10 research outputs found
Robust Approximation of Temporal CSP
A temporal constraint language G is a set of relations with first-order definitions in (Q; = 0, given a (1-e)-satisfiable instance of CSP(G), we can compute an assignment that satisfies at least a (1-f(e))-fraction of constraints in polynomial time. Here, f(e) is some function satisfying f(0)=0 and f(e) goes 0 as e goes 0.
Firstly, we give a qualitative characterization of robust approximability: Assuming the Unique Games Conjecture, we give a
necessary and sufficient condition on G under which CSP(G) admits
robust approximation. Secondly, we give a quantitative characterization of robust approximability: Assuming the Unique Games
Conjecture, we precisely characterize how f(e) depends on e for each
G. We show that our robust approximation algorithms can be run in
almost linear time
Streaming Complexity of Approximating Max 2CSP and Max Acyclic Subgraph
We study the complexity of estimating the optimum value of a Boolean 2CSP (arity two constraint satisfaction problem) in the single-pass streaming setting, where the algorithm is presented the constraints in an arbitrary order. We give a streaming algorithm to estimate the optimum within a factor approaching 2/5 using logarithmic space, with high probability. This beats the trivial factor 1/4 estimate obtained by simply outputting 1/4-th of the total number of constraints.
The inspiration for our work is a lower bound of Kapralov, Khanna, and Sudan (SODA\u2715) who showed that a similar trivial estimate (of factor 1/2) is the best one can do for Max CUT. This lower bound implies that beating a factor 1/2 for Max DICUT (a special case of Max 2CSP), in particular, to distinguish between the case when the optimum is m/2 versus when it is at most (1/4+eps)m, where m is the total number of edges, requires polynomial space. We complement this hardness result by showing that for DICUT, one can distinguish between the case in which the optimum exceeds (1/2+eps)m and the case in which it is close to m/4.
We also prove that estimating the size of the maximum acyclic subgraph of a directed graph, when its edges are presented in a single-pass stream, within a factor better than 7/8 requires polynomial space
Robustly Solvable Constraint Satisfaction Problems
An algorithm for a constraint satisfaction problem is called robust if it
outputs an assignment satisfying at least -fraction of the
constraints given a -satisfiable instance, where
as . Guruswami and
Zhou conjectured a characterization of constraint languages for which the
corresponding constraint satisfaction problem admits an efficient robust
algorithm. This paper confirms their conjecture
Hardness of robust graph isomorphism, Lasserre gaps, and asymmetry of random graphs
Building on work of Cai, F\"urer, and Immerman \cite{CFI92}, we show two
hardness results for the Graph Isomorphism problem. First, we show that there
are pairs of nonisomorphic -vertex graphs and such that any
sum-of-squares (SOS) proof of nonisomorphism requires degree . In
other words, we show an -round integrality gap for the Lasserre SDP
relaxation. In fact, we show this for pairs and which are not even
-isomorphic. (Here we say that two -vertex, -edge graphs
and are -isomorphic if there is a bijection between their
vertices which preserves at least edges.) Our second result is that
under the {\sc R3XOR} Hypothesis \cite{Fei02} (and also any of a class of
hypotheses which generalize the {\sc R3XOR} Hypothesis), the \emph{robust}
Graph Isomorphism problem is hard. I.e.\ for every , there is no
efficient algorithm which can distinguish graph pairs which are
-isomorphic from pairs which are not even
-isomorphic for some universal constant . Along the
way we prove a robust asymmetry result for random graphs and hypergraphs which
may be of independent interest
Approximation Algorithms for CSPs
In this survey, we offer an overview of approximation algorithms for constraint satisfaction problems (CSPs) - we describe main results and discuss various techniques used for solving CSPs
A Blueprint for Semidefinite Relaxations of Binary-Constrained Quadratic Programs Computing tight bounds on NP-hard problems using ADMM
This thesis looks at the solution techniques of two NP-hard, large scale problems, the quadratic assignment problem, QAP, and the side chain positioning, SCP, problem. We summarize existing approaches from and look at the two problems in a unified way using a binary-constrained quadratic program, BCQP. We show how to obtain upper and lower bounds for the BCQP by formulating the semidefinite programming (SDP) relaxation and applying the Alternating Direction Method of Multipliers (ADMM) algorithm to solve it. By unifying the two problems under the umbrella of the BCQP, we better understand why the method is so successful for these two problems and obtain a blueprint for applying ADMM to similar combinatorial optimization problems
SDPs and Robust Satisfiability of Promise CSP
For a constraint satisfaction problem (CSP), a robust satisfaction algorithm
is one that outputs an assignment satisfying most of the constraints on
instances that are near-satisfiable. It is known that the CSPs that admit
efficient robust satisfaction algorithms are precisely those of bounded width,
i.e., CSPs whose satisfiability can be checked by a simple local consistency
algorithm (eg., 2-SAT or Horn-SAT in the Boolean case). While the exact
satisfiability of a bounded width CSP can be checked by combinatorial
algorithms, the robust algorithm is based on rounding a canonical Semidefinite
programming(SDP) relaxation.
In this work, we initiate the study of robust satisfaction algorithms for
promise CSPs, which are a vast generalization of CSPs that have received much
attention recently. The motivation is to extend the theory beyond CSPs, as well
as to better understand the power of SDPs. We present robust SDP rounding
algorithms under some general conditions, namely the existence of particular
high-dimensional Boolean symmetries known as majority or alternating threshold
polymorphisms. On the hardness front, we prove that the lack of such
polymorphisms makes the PCSP hard for all pairs of symmetric Boolean
predicates. Our method involves a novel method to argue SDP gaps via the
absence of certain colorings of the sphere, with connections to sphere Ramsey
theory.
We conjecture that PCSPs with robust satisfaction algorithms are precisely
those for which the feasibility of the canonical SDP implies (exact)
satisfiability. We also give a precise algebraic condition, known as a minion
characterization, of which PCSPs have the latter property.Comment: 62 pages, to appear in STOC 202