11 research outputs found
An upper bound for the chromatic number of line graphs
It was conjectured by Reed [reed98conjecture] that for any graph , the graph's chromatic number is bounded above by , where and are the maximum degree and clique number of , respectively. In this paper we prove that this bound holds if is the line graph of a multigraph. The proof yields a polynomial time algorithm that takes a line graph and produces a colouring that achieves our bound
Goldberg's Conjecture is true for random multigraphs
In the 70s, Goldberg, and independently Seymour, conjectured that for any
multigraph , the chromatic index satisfies , where . We show that their conjecture (in a
stronger form) is true for random multigraphs. Let be the probability
space consisting of all loopless multigraphs with vertices and edges,
in which pairs from are chosen independently at random with
repetitions. Our result states that, for a given ,
typically satisfies . In
particular, we show that if is even and , then
for a typical . Furthermore, for a fixed
, if is odd, then a typical has
for , and
for .Comment: 26 page
Proof of the Goldberg-Seymour Conjecture on Edge-Colorings of Multigraphs
Given a multigraph , the {\em edge-coloring problem} (ECP) is to
color the edges of with the minimum number of colors so that no two
adjacent edges have the same color. This problem can be naturally formulated as
an integer program, and its linear programming relaxation is called the {\em
fractional edge-coloring problem} (FECP). In the literature, the optimal value
of ECP (resp. FECP) is called the {\em chromatic index} (resp. {\em fractional
chromatic index}) of , denoted by (resp. ). Let
be the maximum degree of and let where is the set of all edges of with
both ends in . Clearly, is
a lower bound for . As shown by Seymour, . In the 1970s Goldberg and Seymour independently conjectured
that . Over the
past four decades this conjecture, a cornerstone in modern edge-coloring, has
been a subject of extensive research, and has stimulated a significant body of
work. In this paper we present a proof of this conjecture. Our result implies
that, first, there are only two possible values for , so an analogue
to Vizing's theorem on edge-colorings of simple graphs, a fundamental result in
graph theory, holds for multigraphs; second, although it is -hard in
general to determine , we can approximate it within one of its true
value, and find it exactly in polynomial time when ;
third, every multigraph satisfies , so FECP has a
fascinating integer rounding property
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Stochastic Local Search and the Lovasz Local Lemma
Stochastic Local Search and the Lovasz Local LemmabyFotios IliopoulosDoctor of Philosophy in Computer ScienceUniversity of California, BerkeleyProfessor Alistair Sinclair, ChairThis thesis studies randomized local search algorithms for finding solutions of constraint satisfaction problems inspired by and extending the Lovasz Local Lemma (LLL). The LLL is a powerful probabilistic tool for establishing the existence of objects satisfying certain properties (constraints). As a probability statement it asserts that, given a family of “bad” events, if each bad event is individually not very likely and independent of all but a small number of other bad events, then the probability of avoiding all bad events is strictly positive. In a celebrated breakthrough, Moser and Tardos made the LLL constructive for any product probability measure over explicitly presented variables. Specifically, they proved that whenever the LLL condition holds, their Resample algorithm, which repeatedly selects any occurring bad event and resamples all its variables according to the measure, quickly converges to an object with desired properties. In this dissertation we present a framework that extends the work of Moser and Tardos and can be used to analyze arbitrary, possibly complex, focused local search algorithms, i.e., search algorithms whose process for addressing violated constraints, while local, is more sophisticated than obliviously resampling their variables independently of the current configuration. We give several applications of this framework, notably a new vertex coloring algorithm for graphs with sparse vertex neighborhoods that uses a number of colors that matches the algorithmic barrier for random graphs, and polynomial time algorithms for the celebrated (non-constructive) results of Kahn for the Goldberg-Seymour and List-Edge-Coloring Conjectures.Finally, we introduce a generalization of Kolmogorov’s notion of commutative algorithms, cast as matrix commutativity, and show that their output distribution approximates the so-called “LLL-distribution”, i.e., the distribution obtained by conditioning on avoiding all bad events. This fact allows us to consider questions such as the number of possible distinct final states and the probability that certain portions of the state space are visited by a local search algorithm, extending existing results for the Moser-Tardos algorithm to commutative algorithms
Approximating the chromatic index of multigraphs
It is well known that if G is a multigraph then χ′(G) ≥χ′ *(G):=max{Δ(G),Γ(G)}, where χ′(G) is the chromatic index of G, χ′ *(G) is the fractional chromatic index of G, Δ(G) is the maximum degree of G, and Γ(G)=max{2|E(G[U])|/(|U|-1):U ⊇ V(G),|U|≥3,|U| is odd}. The conjecture that χ′(G)≤max{Δ(G)+1,Γ(G) was made independently by Goldberg (Discret. Anal. 23:3-7, 1973), Anderson (Math. Scand. 40:161-175, 1977), and Seymour (Proc. Lond. Math. Soc. 38:423-460, 1979). Using a probabilistic argument Kahn showed that for any c>0 there exists D>0 such that χ′(G)≤χ′ *(G)+c χ′ *(G) when χ′ *(G)>D. Nishizeki and Kashiwagi proved this conjecture for multigraphs G with χ′(G)>⌊(11Δ(G)+8)/10⌋ and Scheide recently improved this bound to χ′(G)>⌊(15Δ(G)+12)/14⌋. We prove this conjecture for multigraphs G with χ′(G)> ⌊ Δ(G)+ Δ(G)/2 ⌋, improving the above mentioned results. As a consequence, for multigraphs G with χ(G)>Delta(G)+sqrt Δ(G)/2} the answer to a 1964 problem of Vizing is in the affirmative. © 2009 Springer Science+Business Media, LLC.link_to_subscribed_fulltex