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1.
, for the monotone depth of functions in monotone-P. As a result we achieve the separation of the following classes.
1. monotone-NC ≠ monotone-P.
2. For every i≥1, monotone-≠ monotone-.
3. More generally: For any integer function D(n), up to (for some ε>0), we give an explicit example of a monotone Boolean function, that can be computed by polynomial size monotone
Boolean circuits of depth D(n), but that cannot be computed by any (fan-in 2) monotone Boolean circuits of depth less than Const·D(n) (for some constant Const).
Only a separation of monotone- from monotone- was previously known.
Our argument is more general: we define a new class of communication complexity search problems, referred to below as DART
games, and we prove a tight lower bound for the communication complexity of every member of this class. As a result we get
lower bounds for the monotone depth of many functions. In particular, we get the following bounds:
1. For st-connectivity, we get a tight lower bound of . That is, we get a new proof for Karchmer–Wigderson's theorem, as an immediate corollary of our general result.
2. For the k-clique function, with , we get a tight lower bound of Ω(k log n). This lower bound was previously known for k≤ log n [1]. For larger k, however, only a bound of Ω(k) was previously known.
Received: December 19, 1997 相似文献
2.
Alexander A. Sherstov 《Combinatorica》2011,31(5):583-614
We prove an essentially tight lower bound on the unbounded-error communication complexity of every symmetric function, i.e.,
f(x,y)=D(|x∧y|), where D: {0,1,…,n}→{0,1} is a given predicate and x,y range over {0,1}
n
. Specifically, we show that the communication complexity of f is between Θ(k/log5
n) and Θ(k logn), where k is the number of value changes of D in {0,1,…, n}. Prior to this work, the problem was solved only for the parity predicate D (Forster 2001). 相似文献
3.
In this paper we consider the worst case ratio between the capacity of min-cuts and the value of max-flow for multicommodity flow problems. We improve the best known bounds for the min-cut max-flow ratio for multicommodity flows in undirected graphs, by replacing theO(logD) in the bound byO(logk), whereD denotes the sum of all demands, andk demotes the number of commodities. In essence we prove that up to constant factors the worst min-cut max-flow ratios appear in problems where demands are integral and polynomial in the number of commodities.Klein, Rao, Agrawal, and Ravi have previously proved that if the demands and the capacities are integral, then the min-cut max-flow ratio in general undirected graphs is bounded byO(logClogD), whereC denotes the sum of all the capacities. Tragoudas has improved this bound toO(lognlogD), wheren is the number of nodes in the network. Garg, Vazirani and Yannakakis further improved this toO(logklogD). Klein, Plotkin and Rao have proved that for planar networks, the ratio isO(logD).Our result improves the bound for general networks toO(log2
k) and the bound for planar networks toO(logk). In both cases our result implies the first non-trivial bound that is independent of the magnitude of the numbers involved. The method presented in this paper can be used to give polynomial time approximation algorithms to the minimum cuts in the network up to the above factors.Preliminary version appeared in Proceedings of the 25th Annual ACM Symposium on the Theory of Computing, 1993, 691-697.Research supported by U.S. Army Research Office Grant DAAL-03-91-G-0102, and by a grant from Mitsubishi Electric Laboratories.Research supported in part by a Packard Fellowship, an NSF PYI award, a Sloan Fellowship, and by the National Science Foundation, the Air Force Office of Scientific Research, and the Office of Naval Research, through NSF grant DMS-8920550. 相似文献
4.
Oded Goldreich Shafi Goldwasser Eric Lehman Dana Ron Alex Samorodnitsky 《Combinatorica》2000,20(3):301-337
at arguments of its choice, the test always accepts a monotone f, and rejects f with high probability if it is ε-far from being monotone (i.e., every monotone function differs from f on more than an ε fraction of the domain). The complexity of the test is O(n/ε).
The analysis of our algorithm relates two natural combinatorial quantities that can be measured with respect to a Boolean
function; one being global to the function and the other being local to it. A key ingredient is the use of a switching (or sorting) operator on functions.
Received March 29, 1999 相似文献
5.
We show that Closest Substring, one of the most important problems in the field of consensus string analysis, is W[1]-hard when parameterized by the number
k of input strings (and remains so, even over a binary alphabet). This is done by giving a “strongly structure-preserving”
reduction from the graph problem Clique to Closest Substring. This problem is therefore unlikely to be solvable in time O(f(k)•nc) for any function f of k and constant c independent of k, i.e., the combinatorial explosion seemingly inherent to this NP-hard problem cannot be restricted to parameter k. The problem can therefore be expected to be intractable, in any practical sense, for k ≥ 3. Our result supports the intuition that Closest Substring is computationally much harder than the special case of Closest String, althoughb othp roblems are NP-complete. We also prove W[1]-hardness for other parameterizations in the case of unbounded
alphabet size. Our W[1]-hardness result for Closest Substring generalizes to Consensus Patterns, a problem arising in computational biology.
