492 research outputs found
Quantum communication complexity of symmetric predicates
We completely (that is, up to a logarithmic factor) characterize the
bounded-error quantum communication complexity of every predicate
depending only on (). Namely, for a predicate
on let \ell_0(D)\df \max\{\ell : 1\leq\ell\leq n/2\land
D(\ell)\not\equiv D(\ell-1)\} and \ell_1(D)\df \max\{n-\ell : n/2\leq\ell <
n\land D(\ell)\not\equiv D(\ell+1)\}. Then the bounded-error quantum
communication complexity of is equal (again, up to a
logarithmic factor) to . In particular, the
complexity of the set disjointness predicate is . This result
holds both in the model with prior entanglement and without it.Comment: 20 page
On the Fon-der-Flaass Interpretation of Extremal Examples for Turan's (3,4)-problem
In 1941, Turan conjectured that the edge density of any 3-graph without
independent sets on 4 vertices (Turan (3,4)-graph) is >= 4/9(1-o(1)), and he
gave the first example witnessing this bound. Brown (1983) and Kostochka (1982)
found many other examples of this density. Fon-der-Flaass (1988) presented a
general construction that converts an arbitrary -free orgraph
into a Turan (3,4)-graph. He observed that all Turan-Brown-Kostochka
examples result from his construction, and proved the bound >= 3/7(1-o(1)) on
the edge density of any Turan (3,4)-graph obtainable in this way.
In this paper we establish the optimal bound 4/9(1-o(1)) on the edge density
of any Turan (3,4)-graph resulting from the Fon-der-Flaass construction under
any of the following assumptions on the undirected graph underlying the
orgraph :
1. is complete multipartite;
2. The edge density of is >= (2/3-epsilon) for some absolute constant
epsilon>0.
We are also able to improve Fon-der-Flaass's bound to 7/16(1-o(1)) without
any extra assumptions on
Depth-Independent Lower bounds on the Communication Complexity of Read-Once Boolean Formulas
We show lower bounds of and on the
randomized and quantum communication complexity, respectively, of all
-variable read-once Boolean formulas. Our results complement the recent
lower bound of by Leonardos and Saks and
by Jayram, Kopparty and Raghavendra for
randomized communication complexity of read-once Boolean formulas with depth
. We obtain our result by "embedding" either the Disjointness problem or its
complement in any given read-once Boolean formula.Comment: 5 page
On the Number of Pentagons in Triangle-Free Graphs
Using the formalism of flag algebras, we prove that every triangle-free graph
with vertices contains at most cycles of length five.
Moreover, the equality is attained only when is divisible by five and
is the balanced blow-up of the pentagon. We also compute the maximal number of
pentagons and characterize extremal graphs in the non-divisible case provided
is sufficiently large. This settles a conjecture made by Erd\H{o}s in 1984.Comment: 16 pages, accepted to Journal of Combinatorial Theory Ser.
Resolution over Linear Equations and Multilinear Proofs
We develop and study the complexity of propositional proof systems of varying
strength extending resolution by allowing it to operate with disjunctions of
linear equations instead of clauses. We demonstrate polynomial-size refutations
for hard tautologies like the pigeonhole principle, Tseitin graph tautologies
and the clique-coloring tautologies in these proof systems. Using the
(monotone) interpolation by a communication game technique we establish an
exponential-size lower bound on refutations in a certain, considerably strong,
fragment of resolution over linear equations, as well as a general polynomial
upper bound on (non-monotone) interpolants in this fragment.
We then apply these results to extend and improve previous results on
multilinear proofs (over fields of characteristic 0), as studied in
[RazTzameret06]. Specifically, we show the following:
1. Proofs operating with depth-3 multilinear formulas polynomially simulate a
certain, considerably strong, fragment of resolution over linear equations.
2. Proofs operating with depth-3 multilinear formulas admit polynomial-size
refutations of the pigeonhole principle and Tseitin graph tautologies. The
former improve over a previous result that established small multilinear proofs
only for the \emph{functional} pigeonhole principle. The latter are different
than previous proofs, and apply to multilinear proofs of Tseitin mod p graph
tautologies over any field of characteristic 0.
We conclude by connecting resolution over linear equations with extensions of
the cutting planes proof system.Comment: 44 page
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