Stathis Zachos

A decisive characterization of BPP

The complexity class BPP (defined by Gill) contains problems that can be solved in polynomial time with bounded error probability. A new and simple characterization of BPP is given. It is shown that a language L is in BPP iff ( x ∈ L → ∃ + y ∀ zP ( x , y , z )) ∧ ( x ∉ L → ∀ y ∃ + z ¬ P ( x , y , z )) for a polynomial-time predicate P and for | y |, | z | ⩽ poly (| x |). The formula ∃ + yP ( y ) with the random quantifier ∃ + means that the probability Pr({ y | P ( y )}) ⩾+ ɛ for a fixed ɛ. This characterization allows a simple proof that BPP ⊆ ZPP NP , which strengthens the result of (Lautemann, Inform. Process. Lett. 17 (1983), 215–217; Sipser, in “Proceedings, 15th Annu. ACM Sympos. Theory of Comput.,” 1983, pp. 330–335) that BPP ⊆ Σ 2 p ∩ Π 2 p . Several other results about probabilistic classes can be proved using similar techniques, e.g., NP R ⊆ ZPP NP and Σ 2 p ,BPP = Σ 2 p .

Probabilistic quantifiers and games

We consider inclusion relations among a multitude of classical complexity classes and classes with probabilistic components. A key tool is a method for characterizing such classes in terms of the ordinary quantifiers 3 and V together with a quantifier 3+, which means roughly “for most,” applied to polynomial-time predicates. This approach yields a uniform treatment which leads to easier proofs for class-inclusion and hierarchy-collapse results. Furthermore, the method captures some recently introduced game classes and game hierarchies. This survey also includes a charting of class-inclusion and oracle-based separation results. @?I 1988 Academic Press. Inc.

Probabilistic Quantifiers, Adversaries, and Complexity Classes: An Overview

We consider inclusion relations among a multitude of classical complexity classes and classes with probabilistic components. A key tool is a method for characterizing such classes in terms of the ordinary quantifiers ∃ and ∀ together with a quantifier ∃+, which means roughly “for most”, applied to polynomial-time predicates. This approach yields a uniform treatment which leads to easier proofs for class-inclusion and hierarchy-collapse results. Furthermore the method captures some recently introduced game classes and game hierarchies.

Routing and path multicoloring

Abstract In optical networks it is important to make an optimal use of the available bandwidth. Given a set of requests the goal is to satisfy them by using a minimum number of wavelengths. We introduce a variation to this well known problem, by allowing multiple parallel links, in order to be able to satisfy any set of requests even if the available bandwidth is insufficient. In this new approach the goal is to use a minimum number of active links and thus reduce network pricing. In graph-theoretic terms, given a graph, a list of pairs of vertices, and a number of available colors, the goal is to route paths with the given pairs of vertices as endpoints and to find a color assignment to paths that minimizes color collisions over all possible routings and colorings. We present efficient algorithms for simple network topologies. For chains our solutions are optimal; for stars and rings — where it is NP-hard to solve the problem optimally — our solutions are approximate within a factor two of the optimal solution. The key technique involves transformation to edge coloring of bipartite graphs. For rings we also present a 2-approximation algorithm, for a variation of the problem, in which the routing is already prescribed.

Routing and wavelength assignment in multifiber WDM networks with non-uniform fiber cost

Motivated by the increasing importance of multifiber WDM networks we study a routing and wavelength assignment problem in such networks. In this problem the number of wavelengths per fiber is given and the goal is to minimize the cost of fiber links that need to be reserved in order to satisfy a set of communication requests; we introduce a generalized setting where network pricing is non-uniform, that is the cost of hiring a fiber may differ from link to link. We consider two variations: undirected, which corresponds to full-duplex communication, and directed, which corresponds to one-way communication. Moreover, for rings we also study the problem in the case of pre-determined routing. We present exact or constant-ratio approximation algorithms for all the above variations in chain, ring and spider networks.

Satisfying a maximum number of pre-routed requests in all-optical rings

We address the problem of maximizing the number of satisfied requests in all-optical networks that use wavelength division multiplexing . We consider the case where requests are pre-routed and formulate it as the maximum path coloring problem. We study the problem for rings and present a (2/3)-approximation algorithm. Along the way we develop a fast matching technique for a special class of bipartite graphs. By using this technique we achieve an O( n + m log L ) time complexity for our approximation algorithm, where n is the number of nodes, m is the number of requests and L is the maximum load of requests on a single link.

Fiber Cost reduction and Wavelength minimization in multifiber WDM networks

Motivated by the increasing importance of multifiber WDM networks we study two routing and wavelength assignment problems in such networks: n n nFiber Cost Minimization: the number of wavelengths per fiber is given and we want to minimize the cost of fiber links that need to be reserved in order to satisfy a set of communication requests; we introduce a generalized setting where network pricing is nonuniform, that is the cost of hiring a fiber may differ from link to link. n n nWavelength Minimization: the number of available parallel fibers on each link is given and we want to minimize the wavelengths per fiber that are needed in order to satisfy a set of communication requests.

Maximizing the guarded boundary of an Art Gallery is APX-complete

In the Art Gallery problem, given is a polygonal gallery and the goal is to guard the gallerys interior or walls with a number of guards that must be placed strategically in the interior, on walls or on corners of the gallery. Here we consider a more realistic version: exhibits now have size and may have different costs. Moreover the meaning of guarding is relaxed: we use a new concept, that of watching an expensive art item, i.e. overseeing a part of the item. The main result of the paper is that the problem of maximizing the total value of a guarded weighted boundary is APX-complete. This is shown by an appropriate gap-preserving reduction from the MAX-5-OCCURENCE-3-SAT problem. We also show that this technique can be applied to a number of maximization variations of the art gallery problem. In particular we consider the following problems: given a polygon with or without holes and k available guards, maximize a) the length of walls guarded and b) the total cost of paintings watched or overseen. We prove that all the above problems are APX-complete.

Randomized and approximation algorithms for blue-red matching

We introduce the Blue-Red Matching problem: given a graph with red and blue edges, and a bound w, find a maximum matching consisting of at most w edges of each color. We show that Blue-Red Matching is at least as hard as the problem Exact Matching (Papadimitriou and Yannakakis, 1982), for which it is still open whether it can be solved in polynomial time. We present an RNC algorithm for this problem as well as two fast approximation algorithms. We finally show the applicability of our results to the problem of routing and assigning wavelengths to a maximum number of requests in all-optical rings.

Path multicoloring with fewer colors in spiders and caterpillars

We study a recently introduced path coloring problem with applications to wavelength assignment in all-optical networks with multiple fibers. In contrast to classical path coloring, it is, in this setting, possible to assign a color more than once to paths that pass through the same edge; the number of allowed repetitions per edge is given and the goal is to minimize the number of colors used.We present algorithms and hardness results for tree topologies of special interest. Our algorithms achieve approximation ratio of 2 in spiders and 3 in caterpillars, whereas the best algorithm for trees so far, achieves an approximation ratio of 4. We also study the directed version of the problem and show that it admits a 3-approximation algorithm in caterpillars, while it can be solved exactly in spiders.