5 research outputs found
Towards the Characterization of Terminal Cut Functions: a Condition for Laminar Families
We study the following characterization problem. Given a set of terminals
and a -dimensional vector whose coordinates are indexed by
proper subsets of , is there a graph that contains , such that for
all subsets , equals the value of
the min-cut in separating from ? The only known necessary
conditions are submodularity and a special class of linear inequalities given
by Chaudhuri, Subrahmanyam, Wagner and Zaroliagis.
Our main result is a new class of linear inequalities concerning laminar
families, that generalize all previous ones. Using our new class of
inequalities, we can generalize Karger's approximate min-cut counting result to
graphs with terminals
Cut Sparsification and Succinct Representation of Submodular Hypergraphs
In cut sparsification, all cuts of a hypergraph are approximated
within factor by a small hypergraph . This widely applied
method was generalized recently to a setting where the cost of cutting each
is provided by a splitting function, . This
generalization is called a submodular hypergraph when the functions
are submodular, and it arises in machine learning,
combinatorial optimization, and algorithmic game theory. Previous work focused
on the setting where is a reweighted sub-hypergraph of , and measured
size by the number of hyperedges in . We study such sparsification, and
also a more general notion of representing succinctly, where size is
measured in bits.
In the sparsification setting, where size is the number of hyperedges, we
present three results: (i) all submodular hypergraphs admit sparsifiers of size
polynomial in ; (ii) monotone-submodular hypergraphs admit sparsifiers
of size ; and (iii) we propose a new parameter, called
spread, to obtain even smaller sparsifiers in some cases.
In the succinct-representation setting, we show that a natural family of
splitting functions admits a succinct representation of much smaller size than
via reweighted subgraphs (almost by factor ). This large gap is surprising
because for graphs, the most succinct representation is attained by reweighted
subgraphs. Along the way, we introduce the notion of deformation, where
is decomposed into a sum of functions of small description, and we provide
upper and lower bounds for deformation of common splitting functions
Cuts and connectivity in graphs and hypergraphs
In this thesis, we consider cut and connectivity problems on graphs, digraphs, hypergraphs and hedgegraphs.
The main results are the following:
- We introduce a faster algorithm for finding the reduced graph in element-connectivity computations. We also show its application to node separation.
- We present several results on hypergraph cuts, including (a) a near linear time algorithm for finding a (2+epsilon)-approximate min-cut, (b) an algorithm to find a representation of all min-cuts in the same time as finding a single min-cut, (c) a sparse subgraph that preserves connectivity for hypergraphs and (d) a near linear-time hypergraph cut sparsifier.
- We design the first randomized polynomial time algorithm for the hypergraph k-cut problem whose complexity has been open for over 20 years. The algorithm generalizes to hedgegraphs with constant span.
- We address the complexity gap between global vs. fixed-terminal cuts problems in digraphs by presenting a 2-1/448 approximation algorithm for the global bicut problem