Searcharxiv⌕ Search

arXiv subjects

Sophie Spirkl

Publications and source records attributed to Sophie Spirkl.

107 records · Page 6Linked to original sources
arXiv2019

Disproportionate division

We study the disproportionate version of the classical cake-cutting problem: how efficiently can we divide a cake, here $[0,1]$, among $n$ agents with different demands $α_1, α_2, \dots, α_n$ summing to $1$? When all the agents have equal demands of $α_1 = α_2 = \dots = α_n = 1/n$, it is well-known that there exists a fair division with $n-1$ cuts, and this is optimal. For arbitrary demands on the other hand, folklore arguments from algebraic topology show that $O(n\log n)$ cuts suffice, and this has been the state of the art for decades. Here, we improve the state of affairs in two ways: we prove that disproportionate division may always be achieved with $3n-4$ cuts, and give an effective combinatorial procedure to construct such a division. We also offer a topological conjecture that implies that $2n-2$ cuts suffice in general, which would be optimal.

math.CO↗
arXiv2019

Short directed cycles in bipartite digraphs

The Caccetta-Häggkvist conjecture implies that for every integer $k\ge 1$, if $G$ is a bipartite digraph, with $n$ vertices in each part, and every vertex has out-degree more than $n/(k+1)$, then $G$ has a directed cycle of length at most $2k$. If true this is best possible, and we prove this for $k = 1,2,3,4,6$ and all $k\ge 224,539$. More generally, we conjecture that for every integer $k\ge 1$, and every pair of reals $α, β> 0$ with $kα+β>1$, if $G$ is a bipartite digraph with bipartition $(A,B)$, where every vertex in $A$ has out-degree at least $β|B|$, and every vertex in $B$ has out-degree at least $α|A|$, then $G$ has a directed cycle of length at most $2k$. This implies the Caccetta-Häggkvist conjecture (set $β>0$ and very small), and again is best possible for infinitely many pairs $(α,β)$. We prove this for $k = 1,2$, and prove a weaker statement (that $α+β>2/(k+1)$ suffices) for $k=3,4$.

math.CO↗
arXiv2019

$H$-colouring $P_t$-free graphs in subexponential time

A graph is called $P_t$-free if it does not contain the path on $t$ vertices as an induced subgraph. Let $H$ be a multigraph with the property that any two distinct vertices share at most one common neighbour. We show that the generating function for (list) graph homomorphisms from $G$ to $H$ can be calculated in subexponential time $2^{O\left(\sqrt{tn\log(n)}\right)}$ for $n=|V(G)|$ in the class of $P_t$-free graphs $G$. As a corollary, we show that the number of 3-colourings of a $P_t$-free graph $G$ can be found in subexponential time. On the other hand, no subexponential time algorithm exists for 4-colourability of $P_t$-free graphs assuming the Exponential Time Hypothesis. Along the way, we prove that $P_t$-free graphs have pathwidth that is linear in their maximum degree.

cs.DM↗
arXiv2019

Detecting an odd hole

A hole in a graph G is an induced cycle of length at least four; an antihole is a hole in the complement of G. In 2005, Chudnovsky, Cornuejols, Liu, Seymour and Vuskovic showed that it is possible to test in polynomial time whether a graph contains an odd hole or antihole (and thus whether G is perfect). However, the complexity of testing for odd holes has remained open. Indeed, it seemed quite likely that testing for an odd hole was NP-complete: for instance, Bienstock showed that testing if a graph has an odd hole containing a given vertex is NP-complete. In this paper we resolve the question, by giving a polynomial-time algorithm to test whether a graph contains an odd hole. This also gives a new and considerably simpler polynomial-time algorithm that tests for perfection.

math.CO↗
arXiv2018

Caterpillars in Erdős-Hajnal

Let $T$ be a tree such that all its vertices of degree more than two lie on one path, that is, $T$ is a caterpillar subdivision. We prove that there exists $ε>0$ such that for every graph $G$ with $|V(G)|\ge 2$ not containing $T$ as an induced subgraph, either some vertex has at least $ε|V(G)|$ neighbours, or there are two disjoint sets of vertices $A,B$, both of cardinality at least $ε|V(G)|$, where there is no edge joining $A$ and $B$. A consequence is: for every caterpillar subdivision $T$, there exists $c>0$ such that for every graph $G$ containing neither of $T$ and its complement as an induced subgraph, $G$ has a clique or stable set with at least $|V(G)|^c$ vertices. This extends a theorem of Bousquet, Lagoutte and Thomassé [JCTB 2015], who proved the same when $T$ is a path, and a recent theorem of Choromanski, Falik, Liebenau, Patel and Pilipczuk [Electron. J. Combin. 2018], who proved it when $T$ is a "hook".

