459 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
Sparse graphs with bounded induced cycle packing number have logarithmic treewidth
A graph is -free if it does not contain pairwise vertex-disjoint and
non-adjacent cycles. We show that Maximum Independent Set and 3-Coloring in
-free graphs can be solved in quasi-polynomial time. As a main technical
result, we establish that "sparse" (here, not containing large complete
bipartite graphs as subgraphs) -free graphs have treewidth (even, feedback
vertex set number) at most logarithmic in the number of vertices. This is
proven sharp as there is an infinite family of -free graphs without
-subgraph and whose treewidth is (at least) logarithmic.
Other consequences include that most of the central NP-complete problems
(such as Maximum Independent Set, Minimum Vertex Cover, Minimum Dominating Set,
Minimum Coloring) can be solved in polynomial time in sparse -free graphs,
and that deciding the -freeness of sparse graphs is polynomial time
solvable.Comment: 28 pages, 6 figures. v3: improved complexity result
Combinatorics
This is the report on the Oberwolfach workshop on Combinatorics, held 1–7 January 2006. Combinatorics is a branch of mathematics studying families of mainly, but not exclusively, finite or countable structures – discrete objects. The discrete objects considered in the workshop were graphs, set systems, discrete geometries, and matrices. The programme consisted of 15 invited lectures, 18 contributed talks, and a problem session focusing on recent developments in graph theory, coding theory, discrete geometry, extremal combinatorics, Ramsey theory, theoretical computer science, and probabilistic combinatorics
Round and Bipartize for Vertex Cover Approximation
The vertex cover problem is a fundamental and widely studied combinatorial optimization problem. It is known that its standard linear programming relaxation is integral for bipartite graphs and half-integral for general graphs. As a consequence, the natural rounding algorithm based on this relaxation computes an optimal solution for bipartite graphs and a 2-approximation for general graphs. This raises the question of whether one can interpolate the rounding curve of the standard linear programming relaxation in a beyond the worst-case manner, depending on how close the graph is to being bipartite. In this paper, we consider a round-and-bipartize algorithm that exploits the knowledge of an induced bipartite subgraph to attain improved approximation ratios. Equivalently, we suppose that we work with a pair (?, S), consisting of a graph with an odd cycle transversal.
If S is a stable set, we prove a tight approximation ratio of 1 + 1/?, where 2? -1 denotes the odd girth (i.e., length of the shortest odd cycle) of the contracted graph ?? : = ?/S and satisfies ? ? [2,?], with ? = ? corresponding to the bipartite case. If S is an arbitrary set, we prove a tight approximation ratio of (1+1/?) (1 - ?) + 2 ?, where ? ? [0,1] is a natural parameter measuring the quality of the set S. The technique used to prove tight improved approximation ratios relies on a structural analysis of the contracted graph ??, in combination with an understanding of the weight space where the fully half-integral solution is optimal. Tightness is shown by constructing classes of weight functions matching the obtained upper bounds. As a byproduct of the structural analysis, we also obtain improved tight bounds on the integrality gap and the fractional chromatic number of 3-colorable graphs. We also discuss algorithmic applications in order to find good odd cycle transversals, connecting to the MinUncut and Colouring problems. Finally, we show that our analysis is optimal in the following sense: the worst case bounds for ? and ?, which are ? = 2 and ? = 1 - 4/n, recover the integrality gap of 2 - 2/n of the standard linear programming relaxation, where n is the number of vertices of the graph
Graph Theory
Highlights of this workshop on structural graph theory included new developments on graph and matroid minors, continuous structures arising as limits of finite graphs, and new approaches to higher graph connectivity via tree structures
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