Expected Polynomial Time Algorithm Research Articles

An expected polynomial time algorithm for coloring 2-colorable 3-graphs

Graphs and Algorithms We present an algorithm that for 2-colorable 3-uniform hypergraphs, finds a 2-coloring in average running time O (n(5) log(2) n).

An expected polynomial time algorithm for coloring 2-colorable 3-graphs

Abstract We present an algorithm that colors a random 2-colorable 3-uniform hypergraph optimally in expected running time O ( n 5 log 2 n ) .

Generating unlabeled connected cubic planar graphs uniformly at random

AbstractWe present an expected polynomial time algorithm to generate an unlabeled connected cubic planar graph uniformly at random. We first consider rooted connected cubic planar graphs, i.e., we count connected cubic planar graphs up to isomorphisms that fix a certain directed edge. Based on decompositions along the connectivity structure, we derive recurrence formulas for the exact number of rooted cubic planar graphs. This leads to rooted 3‐connected cubic planar graphs, which have a unique embedding on the sphere. Special care has to be taken for rooted graphs that have a sense‐reversing automorphism. Therefore we introduce the concept of colored networks, which stand in bijective correspondence to rooted 3‐connected cubic planar graphs with given symmetries. Colored networks can again be decomposed along the connectivity structure. For rooted 3‐connected cubic planar graphs embedded in the plane, we switch to the dual and count rooted triangulations. Since all these numbers can be evaluated in polynomial time using dynamic programming, rooted connected cubic planar graphs can be generated uniformly at random in polynomial time by inverting the decomposition along the connectivity structure. To generate connected cubic planar graphs without a root uniformly at random, we apply rejection sampling and obtain an expected polynomial time algorithm. © 2008 Wiley Periodicals, Inc. Random Struct. Alg., 2008

Correlated algebraic-geometric codes: Improved list decoding over bounded alphabets

We define a new family of error-correcting codes based on algebraic curves over finite fields, and develop efficient list decoding algorithms for them. Our codes extend the class of algebraic-geometric (AG) codes via a (nonobvious) generalization of the approach in the recent breakthrough work of Parvaresh and Vardy (2005). Our work shows that the PV framework applies to fairly general settings by elucidating the key algebraic concepts underlying it. Also, more importantly, AG codes of arbitrary block length exist over fixed alphabets Σ, thus enabling us to establish new trade-offs between the list decoding radius and rate over a bounded alphabet size. The work of Parvaresh and Vardy (2005) was extended in Guruswami and Rudra (2006) to give explicit codes that achieve the list decoding capacity (optimal trade-off between rate and fraction of errors corrected) over large alphabets. A similar extension of this work along the lines of Guruswami and Rudra could have substantial impact. Indeed, it could give better trade-offs than currently known over a fixed alphabet (say, GF(2 12 )), which in turn, upon concatenation with a fixed, well-understood binary code, could take us closer to the list decoding capacity for binary codes. This may also be a promising way to address the significant complexity drawback of the result of Guruswami and Rudra, and to enable approaching capacity with bounded list size independent of the block length (the list size and decoding complexity in their work are both nΩ(1/e ) where e is the distance to capacity). Similar to algorithms for AG codes from Guruswami and Sudan (1999) and (2001), our encoding/decoding algorithms run in polynomial time assuming a natural polynomial-size representation of the code. For codes based on a specific optimal algebraic curve, we also present an expected polynomial time algorithm to construct the requisite representation. This in turn fills an important void in the literature by presenting an efficient construction of the representation often assumed in the list decoding algorithms for AG codes.

Zero Knowledge and the Chromatic Number

We present a new technique, inspired by zero-knowledge proof systems, for proving lower bounds on approximating the chromatic number of a graph. To illustrate this technique we present simple reductions frommax-3-coloringandmax-3-sat, showing that it is hard to approximate the chromatic number withinΩ(Nδ) for someδ>0. We then apply our technique in conjunction with the probabilistically checkable proofs of Håstad and show that it is hard to approximate the chromatic number to withinΩ(N1−ε) for anyε>0, assuming NP⊈ZPP. Here, ZPP denotes the class of languages decidable by a random expected polynomial-time algorithm that makes no errors. Our result matches (up to low order terms) the known gap for approximating the size of the largest independent set. PreviousO(Nδ) gaps for approximating the chromatic number (such as those by Lund and Yannakakis, and by Furer) did not match the gap for independent set nor extend beyondΩ(N1/2−ε).

Oracles and Queries That Are Sufficient for Exact Learning

We show that the class of all circuits is exactly learnable in randomized expected polynomial time using weak subset and weak superset queries. This is a consequence of the following result which we consider to be of independent interest: circuits are exactly learnable in randomized expected polynomial time with equivalence queries and the aid of anNP-oracle. We also show that circuits are exactly learnable in deterministic polynomial time with equivalence queries and a ∑P3-oracle. The hypothesis class for the above learning algorithms is the class of circuits of larger—but polynomially related—size. Also, the algorithms can be adapted to learn the class of DNF formulas with hypothesis class consisting of depth-3 ∧-∨-∧ formulas (by the work of Angluin this is optimal in the sense that the hypothesis class cannot be reduced to DNF formulas, i.e., depth-2 ∨-∧ formulas). We also investigate the power of anNP-oracle in the context of learning withmembershipqueries. We show that there are deterministic learning algorithms that use membership queries and anNP-oracle to learn: monotone boolean functions in time polynomial in the DNF size and CNF size of the target formula; and the class ofO(logn)-DNF∩O(logn)-CNF formulas in time polynomial inn. We also show that, with anNP-oracle and membership queries, there is a randomized expected polynomial time algorithm that learns any class that is learnable from membership queries with unlimited computational power. Using similar techniques, we show the following both for membership and for equivalence queries (when the hypotheses allowed are precisely the concepts in the class); any class learnable with unbounded ?computational-power is learnable in deterministic polynomial time with a ∑:p5-oracle. Furthermore, we identify the combinatorial properties that completely determine learnability in this information-theoretic sense. Finally we point out a consequence of our result in structural complexity theory showing that if everyNPset has polynomial-size circuits then the polynomial hierarchy collapses to ZPPNP.