math.CO↗
arXiv2018

Proof of the Kalai-Meshulam conjecture

Let $G$ be a graph, and let $f_G$ be the sum of $(-1)^{|A|}$, over all stable sets $A$. If $G$ is a cycle with length divisible by three, then $f_G= \pm 2$. Motivated by topological considerations, G. Kalai and R. Meshulam made the conjecture that,if no induced cycle of a graph $G$ has length divisible by three, then $|f_G|\le 1$. We prove this conjecture.

math.CO↗
arXiv2018

Four-coloring $P_6$-free graphs. I. Extending an excellent precoloring

This is the first paper in a series whose goal is to give a polynomial time algorithm for the $4$-coloring problem and the $4$-precoloring extension problem restricted to the class of graphs with no induced six-vertex path, thus proving a conjecture of Huang. Combined with previously known results this completes the classification of the complexity of the $4$-coloring problem for graphs with a connected forbidden induced subgraph. In this paper we give a polynomial time algorithm that determines if a special kind of precoloring of a $P_6$-free graph has a precoloring extension, and constructs such an extension if one exists. Combined with the main result of the second paper of the series, this gives a complete solution to the problem.

math.CO↗
arXiv2018

List-three-coloring graphs with no induced $P_6+rP_3$

For an integer $r$, the graph $P_6+rP_3$ has $r+1$ components, one of which is a path on $6$ vertices, and each of the others is a path on $3$ vertices. In this paper we provide a polynomial-time algorithm to test if a graph with no induced subgraph isomorphic to $P_6+rP_3$ is three-colorable. We also solve the list version of this problem, where each vertex is assigned a list of possible colors, which is a subset of $\{1,2,3\}$.

math.CO↗
arXiv2018

Towards Erdos-Hajnal for graphs with no 5-hole

The Erdos-Hajnal conjecture says that for every graph $H$ there exists $c>0$ such that $\max(α(G),ω(G))\ge n^c$ for every $H$-free graph $G$ with $n$ vertices, and this is still open when $H=C_5$. Until now the best bound known on $\max(α(G),ω(G))$ for $C_5$-free graphs was the general bound of Erdos and Hajnal, that for all $H$, $\max(α(G),ω(G))\ge 2^{Ω(\sqrt{\log n })}$ if $G$ is $H$-free. We improve this when $H=C_5$ to $\max(α(G),ω(G))\ge 2^{Ω(\sqrt{\log n \log \log n})}.$

math.CO↗
arXiv2018

Four-coloring $P_6$-free graphs. II. Finding an excellent precoloring

This is the second paper in a series of two. The goal of the series is to give a polynomial time algorithm for the $4$-coloring problem and the $4$-precoloring extension problem restricted to the class of graphs with no induced six-vertex path, thus proving a conjecture of Huang. Combined with previously known results this completes the classification of the complexity of the $4$-coloring problem for graphs with a connected forbidden induced subgraph. In this paper we give a polynomial time algorithm that starts with a $4$-precoloring of a graph with no induced six-vertex path, and outputs a polynomial-size collection of so-called excellent precolorings. Excellent precolorings are easier to handle than general ones, and, in addition, in order to determine whether the initial precoloring can be extended to the whole graph, it is enough to answer the same question for each of the excellent precolorings in the collection. The first paper in the series deals with excellent precolorings, thus providing a complete solution to the problem.

