In this chapter, we discuss approximation algorithms for optimization problems. An optimization problem consists in finding the best (cheapest, heaviest, etc.) element in a large set P, called the feasible region and usually specified implicitly, where the quality of elements of the set are evaluated using a function f(x), the objective function, usually something fairly simple. The element tha...

We consider the problem of synthesizing feasible signals in a Hilbert space in the presence of inconsistent convex constraints, some of which must imperatively be satisfied. This problem is formalized as that of minimizing a convex objective measuring the amount of violation of the soft constraints over the intersection of the sets associated with the hard ones. The resulting convex optimizatio...

The 3sum problem is defined as follows: given a set S of n integers, do there exist three elements {a, b, c} ∈ S such that a + b + c = 0? This is a linear satisfiability problem. 3sum can be solved using a simple algorithm with Θ(n) runtime (sort S, then test 3-tuples intelligently) and it is widely believed that Ω(n) is the lower bound for the worst-case runtime of any solution to the 3sum pro...

We study the effects of faulty data on NP-hard sets. We consider hard sets for several polynomial time reductions, add corrupt data and then analyze whether the resulting sets are still hard for NP. We explain that our results are related to a weakened deterministic variant of the notion of program self-correction by Blum, Luby, and Rubinfeld. Among other results, we use the Left-Set technique ...

We give the first representation-independent hardness result for agnostically learning halfspaces with respect to the Gaussian distribution. We reduce from the problem of learning sparse parities with noise with respect to the uniform distribution on the hypercube (sparse LPN), a notoriously hard problem in theoretical computer science and show that any algorithm for agnostically learning halfs...

DNA experiments are proposed to solve the famous "SAT" problem of computer science. This is a special case of a more general method that can solve NP-complete problems. The advantage of these results is the huge parallelism inherent in DNA-based computing. It has the potential to yield vast speedups over conventional electronic-based computers for such search problems.

We look at problems involving finding subgraphs in larger graphs, such as the maximum clique problem, the subgraph isomorphism problem, and the maximum common subgraph problem. We investigate variable and value ordering heuristics, different inference strategies, intelligent backtracking search (backjumping), and bitand thread-parallelism to exploit modern hardware.

This paper discusses polynomial-time reductions from Hamiltonian Circuit (HC), k-Vertex Coloring (k-VC), and k-Clique Problems to Satis ability Problem (SAT) which are e cient in the number of Boolean variables needed in SAT. We rst present a basic type of reductions that need (n 1) log(n 1), (n 1) log k, and k logn variables for HC, k-VC and k-Clique, respectively. Several heuristics can reduc...

Longprt, L. and A.L. Selman, Hard promise problems and nonuniform complexity, Theoretical Computer Science 115 (1993) 277-290. For every recursive set A, let PP-A denote the following promise problem: input x and y promise (u~A)@(ysA) property XE A. We show that if L is a solution of PP-A, then A~p~/Poly. From this result, it follows that if A is <F-hard for NP, then all solutions of PP-A are h...

Recently proposed end-to-end independently-verifiable (E2E) voting schemes provide encrypted paper receipts to voters, who may later check that these receipts are in the electronic ballot box. Unfortunately, few voters are likely to follow up on the voting process after leaving the voting site; as a result, few receipts will be checked. This paper describes an enhancement to E2E schemes that do...