Ravi B. Boppana

Does co-NP have short interactive proofs?

Abstract Babai (1985) and Goldwasser,Micali and Rackoff (1985) introduced two probabilistic extensions of the complexity class NP. The two complexity classes, denoted AM[Q] and IP[Q] respectively, are defined using randomized interactive proofs between a prover and a verifier. Goldwasser and Sipser (1986) proved that the two classes are equal. We prove that if the complexity class co -NP is contained in IP[k] for some constant k (i.e., if every language in co -NP has a short interactive proof), then the polynomial-time hierarchy collapses to the second level. As a corollary, we show that if the Graph Isomorphism problem is NP-complete, then the polynomial-time hierarchy collapses.

Eigenvalues and graph bisection: An average-case analysis

Graph Bisection is the problem of partitioning the vertices of a graph into two equal-size pieces so as to minimize the number of edges between the two pieces. This paper presents an algorithm that will, for almost all graphs in a certain class, output the minimum-size bisection. Furthermore the algorithm will yield, for almost all such graphs, a proof that the bisection is optimal. The algorithm is based on computing eigenvalues and eigenvectors of matrices associated with the graph.

The monotone circuit complexity of Boolean functions

Recently, Razborov obtained superpolynomial lower bounds for monotone circuits that cliques in graphs. In particular, Razborov showed that detecting cliques of sizes in a graphm vertices requires monotone circuits of size Ω(ms/(logm)2s) for fixeds, and sizemΩ(logm) form/4].In this paper we modify the arguments of Razborov to obtain exponential lower bounds for circuits. In particular, detecting cliques of size (1/4) (m/logm)2/3 requires monotone circuits exp (Ω((m/logm)1/3)). For fixeds, any monotone circuit that detects cliques of sizes requiresm)s) AND gates. We show that even a very rough approximation of the maximum clique of a graph requires superpolynomial size monotone circuits, and give lower bounds for some Boolean functions. Our best lower bound for an NP function ofn variables is exp (Ω(n1/4 · (logn)1/2)), improving a recent result of exp (Ω(n1/8-ε)) due to Andreev.

Approximating maximum independent sets by excluding subgraphs

An approximation algorithm for the maximum independent set problem is given, improving the best performance guarantee known toO(n/(logn)2). We also obtain the same performance guarantee for graph coloring. The results can be combined into a surprisingly strongsimultaneous performance guarantee for the clique and coloring problems.The framework ofsubgraph-excluding algorithms is presented. We survey the known approximation algorithms for the independent set (clique), coloring, and vertex cover problems and show how almost all fit into that framework. We show that among subgraph-excluding algorithms, the ones presented achieve the optimal asymptotic performance guarantees.

The complexity of finite functions

Publisher Summary This chapter discusses the complexity of finite functions. A deterministic Turing machine consists of a finite control and a finite collection of tapes each with a head for reading and writing. The finite control is a finite collection of states. A tape is an infinite list of cells each containing a symbol. Initially, all tapes have blanks except for the first, which contains the input string. Once started, the machine goes from state to state, reading the symbols under the heads, writing new ones, and moving the heads. The exact action taken is governed by the current state, the symbols read, and the next-move function of the machine. This continues until a designated halt state is entered. The machine indicates its output by the halting condition of the tapes. In a nondeterministic Turing machine, the next-move function is multivalued. There can be several computations on a given input and several output values.

The average sensitivity of bounded-depth circuits

Abstract The average sensitivity of a Boolean circuit is the expected number of input bits that, when flipped, change the output of the circuit, starting with a random input setting. We show that unbounded-fanin circuits of depth d and size s have average sensitivity O(log s)d−1. This bound is asymptotically tight.

Approximating Maximum Independent Sets by Excluding Subgraphs

An approximation algorithm for the maximum independent set problem is given, improving the best performance guarantee known to \({\cal O}\)(n/(log n)2). We also obtain the same performance guarantee for graph coloring. The results can be combined into a surprisingly strong simultaneous performance guarantee for the clique and coloring problems.

Optimal separations between concurrent-write parallel machines

We obtain tight bounds on the relative powers of the Priority and Common models of parallel random-access machines (PRAMs). Specifically we prove that:<list><item>The Element Distinctness function of <italic>n</italic> integers, though solvable in constant time on a Priority PRAM with <italic>n</italic> processors, requires &OHgr;(<italic>A</italic>(<italic>n,p</italic>)) time to solve on a Common PRAM with <italic>p</italic> ≥ <italic>n</italic> processors, where <italic>A</italic>(<italic>n</italic>,<italic>p</italic>) = <italic>n</italic> log <italic>n</italic>/<italic>p</italic> log (<italic>n</italic>/<italic>p</italic> log <italic>n</italic> + 1). </item><item>One step of a Priority PRAM with <italic>n</italic> processors can be simulated on a Common PRAM with <italic>p</italic> processors in <italic>&Ogr;</italic>(<italic>A</italic>(<italic>n</italic>,<italic>p</italic>)) steps. </item></list> As an example, the results show that the time separation between Priority and Common PRAMs each with <italic>n</italic> processors is &THgr;(log <italic>n</italic>/log log <italic>n</italic>).

Amplification of probabilistic boolean formulas

The amplification of probabilistic Boolean formulas refers to combining independent copies of such formulas to reduce the error probability. Les Valiant used the amplification method to produce monotone Boolean formulas of size O(n5.3) for the majority function of n variables. In this paper we show that the amount of amplification that Valiant obtained is optimal. In addition, using the amplification method we give an O(k4.3 n log n) upper bound for the size of monotone formulas computing the kth threshold function of n variables.

Threshold functions and bounded deptii monotone circuits

We prove an exponential lower bound for the majority function on constant depth monotone circuits, solving an open problem of A. Yaos.. In particular, we prove that computing majority on depth <italic>d</italic> monotone circuits requires expΩ(<italic>n</italic><supscrpt>1/(d-1)</supscrpt>) size. Using this result we also get exponential lower bounds for other problems, such as connectivity and cliques.