math.CO↗
arXiv2017

Piercing axis-parallel boxes

Let $\F$ be a finite family of axis-parallel boxes in $\R^d$ such that $\F$ contains no $k+1$ pairwise disjoint boxes. We prove that if $\F$ contains a subfamily $\M$ of $k$ pairwise disjoint boxes with the property that for every $F\in \F$ and $M\in \M$ with $F \cap M \neq \emptyset$, either $F$ contains a corner of $M$ or $M$ contains $2^{d-1}$ corners of $F$, then $\F$ can be pierced by $O(k)$ points. One consequence of this result is that if $d=2$ and the ratio between any of the side lengths of any box is bounded by a constant, then $\F$ can be pierced by $O(k)$ points. We further show that if for each two intersecting boxes in $\F$ a corner of one is contained in the other, then $\F$ can be pierced by at most $O(k\log\log(k))$ points, and in the special case where $\F$ contains only cubes this bound improves to $O(k)$.

math.CO↗
arXiv2017

Colouring perfect graphs with bounded clique number

A graph is perfect if the chromatic number of every induced subgraph equals the size of its largest clique, and an algorithm of Grötschel, Lovász, and Schrijver from 1988 finds an optimal colouring of a perfect graph in polynomial time. But this algorithm uses the ellipsoid method, and it is a well-known open question to construct a "combinatorial" polynomial-time algorithm that yields an optimal colouring of a perfect graph. A skew partition in $G$ is a partition $(A,B)$ of $V(G)$ such that $G[A]$ is not connected and $\bar{G}[B]$ is not connected, where $\bar{G}$ denotes the complement graph ; and it is balanced if an additional parity condition of paths in $G$ and $\bar{G}$ is satisfied. In this paper we first give a polynomial-time algorithm that, with input a perfect graph, outputs a balanced skew partition if there is one. Then we use this to obtain a combinatorial algorithm that finds an optimal colouring of a perfect graph with clique number $k$, in time that is polynomial for fixed $k$.

math.CO↗
arXiv2016

Approximately coloring graphs without long induced paths

It is an open problem whether the 3-coloring problem can be solved in polynomial time in the class of graphs that do not contain an induced path on $t$ vertices, for fixed $t$. We propose an algorithm that, given a 3-colorable graph without an induced path on $t$ vertices, computes a coloring with $\max\{5,2\lceil{\frac{t-1}{2}}\rceil-2\}$ many colors. If the input graph is triangle-free, we only need $\max\{4,\lceil{\frac{t-1}{2}}\rceil+1\}$ many colors. The running time of our algorithm is $O((3^{t-2}+t^2)m+n)$ if the input graph has $n$ vertices and $m$ edges.

math.CO↗
arXiv2014

Fast Prefix Adders for Non-Uniform Input Arrival Times

We consider the problem of constructing fast and small parallel prefix adders for non-uniform input arrival times. This problem arises whenever the adder is embedded into a more complex circuit, e. g. a multiplier. Most previous results are based on representing binary carry-propagate adders as so-called parallel prefix graphs, in which pairs of generate and propagate signals are combined using complex gates known as prefix gates. Adders constructed in this model usually minimize the delay in terms of these prefix gates. However, the delay in terms of logic gates can be worse by a factor of two. In contrast, we aim to minimize the delay of the underlying logic circuit directly. We prove a lower bound on the delay of a carry bit computation achievable by any prefix carry bit circuit and develop an algorithm that computes a prefix carry bit circuit with optimum delay up to a small additive constant. Furthermore, we use this algorithm to construct a small parallel prefix adder. Compared to existing algorithms we simultaneously improve the delay and size guarantee, as well as the running time for constructing prefix carry bit and adder circuits.

cs.AR↗
arXiv2014

The guillotine approach for TSP with neighborhoods revisited

The Euclidean TSP with neighborhoods (TSPN) is the following problem: Given a set R of k regions, find a shortest tour that visits at least one point from each region. We study the special cases of disjoint, connected, alpha-fat regions (i.e., every region P contains a disk of diameter diam(P)/alpha) and disjoint unit disks. For the latter, Dumitrescu and Mitchell proposed an algorithm based on Mitchell's guillotine subdivision approach for the Euclidean TSP and claimed it to be a PTAS. However, their proof contains a severe gap, which we will close in the following. Bodlaender et al. remark that their techniques for the minimum corridor connection problem carry over to the TSPN and yield an alternative PTAS for this problem. For disjoint connected alpha-fat regions of varying size, Mitchell proposed a slightly different PTAS candidate. We will expose several further problems and gaps in this approach. Some of them we can close, but overall, for alpha-fat regions, the existence of a PTAS for the TSPN remains open.

cs.CG↗