11 research outputs found
Integer programs with bounded subdeterminants and two nonzeros per row
We give a strongly polynomial-time algorithm for integer linear programs
defined by integer coefficient matrices whose subdeterminants are bounded by a
constant and that contain at most two nonzero entries in each row. The core of
our approach is the first polynomial-time algorithm for the weighted stable set
problem on graphs that do not contain more than vertex-disjoint odd cycles,
where is any constant. Previously, polynomial-time algorithms were only
known for (bipartite graphs) and for .
We observe that integer linear programs defined by coefficient matrices with
bounded subdeterminants and two nonzeros per column can be also solved in
strongly polynomial-time, using a reduction to -matching
Weak Coloring Numbers of Intersection Graphs
Weak and strong coloring numbers are generalizations of the degeneracy of a
graph, where for each natural number , we seek a vertex ordering such every
vertex can (weakly respectively strongly) reach in steps only few vertices
with lower index in the ordering. Both notions capture the sparsity of a graph
or a graph class, and have interesting applications in the structural and
algorithmic graph theory. Recently, the first author together with McCarty and
Norin observed a natural volume-based upper bound for the strong coloring
numbers of intersection graphs of well-behaved objects in , such
as homothets of a centrally symmetric compact convex object, or comparable
axis-aligned boxes.
In this paper, we prove upper and lower bounds for the -th weak coloring
numbers of these classes of intersection graphs. As a consequence, we describe
a natural graph class whose strong coloring numbers are polynomial in , but
the weak coloring numbers are exponential. We also observe a surprising
difference in terms of the dependence of the weak coloring numbers on the
dimension between touching graphs of balls (single-exponential) and hypercubes
(double-exponential)
The ?-t-Net Problem
We study a natural generalization of the classical ?-net problem (Haussler - Welzl 1987), which we call the ?-t-net problem: Given a hypergraph on n vertices and parameters t and ? ? t/n, find a minimum-sized family S of t-element subsets of vertices such that each hyperedge of size at least ? n contains a set in S. When t=1, this corresponds to the ?-net problem.
We prove that any sufficiently large hypergraph with VC-dimension d admits an ?-t-net of size O((1+log t)d/? log 1/?). For some families of geometrically-defined hypergraphs (such as the dual hypergraph of regions with linear union complexity), we prove the existence of O(1/?)-sized ?-t-nets.
We also present an explicit construction of ?-t-nets (including ?-nets) for hypergraphs with bounded VC-dimension. In comparison to previous constructions for the special case of ?-nets (i.e., for t=1), it does not rely on advanced derandomization techniques. To this end we introduce a variant of the notion of VC-dimension which is of independent interest
Towards Efficient Private Distributed Computation on Unbounded Input Streams
In the problem of private ``swarm\u27\u27 computing, agents wish to securely and
distributively perform a computation on common inputs, in such a way
that even if the entire memory contents of some of them are exposed,
no information is revealed about the state of the computation.
Recently, Dolev, Garay, Gilboa and Kolesnikov [ICS 2011] considered this
problem in the setting of information-theoretic security, showing how to perform such computations on input streams of {\em unbounded length}. The cost of their solution, however, is exponential in the size of the Finite State Automaton (FSA) computing the function.
In this work we are interested in efficient (i.e., polynomial time)
computation in the above model, at the expense of {\em minimal}
additional assumptions. Relying on the existence of one-way functions,
we show how to process unbounded inputs (but of course, polynomial in the security parameter) at a cost {\em linear} in , the number of FSA states. In
particular, our algorithms achieve the following:
\begin{tiret}
\item In the case of -reconstruction (i.e., in which all
agents participate in the reconstruction of the distributed
computation) and at most agents are corrupted, the agent
storage, the time required to process each input symbol, and the time
complexity for reconstruction are all .
\item
In the case of -reconstruction (where only agents
take part in the reconstruction) and at most agents are corrupted,
the agents\u27 storage and time required to process each input symbol are .
The complexity of reconstruction is .
\end{tiret}
We achieve the above through a carefully orchestrated use of pseudo-random generators and secret-sharing, and in particular a novel share
re-randomization technique which might be of independent interest
Almost all string graphs are intersection graphs of plane convex sets
A string graph is the intersection graph of a family of continuous arcs in the plane. The intersection graph of a family of plane convex sets is a string graph, but not all string graphs can be obtained in this way. We prove the following structure theorem conjectured by Janson and Uzzell: The vertex set of almost all string graphs on n vertices can be partitioned into five cliques such that some pair of them is not connected by any edge (n→∞). We also show that every graph with the above property is an intersection graph of plane convex sets. As a corollary, we obtain that almost all string graphs on n vertices are intersection graphs of plane convex sets