13 research outputs found
On the Optimality of a Class of LP-based Algorithms
In this paper we will be concerned with a class of packing and covering
problems which includes Vertex Cover and Independent Set. Typically, one can
write an LP relaxation and then round the solution. In this paper, we explain
why the simple LP-based rounding algorithm for the \\VC problem is optimal
assuming the UGC. Complementing Raghavendra's result, our result generalizes to
a class of strict, covering/packing type CSPs
Improved NP-Inapproximability for 2-Variable Linear Equations
An instance of the 2-Lin(2) problem is a system of equations of the form "x_i + x_j = b (mod 2)". Given such a system in which it\u27s possible to satisfy all but an epsilon fraction of the equations, we show it is NP-hard to satisfy all but a C*epsilon fraction of the equations, for any C < 11/8 = 1.375 (and any 0 < epsilon <= 1/8). The previous best result, standing for over 15 years, had 5/4 in place of 11/8. Our result provides the best known NP-hardness even for the Unique Games problem, and it also holds for the special case of Max-Cut. The precise factor 11/8 is unlikely to be best possible; we also give a conjecture concerning analysis of Boolean functions which, if true, would yield a larger hardness factor of 3/2.
Our proof is by a modified gadget reduction from a pairwise-independent predicate. We also show an inherent limitation to this type of gadget reduction. In particular, any such reduction can never establish a hardness factor C greater than 2.54. Previously, no such limitation on gadget reductions was known
Approximability and Mathematical Relaxations
The thesis ascertains the approximability of classic combinatorial
optimization problems using mathematical relaxations. The general
flavor of results in the thesis is: a problem P is hard to
approximate to a factor better than one obtained from the R
relaxation, unless the Unique Games Conjecture is false.
Almost optimal inapproximability is shown for a wide set of problems
including Metric Labeling, Max. Acyclic Subgraph, various packing
and covering problems. The key new idea in this thesis is in
coverting hard instances of relaxations (a.k.a integrality gap
instances) into a proof of inapproximability (assuming the UGC). In
most cases, the hard instances were discovered prior to this work;
our results imply that these hard instances are possibly strong
bottlenecks in designing approximation algorithms of better quality
for these problems.
For ordering problems such as Max Acyclic Subgraph and
Feedback Arc Set such hard instances were previously unknown.
For these problems, we construct such
hard instance by using the reduction designed to prove the
inapproximability. The hard instances show that all ordering
problems are hard to approximate to a factor larger than the
expected fraction satisfied by a random ordering: i.e., all ordering
CSPs are approximation resistant.
Techniques involve using mathematical relaxations to obtain local
distributions, converting them into low degree functions defined
over the boolean cube and using the invariance principle to analyse
such function.
I believe the thesis will be a good reference, both for the results
proven therein, and for the framework designed in ascertaining
approximability from mathematical relaxations
On LP-based Approximability for Strict CSPs
In a beautiful result, Raghavendra established optimal Unique Games Conjecture (UGC)-based inapproximability for a large class of constraint satisfaction problems (CSPs). In the class of CSPs he considers, of which Maximum Cut is a prominent example, the goal is to find an assignment which maximizes a weighted fraction of constraints satisfied. He gave a generic semi-definite program (SDP) for this class of problems and showed how the approximability of each problem is determined by the corresponding SDP (upto an arbitrarily small additive error) assuming the UGC. He noted that his techniques do no apply to CSPs with strict constraints (all of which must be satisfied) such as Vertex Cover. In this paper we address the approximability of these strict-CSPs. In the class of CSPs we consider, one is given a set of constraints over a set of variables, and a cost function over the assignments, the goal is to find an assignment to the variables of minimum cost which satisfies all the constraints. We present a generic linear program (LP) for a large class of strict-CSPs and give a systematic way to convert integrality gaps for this LP into UGC-based inapproximability results. Some important problems whose approximability our framework captures are Vertex Cover, Hypergraph Vertex Cover, k-partite-Hypergraph Vertex Cover, Independent Set and other covering and packing problems over q-ary alphabets, and a scheduling problem. For the covering and packing problems, which occur quite commonly in practice as well, we provide a matching rounding algorithm, thus settling their approximability upto an arbitrarily small additive error