We consider a version of the game “Twenty Questions” played on the set {0, · · · ,N − 1} where the player giving answersmay lie in her answers. The questioner is allowedQ questions and the responder may lie in up to rQ of the answers, for some fixed and previously known fraction r. Under various models of this game and different question classes, we give precise conditions (i.e. tight bounds on...

We investigate finite-rank intersection type systems, analyzing the complexity of their type inference problems and their relation to the problem of recognizing semantically equivalent terms. Intersection types allow something of type τ1 ∧ τ2 to be used in some places at type τ1 and in other places at type τ2. A finite-rank intersection type system bounds how deeply the ∧ can appear in type exp...

For various polynomial-time reducibilities r, this paper asks whether being r-reducible to a sparse set is a broader notion than being r-equivalent to a sparse set. Although distinguishing equivalence and reducibility to sparse sets, for many-one or 1-truth-table reductions, would imply that P 6= NP, this paper shows that for k-truth-table reductions, k 2, equivalence and reducibility to sparse...

We give fast new approximation algorithms for the problem of choosing k planar points out of n to minimize the length of their minimum spanning tree (equivalently, of their traveling salesman tour or Steiner tree). For any x ≤ k, we can find an approximation achieving approximation ratio O(log k/ log x) in time O(n log n+ 2kn log k). In particular, we get an approximation with ratio O(log k/ lo...

Some natural problems seem not to be complete for any of the complexity classes we have seen so far. For example, consider the problem of taking as input a graph G and a number k, and deciding whether k is exactly the length of the shortest traveling salesperson’s tour. This is clearly related to the problem discussed in Chapter 28, Section 3, but in contrast to , it seems not to belong to NP, ...

Some natural problems seem not to be complete for any of the complexity classes we have seen so far. For example, consider the problem of taking as input a graph G and a number k, and deciding whether k is exactly the length of the shortest traveling salesperson’s tour. This is clearly related to the TSP problem discussed in Chapter 28, Section 3, but in contrast to TSP, it seems not to belong ...

A modular query consists of asking how many (modulo m) of k strings belong to a fixed NP language. Modular queries provide a form of restricted access to an NP oracle. For each k and m, we consider the class of languages accepted by NP machines that ask a single modular query. Han and Thierauf [HT95] showed that these classes coincide with levels of the Boolean hierarchy when m is even or k ≤ 2...

Continuing a line of investigation that has studied the function classes #P [Val79b], #SAC [Val79a, Vin91, AJMV], #L [AJ93b, Vin91, AO94], and #NC [CMTV96], we study the class of functions #AC. One way to define #AC is as the class of functions computed by constant-depth polynomial-size arithmetic circuits of unbounded fan-in addition and multiplication gates. In contrast to the preceding ∗Part...

This paper presents a unified and simple treatment of basic questions concerning two computational models: multiparty communication complexity and polynomials over GF(2). The key is the use of (known) norms on Boolean functions, which capture their proximity to each of these models (and are closely related to property testers of this proximity). The main contributions are new XOR lemmas. We sho...