120,808 research outputs found
Sum-of-squares lower bounds for planted clique
Finding cliques in random graphs and the closely related "planted" clique
variant, where a clique of size k is planted in a random G(n, 1/2) graph, have
been the focus of substantial study in algorithm design. Despite much effort,
the best known polynomial-time algorithms only solve the problem for k ~
sqrt(n).
In this paper we study the complexity of the planted clique problem under
algorithms from the Sum-of-squares hierarchy. We prove the first average case
lower bound for this model: for almost all graphs in G(n,1/2), r rounds of the
SOS hierarchy cannot find a planted k-clique unless k > n^{1/2r} (up to
logarithmic factors). Thus, for any constant number of rounds planted cliques
of size n^{o(1)} cannot be found by this powerful class of algorithms. This is
shown via an integrability gap for the natural formulation of maximum clique
problem on random graphs for SOS and Lasserre hierarchies, which in turn follow
from degree lower bounds for the Positivestellensatz proof system.
We follow the usual recipe for such proofs. First, we introduce a natural
"dual certificate" (also known as a "vector-solution" or "pseudo-expectation")
for the given system of polynomial equations representing the problem for every
fixed input graph. Then we show that the matrix associated with this dual
certificate is PSD (positive semi-definite) with high probability over the
choice of the input graph.This requires the use of certain tools. One is the
theory of association schemes, and in particular the eigenspaces and
eigenvalues of the Johnson scheme. Another is a combinatorial method we develop
to compute (via traces) norm bounds for certain random matrices whose entries
are highly dependent; we hope this method will be useful elsewhere
Online choosability of graphs
We study several problems in graph coloring. In list coloring, each vertex has a set of available colors and must be assigned a color from this set so that adjacent vertices receive distinct colors; such a coloring is an -coloring, and we then say that is -colorable. Given a graph and a function , we say that is -choosable if is -colorable for any list assignment such that for all . When for all and is -choosable, we say that is -choosable. The least such that is -choosable is the choice number, denoted . We focus on an online version of this problem, which is modeled by the Lister/Painter game.
The game is played on a graph in which every vertex has a positive number of tokens. In each round, Lister marks a nonempty subset of uncolored vertices, removing one token at each marked vertex. Painter responds by selecting a subset of that forms an independent set in . A color distinct from those used on previous rounds is given to all vertices in . Lister wins by marking a vertex that has no tokens, and Painter wins by coloring all vertices in . When Painter has a winning strategy, we say that is -paintable. If for all and is -paintable, then we say that is -paintable. The least such that is -paintable is the paint number, denoted \pa(G).
In Chapter 2, we develop useful tools for studying the Lister/Painter game. We study the paintability of graph joins and of complete bipartite graphs. In particular, \pa(K_{k,r})\le k if and only if .
In Chapter 3, we study the Lister/Painter game with the added restriction that the proper coloring produced by Painter must also satisfy some property . The main result of Chapter 3 provides a general method to give a winning strategy for Painter when a strategy for the list coloring problem is already known. One example of a property is that of having an -dynamic coloring, where a proper coloring is -dynamic if each vertex has at least distinct colors in its neighborhood. For any graph and any , we give upper bounds on how many tokens are necessary for Painter to produce an -dynamic coloring of . The upper bounds are in terms of and the genus of a surface on which embeds.
In Chapter 4, we study a version of the Lister/Painter game in which Painter must assign colors to each vertex so that adjacent vertices receive disjoint color sets. We characterize the graphs in which tokens is sufficient to produce such a coloring. We strengthen Brooks' Theorem as well as Thomassen's result that planar graphs are 5-choosable.
In Chapter 5, we study sum-paintability. The sum-paint number of a graph , denoted \spa(G), is the least over all such that is -paintable. We prove the easy upper bound: \spa(G)\le|V(G)|+|E(G)|. When \spa(G)=|V(G)|+|E(G)|, we say that is sp-greedy. We determine the sum-paintability of generalized theta-graphs. The generalized theta-graph consists of two vertices joined by paths of lengths \VEC \ell1k. We conjecture that outerplanar graphs are sp-greedy and prove several partial results toward this conjecture.
In Chapter 6, we study what happens when Painter is allowed to allocate tokens as Lister marks vertices. The slow-coloring game is played by Lister and Painter on a graph . Lister marks a nonempty set of uncolored vertices and scores 1 point for each marked vertex. Painter colors all vertices in an independent subset of the marked vertices with a color distinct from those used previously in the game. The game ends when all vertices have been colored. The sum-color cost of a graph , denoted \scc(G), is the maximum score Lister can guarantee in the slow-coloring game on . We prove several general lower and upper bounds for \scc(G). In more detail, we study trees and prove sharp upper and lower bounds over all trees with vertices. We give a formula to determine \scc(G) exactly when . Separately, we prove that \scc(G)=\spa(G) if and only if is a disjoint union of cliques. Lastly, we give lower and upper bounds on \scc(K_{r,s})
Faster Algorithms for Rectangular Matrix Multiplication
Let {\alpha} be the maximal value such that the product of an n x n^{\alpha}
matrix by an n^{\alpha} x n matrix can be computed with n^{2+o(1)} arithmetic
operations. In this paper we show that \alpha>0.30298, which improves the
previous record \alpha>0.29462 by Coppersmith (Journal of Complexity, 1997).
More generally, we construct a new algorithm for multiplying an n x n^k matrix
by an n^k x n matrix, for any value k\neq 1. The complexity of this algorithm
is better than all known algorithms for rectangular matrix multiplication. In
the case of square matrix multiplication (i.e., for k=1), we recover exactly
the complexity of the algorithm by Coppersmith and Winograd (Journal of
Symbolic Computation, 1990).
These new upper bounds can be used to improve the time complexity of several
known algorithms that rely on rectangular matrix multiplication. For example,
we directly obtain a O(n^{2.5302})-time algorithm for the all-pairs shortest
paths problem over directed graphs with small integer weights, improving over
the O(n^{2.575})-time algorithm by Zwick (JACM 2002), and also improve the time
complexity of sparse square matrix multiplication.Comment: 37 pages; v2: some additions in the acknowledgment
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