of the near minimum cuts of any graph, which may be of independent interest.
• Approximating Graphic TSP by Matchings Authors: Tobias Moemke and Ola Svensson
We present a framework for approximating the metric TSP based on a novel use of matchings. Traditionally, matchings have been used to add edges in order to make a given graph Eulerian, whereas our approach also allows for the removal of certain edges leading to a decreased cost. For the TSP on graphic metrics (graph-TSP), the approach yields a 1.461-approximation algorithm with respect to the Held-Karp lower bound. For graph-TSP restricted to a class of graphs that contains degree three bounded and claw-free graphs, we show that the integrality gap of the Held-Karp relaxation matches the conjectured ratio 4/3. The framework allows for generalizations in a natural way and also leads to a 1.586-approximation algorithm for the traveling salesman path problem on graphic metrics where the start and end vertices are prespecified.