50 research outputs found
The Minrank of Random Graphs
The minrank of a graph is the minimum rank of a matrix that can be
obtained from the adjacency matrix of by switching some ones to zeros
(i.e., deleting edges) and then setting all diagonal entries to one. This
quantity is closely related to the fundamental information-theoretic problems
of (linear) index coding (Bar-Yossef et al., FOCS'06), network coding and
distributed storage, and to Valiant's approach for proving superlinear circuit
lower bounds (Valiant, Boolean Function Complexity '92).
We prove tight bounds on the minrank of random Erd\H{o}s-R\'enyi graphs
for all regimes of . In particular, for any constant ,
we show that with high probability,
where is chosen from . This bound gives a near quadratic
improvement over the previous best lower bound of (Haviv and
Langberg, ISIT'12), and partially settles an open problem raised by Lubetzky
and Stav (FOCS '07). Our lower bound matches the well-known upper bound
obtained by the "clique covering" solution, and settles the linear index coding
problem for random graphs.
Finally, our result suggests a new avenue of attack, via derandomization, on
Valiant's approach for proving superlinear lower bounds for logarithmic-depth
semilinear circuits
The chromatic number and rank of the complements of the Kasami graphs
AbstractWe determine the rank and chromatic number of the complements of all Kasami graphs, some of which form an infinite family of counterexamples to the now disproven rank-coloring conjecture
Clique versus Independent Set
Yannakakis' Clique versus Independent Set problem (CL-IS) in communication
complexity asks for the minimum number of cuts separating cliques from stable
sets in a graph, called CS-separator. Yannakakis provides a quasi-polynomial
CS-separator, i.e. of size , and addresses the problem of
finding a polynomial CS-separator. This question is still open even for perfect
graphs. We show that a polynomial CS-separator almost surely exists for random
graphs. Besides, if H is a split graph (i.e. has a vertex-partition into a
clique and a stable set) then there exists a constant for which we find a
CS-separator on the class of H-free graphs. This generalizes a
result of Yannakakis on comparability graphs. We also provide a
CS-separator on the class of graphs without induced path of length k and its
complement. Observe that on one side, is of order
resulting from Vapnik-Chervonenkis dimension, and on the other side, is
exponential.
One of the main reason why Yannakakis' CL-IS problem is fascinating is that
it admits equivalent formulations. Our main result in this respect is to show
that a polynomial CS-separator is equivalent to the polynomial
Alon-Saks-Seymour Conjecture, asserting that if a graph has an edge-partition
into k complete bipartite graphs, then its chromatic number is polynomially
bounded in terms of k. We also show that the classical approach to the stubborn
problem (arising in CSP) which consists in covering the set of all solutions by
instances of 2-SAT is again equivalent to the existence of a
polynomial CS-separator
Quantum Graph Parameters
This dissertation considers some of the advantages, and limits, of applying quantum computing to solve two important graph problems. The first is estimating a graph\u27s quantum chromatic number. The quantum chromatic number is the minimum number of colors necessary in a two-player game where the players cannot communicate but share an entangled state and must convince a referee with probability one that they have a proper vertex coloring. We establish several spectral lower bounds for the quantum chromatic number. These lower bounds extend the well-known Hoffman lower bound for the classical chromatic number. The second is the Pattern Matching on Labeled Graphs Problem (PMLG). Here the objective is to match a string (called a pattern) P to a walk in an edge labeled graph G = (V, E). In addition to providing a new quantum algorithm for PMLG, this work establishes conditional lower bounds on the time complexity of any quantum algorithm for PMLG. These include a conditional lower bound based on the recently proposed NC-QSETH and a reduction from the Longest Common Subsequence Problem (LCS). For PMLG where substitutions are allowed to the pattern, our results demonstrate that (i) a quantum algorithm running in time O(|E|m1-ε + |E|1-εm) for any constant ε \u3e 0 would provide an algorithm for LCS on two strings X and Y running in time Õ(|X||Y|1-ε + |X|1-ε|Y|), which is better than any known quantum algorithm for LCS, and (ii) a quantum algorithm running in time O(|E|m½-ε + |E|½-εm) would violate NC-QSETH. Results (i) and (ii) hold even when restricted to binary alphabets for P and the edge labels in G. Our quantum algorithm is for all versions of PMLG (exact, only substitutions, and substitutions/insertions/deletions) and runs in time Õ(√|V||E|· m), making it an improvement over the classical O(|E|m) time algorithm when the graph is non-sparse