Certified Algorithms: Worst-Case Analysis and Beyond

In this paper, we introduce the notion of a certified algorithm. Certified algorithms provide worst-case and beyond-worst-case performance guarantees. First, a ?-certified algorithm is also a ?-approximation algorithm - it finds a ?-approximation no matter what the input is. Second, it exactly solves ?-perturbation-resilient instances (?-perturbation-resilient instances model real-life instances). Additionally, certified algorithms have a number of other desirable properties: they solve both maximization and minimization versions of a problem (e.g. Max Cut and Min Uncut), solve weakly perturbation-resilient instances, and solve optimization problems with hard constraints. In the paper, we define certified algorithms, describe their properties, present a framework for designing certified algorithms, provide examples of certified algorithms for Max Cut/Min Uncut, Minimum Multiway Cut, k-medians and k-means. We also present some negative results

Nonuniform Graph Partitioning with Unrelated Weights

We give a bi-criteria approximation algorithm for the Minimum Nonuniform Partitioning problem, recently introduced by Krauthgamer, Naor, Schwartz and Talwar (2014). In this problem, we are given a graph $G=(V,E)$ on $n$ vertices and $k$ numbers $\rho_1,\dots, \rho_k$. The goal is to partition the graph into $k$ disjoint sets $P_1,\dots, P_k$ satisfying $|P_i|\leq \rho_i n$ so as to minimize the number of edges cut by the partition. Our algorithm has an approximation ratio of $O(\sqrt{\log n \log k})$ for general graphs, and an $O(1)$ approximation for graphs with excluded minors. This is an improvement upon the $O(\log n)$ algorithm of Krauthgamer, Naor, Schwartz and Talwar (2014). Our approximation ratio matches the best known ratio for the Minimum (Uniform) $k$-Partitioning problem. We extend our results to the case of "unrelated weights" and to the case of "unrelated $d$-dimensional weights". In the former case, different vertices may have different weights and the weight of a vertex may depend on the set $P_i$ the vertex is assigned to. In the latter case, each vertex $u$ has a $d$-dimensional weight $r(u,i) = (r_1(u,i), \dots, r_d(u,i))$ if $u$ is assigned to $P_i$. Each set $P_i$ has a $d$-dimensional capacity $c(i) = (c_1(i),\dots, c_d(i))$. The goal is to find a partition such that $\sum_{u\in {P_i}} r(u,i) \leq c(i)$ coordinate-wise

Bilu-Linial Stable Instances of Max Cut and Minimum Multiway Cut

We investigate the notion of stability proposed by Bilu and Linial. We obtain an exact polynomial-time algorithm for $\gamma$-stable Max Cut instances with $\gamma \geq c\sqrt{\log n}\log\log n$ for some absolute constant $c > 0$. Our algorithm is robust: it never returns an incorrect answer; if the instance is $\gamma$-stable, it finds the maximum cut, otherwise, it either finds the maximum cut or certifies that the instance is not $\gamma$-stable. We prove that there is no robust polynomial-time algorithm for $\gamma$-stable instances of Max Cut when $\gamma < \alpha_{SC}(n/2)$, where $\alpha_{SC}$ is the best approximation factor for Sparsest Cut with non-uniform demands. Our algorithm is based on semidefinite programming. We show that the standard SDP relaxation for Max Cut (with $\ell_2^2$ triangle inequalities) is integral if $\gamma \geq D_{\ell_2^2\to \ell_1}(n)$, where $D_{\ell_2^2\to \ell_1}(n)$ is the least distortion with which every $n$ point metric space of negative type embeds into $\ell_1$. On the negative side, we show that the SDP relaxation is not integral when $\gamma < D_{\ell_2^2\to \ell_1}(n/2)$. Moreover, there is no tractable convex relaxation for $\gamma$-stable instances of Max Cut when $\gamma < \alpha_{SC}(n/2)$. That suggests that solving $\gamma$-stable instances with $\gamma =o(\sqrt{\log n})$ might be difficult or impossible. Our results significantly improve previously known results. The best previously known algorithm for $\gamma$-stable instances of Max Cut required that $\gamma \geq c\sqrt{n}$ (for some $c > 0$) [Bilu, Daniely, Linial, and Saks]. No hardness results were known for the problem. Additionally, we present an algorithm for 4-stable instances of Minimum Multiway Cut. We also study a relaxed notion of weak stability.Comment: 24 page

How to Play Unique Games against a Semi-Random Adversary

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. In our semi-random model, one of the steps is random, and all other steps are adversarial. 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. Our result holds only for epsilon less than some absolute constant. We prove that if epsilon > 1/2, then the problem is hard in one of the models, the result assumes the 2-to-2 conjecture. Finally, we study semi-random instances of Unique Games that are at most (1-epsilon) satisfiable. We present an algorithm that with high probability, 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

Eastern Borderlands as Europe-Makers: (How) Can neighbours redefine the EU?

A general and strategic effect of EU’s Association Agreements and DCFTAs with Georgia, Moldova and Ukraine has been the extension of the concept of Europe and its wider opening to neighborhoods and margins. It is on this basis that a European normative order can be differentiated from both the ‘Russian world’ and Eurasian geopolitical space. However this paper argues that the process of association is not a unilateral move, but a multilateral and reciprocal development; it is a way for Europe to know more about itself, and to politically redefine itself. The neighbourhood policy causes controversial effects on the EU. On the one hand, it consolidates the liberal minded groups within European societies eager to see the EU as a promoter of values of freedom and civic liberties to be projected eastwards and defended in EU’s neighborhood. On the other hand, the problems of practical implementation tend to solidify sceptical groups in both the EU and its associated neighbours that contest not only the deepening of EU’s engagement with Ukraine, but EU ’s normative project as a whole. The implementation of the joint strategy of the EU and its close neighbours faces a challenge of finding a proper balance between two dominant – yet to a large extent contradictory – approaches. One consists of capitalizing on these countries’ status as victims of Russia’s policies, countries whose very existence is under threat, which implies support and help from the EU. Another, requiring much more consistent efforts, is for the associated neighbouring states to emerge as positive showcases of transition, and useful partners contributing not only to the transformation process in post-Soviet area, but also to EU’s and NATO’s security. The recent three years made clear that the former alone does not guarantee to Ukraine, Georgia or Moldova a fully-fledged European voice

A bi-criteria approximation algorithm for kk Means

We consider the classical $k$-means clustering problem in the setting bi-criteria approximation, in which an algoithm is allowed to output $\beta k > k$ clusters, and must produce a clustering with cost at most $\alpha$ times the to the cost of the optimal set of $k$ clusters. We argue that this approach is natural in many settings, for which the exact number of clusters is a priori unknown, or unimportant up to a constant factor. We give new bi-criteria approximation algorithms, based on linear programming and local search, respectively, which attain a guarantee $\alpha(\beta)$ depending on the number $\beta k$ of clusters that may be opened. Our gurantee $\alpha(\beta)$ is always at most $9 + \epsilon$ and improves rapidly with $\beta$ (for example: $\alpha(2)<2.59$, and $\alpha(3) < 1.4$). Moreover, our algorithms have only polynomial dependence on the dimension of the input data, and so are applicable in high-dimensional settings

Biopolitical conservatism and “pastoral power”: a Russia – Georgia meeting point.

The paper applies the concept of biopolitics to the analysis of Russia's relations with Georgia. It highlights the centrality of Orthodoxy for Russia's "soft power" and religious diplomacy