Searcharxiv⌕ Search

arXiv subjects

Konstantin Makarychev

Publications and source records attributed to Konstantin Makarychev.

53 records · Page 3Linked to original sources
arXiv2014

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 $ρ_1,\dots, ρ_k$. The goal is to partition the graph into $k$ disjoint sets $P_1,\dots, P_k$ satisfying $|P_i|\leq ρ_i n$ 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.

cs.DS↗
arXiv2014

Precedence-constrained Scheduling of Malleable Jobs with Preemption

Scheduling jobs with precedence constraints on a set of identical machines to minimize the total processing time (makespan) is a fundamental problem in combinatorial optimization. In practical settings such as cloud computing, jobs are often malleable, i.e., can be processed on multiple machines simultaneously. The instantaneous processing rate of a job is a non-decreasing function of the number of machines assigned to it (we call it the processing function). Previous research has focused on practically relevant concave processing functions, which obey the law of diminishing utility and generalize the classical (non-malleable) problem. Our main result is a $(2+ε)$-approximation algorithm for concave processing functions (for any $ε> 0$), which is the best possible under complexity theoretic assumptions. The approximation ratio improves to $(1 + ε)$ for the interesting and practically relevant special case of power functions, i.e., $p_j(z) = c_j \cdot z^γ$.

cs.DS↗
arXiv2014

Maximum Quadratic Assignment Problem: Reduction from Maximum Label Cover and LP-based Approximation Algorithm

We show that for every positive $ε> 0$, unless NP $\subset$ BPQP, it is impossible to approximate the maximum quadratic assignment problem within a factor better than $2^{\log^{1-ε} n}$ by a reduction from the maximum label cover problem. Our result also implies that Approximate Graph Isomorphism is not robust and is in fact, $1 - ε$ vs $ε$ hard assuming the Unique Games Conjecture. Then, we present an $O(\sqrt{n})$-approximation algorithm for the problem based on rounding of the linear programming relaxation often used in the state of the art exact algorithms.

cs.CC↗
arXiv2014

Online Algorithms for Machine Minimization

In this paper, we consider the online version of the machine minimization problem (introduced by Chuzhoy et al., FOCS 2004), where the goal is to schedule a set of jobs with release times, deadlines, and processing lengths on a minimum number of identical machines. Since the online problem has strong lower bounds if all the job parameters are arbitrary, we focus on jobs with uniform length. Our main result is a complete resolution of the deterministic complexity of this problem by showing that a competitive ratio of $e$ is achievable and optimal, thereby improving upon existing lower and upper bounds of 2.09 and 5.2 respectively. We also give a constant-competitive online algorithm for the case of uniform deadlines (but arbitrary job lengths); to the best of our knowledge, no such algorithm was known previously. Finally, we consider the complimentary problem of throughput maximization where the goal is to maximize the sum of weights of scheduled jobs on a fixed set of identical machines (introduced by Bar-Noy et al. STOC 1999). We give a randomized online algorithm for this problem with a competitive ratio of e/e-1; previous results achieved this bound only for the case of a single machine or in the limit of an infinite number of machines.

cs.DM↗
arXiv2013

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 $γ$-stable Max Cut instances with $γ\geq c\sqrt{\log n}\log\log n$ for some absolute constant $c > 0$. Our algorithm is robust: it never returns an incorrect answer; if the instance is $γ$-stable, it finds the maximum cut, otherwise, it either finds the maximum cut or certifies that the instance is not $γ$-stable. We prove that there is no robust polynomial-time algorithm for $γ$-stable instances of Max Cut when $γ< α_{SC}(n/2)$, where $α_{SC}$ 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 $γ\geq D_{\ell_2^2\to \ell_1}(n)$, 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 $γ< D_{\ell_2^2\to \ell_1}(n/2)$. Moreover, there is no tractable convex relaxation for $γ$-stable instances of Max Cut when $γ< α_{SC}(n/2)$. That suggests that solving $γ$-stable instances with $γ=o(\sqrt{\log n})$ might be difficult or impossible. Our results significantly improve previously known results. The best previously known algorithm for $γ$-stable instances of Max Cut required that $γ\geq c\sqrt{n}$ (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.