* An extended abstract of this paper was presented at the 19th International Symposium on Theoretical Aspects of Computer
Science (STACS 2002), Springer-Verlag, LNCS 2285, pages 262–273, held in Juan-Les-Pins, France, March 14–16, 2002.
† Work was supported by the Deutsche Forschungsgemeinschaft (DFG), research project “OPAL” (optimal solutions for hard problems
in computational biology), NI 369/2.
‡ Work was done while the author was with Wilhelm-Schickard-Institut für Informatik, Universit?t Tübingen. Work was partially
supported by the Deutsche Forschungsgemeinschaft (DFG), Emmy Noether research group “PIAF” (fixed-parameter algorithms), NI
369/4. 相似文献
6.
A. A. Razborov 《Combinatorica》1990,10(1):81-93
We present some criteria for obtaining lower bounds for the formula size of Boolean functions. In the monotone case we get the boundn
(logn) for the function MINIMUM COVER using methods considerably simpler than all previously known. In the general case we are only able to prove that the criteria yield an exponential lower bound when applied to almost all functions. Some connections with graph complexity and communication complexity are also given. 相似文献
7.
Dedicated to the Memory of Paul Erdős
We generalize the multiparty communication model of Chandra, Furst, and Lipton (1983) to functions with b-bit output (b = 1 in the CFL model). We allow the players to receive up to b - 1 bits of information from an all-powerful benevolent Helper who can see all the input. Extending results of Babai, Nisan,
and Szegedy (1992) to this model, we construct families of explicit functions for which bits of communication are required to find the "missing bit", where n is the length of each player's input and k is the number of players. As a consequence we settle the problem of separating the one-way vs. multiround communication complexities
(in the CFL sense) for players, extending a result of Nisan and Wigderson (1991) who demonstrated this separation for k = 3 players. As a by-product we obtain lower bounds for the multiparty complexity (in the CFL sense) of new families of explicit boolean functions (not derivable
from BNS). The proofs exploit the interplay between two concepts of multicolor discrepancy; discrete Fourier analysis is the
basic tool. We also include an unpublished lower bound by A. Wigderson regarding the one-way complexity of the 3-party pointer
jumping function.
Received November 12, 1998
RID="*"
ID="*" Supported in part by NSA grant MSPR-96G-184.
RID="†"
ID="†" Supported in part by an NSF Graduate Fellowship. 相似文献
8.
The complexity of computing the Tutte polynomialT(M,x,y) is determined for transversal matroidM and algebraic numbersx andy. It is shown that for fixedx andy the problem of computingT(M,x,y) forM a transversal matroid is #P-complete unless the numbersx andy satisfy (x−1)(y−1)=1, in which case it is polynomial-time computable. In particular, the problem of counting bases in a transversal matroid,
and of counting various types of “matchable” sets of nodes in a bipartite graph, is #P-complete. 相似文献
9.
Non-Separating Paths in 4-Connected Graphs 总被引:2,自引:0,他引:2
In 1975, Lovász conjectured that for any positive integer k, there exists a minimum positive integer f(k) such that, for any two vertices x, y in any f(k)-connected graph G, there is a path P from x to y in G such that G–V(P) is k-connected. A result of Tutte implies f(1) = 3. Recently, f(2) = 5 was shown by Chen et al. and, independently, by Kriesell. In this paper, we show that f(2) = 4 except for double wheels.Received October 17, 2003 相似文献
10.
Suppose that we are given a function f : (0, 1)→(0,1) and, for some unknown p∈(0, 1), a sequence of independent tosses of a p-coin (i.e., a coin with probability p of “heads”). For which functions f is it possible to simulate an f(p)-coin? This question was raised by S. Asmussen and J. Propp. A simple simulation scheme for the constant function f(p)≡1/2 was described by von Neumann (1951); this scheme can be easily implemented using a finite automaton. We prove that in
general, an f(p)-coin can be simulated by a finite automaton for all p ∈ (0, 1), if and only if f is a rational function over ℚ. We also show that if an f(p)-coin can be simulated by a pushdown automaton, then f is an algebraic function over ℚ; however, pushdown automata can simulate f(p)-coins for certain nonrational functions such as
. These results complement the work of Keane and O’Brien (1994), who determined the functions f for which an f(p)-coin can be simulated when there are no computational restrictions on the simulation scheme.
* Supported by a Miller Fellowship.
† Supported in part by NSF Grant DMS-0104073 and by a Miller Professorship.
‡ This work is supported under a National Science Foundation Graduate Research Fellowship. 相似文献
11.
M. Eshaghi Gordji H. Khodaei 《Nonlinear Analysis: Theory, Methods & Applications》2009,71(11):5629-5643
In this paper, we achieve the general solution and the generalized Hyers–Ulam–Rassias stability of the following functional equation
f(x+ky)+f(x−ky)=k2f(x+y)+k2f(x−y)+2(1−k2)f(x)