cs.DS↗
arXiv2013

Approximation Algorithm for Sparsest k-Partitioning

Given a graph $G$, the sparsest-cut problem asks to find the set of vertices $S$ which has the least expansion defined as $$ϕ_G(S) := \frac{w(E(S,\bar{S}))}{\min \set{w(S), w(\bar{S})}}, $$ where $w$ is the total edge weight of a subset. Here we study the natural generalization of this problem: given an integer $k$, compute a $k$-partition $\set{P_1, \ldots, P_k}$ of the vertex set so as to minimize $$ ϕ_k(\set{P_1, \ldots, P_k}) := \max_i ϕ_G(P_i). $$ Our main result is a polynomial time bi-criteria approximation algorithm which outputs a $(1 - \e)k$-partition of the vertex set such that each piece has expansion at most $O_{\varepsilon}(\sqrt{\log n \log k})$ times $OPT$. We also study balanced versions of this problem.

cs.DS↗
arXiv2013

Local Search is Better than Random Assignment for Bounded Occurrence Ordering k-CSPs

We prove that the Bounded Occurrence Ordering k-CSP Problem is not approximation resistant. We give a very simple local search algorithm that always performs better than the random assignment algorithm. Specifically, the expected value of the solution returned by the algorithm is at least Alg > Avg + a(B,k) (Opt - Avg), where "Opt" is the value of the optimal solution; "Avg" is the expected value of the random solution; and a(B,k)=Omega_k(B^{-(k+O(1))} is a parameter depending only on "k" (the arity of the CSP) and "B" (the maximum number of times each variable is used in constraints). The question whether bounded occurrence ordering k-CSPs are approximation resistant was raised by Guruswami and Zhou (APPROX 2012) who recently showed that bounded occurrence 3-CSPs and "monotone" k-CSPs admit a non-trivial approximation.

cs.DS↗
arXiv2012

Approximation Algorithm for Non-Boolean MAX k-CSP

In this paper, we present a randomized polynomial-time approximation algorithm for k-CSPd. In k-CSPd, we are given a set of predicates of arity k over an alphabet of size d. Our goal is to find an assignment that maximizes the number of satisfied constraints. Our algorithm has approximation factor Omega(kd/d^k) (when k > Ω(log d)). This bound is asymptotically optimal assuming the Unique Games Conjecture. The best previously known algorithm has approximation factor Omega(k log d/d^k). We also give an approximation algorithm for the boolean MAX k-CSP2 problem with a slightly improved approximation guarantee.

cs.DS↗
arXiv2012

Approximation Algorithms for Semi-random Graph Partitioning Problems

In this paper, we propose and study a new semi-random model for graph partitioning problems. We believe that it captures many properties of real--world instances. The model is more flexible than the semi-random model of Feige and Kilian and planted random model of Bui, Chaudhuri, Leighton and Sipser. We develop a general framework for solving semi-random instances and apply it to several problems of interest. We present constant factor bi-criteria approximation algorithms for semi-random instances of the Balanced Cut, Multicut, Min Uncut, Sparsest Cut and Small Set Expansion problems. We also show how to almost recover the optimal solution if the instance satisfies an additional expanding condition. Our algorithms work in a wider range of parameters than most algorithms for previously studied random and semi-random models. Additionally, we study a new planted algebraic expander model and develop constant factor bi-criteria approximation algorithms for graph partitioning problems in this model.

cs.DS↗
arXiv2011

Min-Max Graph Partitioning and Small Set Expansion

We study graph partitioning problems from a min-max perspective, in which an input graph on n vertices should be partitioned into k parts, and the objective is to minimize the maximum number of edges leaving a single part. The two main versions we consider are where the k parts need to be of equal-size, and where they must separate a set of k given terminals. We consider a common generalization of these two problems, and design for it an $O(\sqrt{\log n\log k})$-approximation algorithm. This improves over an $O(\log^2 n)$ approximation for the second version, and roughly $O(k\log n)$ approximation for the first version that follows from other previous work. We also give an improved O(1)-approximation algorithm for graphs that exclude any fixed minor. Our algorithm uses a new procedure for solving the Small-Set Expansion problem. In this problem, we are given a graph G and the goal is to find a non-empty set $S\subseteq V$ of size $|S| \leq ρn$ with minimum edge-expansion. We give an $O(\sqrt{\log{n}\log{(1/ρ)}})$ bicriteria approximation algorithm for the general case of Small-Set Expansion, and O(1) approximation algorithm for graphs that exclude any fixed minor.

cs.DS↗
arXiv2011

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.

cs.DS↗
arXiv2011

On Parsimonious Explanations for 2-D Tree- and Linearly-Ordered Data

This paper studies the "explanation problem" for tree- and linearly-ordered array data, a problem motivated by database applications and recently solved for the one-dimensional tree-ordered case. In this paper, one is given a matrix A whose rows and columns have semantics: special subsets of the rows and special subsets of the columns are meaningful, others are not. A submatrix in A is said to be meaningful if and only if it is the cross product of a meaningful row subset and a meaningful column subset, in which case we call it an "allowed rectangle." The goal is to "explain" A as a sparse sum of weighted allowed rectangles. Specifically, we wish to find as few weighted allowed rectangles as possible such that, for all i,j, a_{ij} equals the sum of the weights of all rectangles which include cell (i,j). In this paper we consider the natural cases in which the matrix dimensions are tree-ordered or linearly-ordered. In the tree-ordered case, we are given a rooted tree T1 whose leaves are the rows of A and another, T2, whose leaves are the columns. Nodes of the trees correspond in an obvious way to the sets of their leaf descendants. In the linearly-ordered case, a set of rows or columns is meaningful if and only if it is contiguous. For tree-ordered data, we prove the explanation problem NP-Hard and give a randomized 2-approximation algorithm for it. For linearly-ordered data, we prove the explanation problem NP-Hard and give a 2.56-approximation algorithm. To our knowledge, these are the first results for the problem of sparsely and exactly representing matrices by weighted rectangles.

cs.DS↗
arXiv2010

Improved Approximation for the Directed Spanner Problem

We prove that the size of the sparsest directed k-spanner of a graph can be approximated in polynomial time to within a factor of $\tilde{O}(\sqrt{n})$, for all k >= 3. This improves the $\tilde{O}(n^{2/3})$-approximation recently shown by Dinitz and Krauthgamer.

cs.DS↗
arXiv2010

Metric Extension Operators, Vertex Sparsifiers and Lipschitz Extendability

We study vertex cut and flow sparsifiers that were recently introduced by Moitra, and Leighton and Moitra. We improve and generalize their results. We give a new polynomial-time algorithm for constructing O(log k / log log k) cut and flow sparsifiers, matching the best existential upper bound on the quality of a sparsifier, and improving the previous algorithmic upper bound of O(log^2 k / log log k). We show that flow sparsifiers can be obtained from linear operators approximating minimum metric extensions. We introduce the notion of (linear) metric extension operators, prove that they exist, and give an exact polynomial-time algorithm for finding optimal operators. We then establish a direct connection between flow and cut sparsifiers and Lipschitz extendability of maps in Banach spaces, a notion studied in functional analysis since 1930s. Using this connection, we prove a lower bound of Omega(sqrt{log k/log log k}) for flow sparsifiers and a lower bound of Omega(sqrt{log k}/log log k) for cut sparsifiers. We show that if a certain open question posed by Ball in 1992 has a positive answer, then there exist \tilde O(sqrt{log k}) cut sparsifiers. On the other hand, any lower bound on cut sparsifiers better than \tilde Omega(sqrt{log k}) would imply a negative answer to this question.

cs.DS↗
arXiv2009

How to Play Unique Games on Expanders

In this note we improve a recent result by Arora, Khot, Kolla, Steurer, Tulsiani, and Vishnoi on solving the Unique Games problem on expanders. Given a $(1-\varepsilon)$-satisfiable instance of Unique Games with the constraint graph $G$, our algorithm finds an assignment satisfying at least a $1- C \varepsilon/h_G$ fraction of all constraints if $\varepsilon < c λ_G$ where $h_G$ is the edge expansion of $G$, $λ_G$ is the second smallest eigenvalue of the Laplacian of $G$, and $C$ and $c$ are some absolute constants.

cs.DS↗