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Register a new userAssigning times to minimise reachability in temporal graphs
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Kitty Meeks
Publication date
2018
fields
Informatics Engineering
and research's language is
English
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Temporal graphs (in which edges are active at specified times) are of particular relevance for spreading processes on graphs, e.g.~the spread of disease or dissemination of information. Motivated by real-world applications, modification of static graphs to control this spread has proven a rich topic for previous research. Here, we introduce a new type of modification for temporal graphs: the number of active times for each edge is fixed, but we can change the relative order in which (sets of) edges are active. We investigate the problem of determining an ordering of edges that minimises the maximum number of vertices reachable from any single starting vertex; epidemiologically, this corresponds to the worst-case number of vertices infected in a single disease outbreak. We study t
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The first-fit coloring is a heuristic that assigns to each vertex, arriving in a specified order $sigma$, the smallest available color. The problem Grundy Coloring asks how many colors are needed for the most adversarial vertex ordering $sigma$, i.e., the maximum number of colors that the first-fit coloring requires over all possible vertex orderings. Since its inception by Grundy in 1939, Grundy Coloring has been examined for its structural and algorithmic aspects. A brute-force $f(k)n^{2^{k-1}}$-time algorithm for Grundy Coloring on general graphs is not difficult to obtain, where $k$ is the number of colors required by the most adversarial vertex ordering. It was asked several times whether the dependency on $k$ in the exponent of $n$ can be avoided or reduced, and its answer seemed elusive until now. We prove that Grundy Coloring is W[1]-hard and the brute-force algorithm is essentially optimal under the Exponential Time Hypothesis, thus settling this question by the negative. The key ingredient in our W[1]-hardness proof is to use so-called half-graphs as a building block to transmit a color from one vertex to another. Leveraging the half-graphs, we also prove that b-Chromatic Core is W[1]-hard, whose parameterized complexity was posed as an open question by Panolan et al. [JCSS 17]. A natural follow-up question is, how the parameterized complexity changes in the absence of (large) half-graphs. We establish fixed-parameter tractability on $K_{t,t}$-free graphs for b-Chromatic Core and Partial Grundy Coloring, making a step toward answering this question. The key combinatorial lemma underlying the tractability result might be of independent interest.
The problem of graph Reachability is to decide whether there is a path from one vertex to another in a given graph. In this paper, we study the Reachability problem on three distinct graph families - intersection graphs of Jordan regions, unit contact disk graphs (penny graphs), and chordal graphs. For each of these graph families, we present space-efficient algorithms for the Reachability problem. For intersection graphs of Jordan regions, we show how to obtain a good vertex separator in a space-efficient manner and use it to solve the Reachability in polynomial time and $O(m^{1/2}log n)$ space, where $n$ is the number of Jordan regions, and $m$ is the total number of crossings among the regions. We use a similar approach for chordal graphs and obtain a polynomial-time and $O(m^{1/2}log n)$ space algorithm, where $n$ and $m$ are the number of vertices and edges, respectively. However, we use a more involved technique for unit contact disk graphs (penny graphs) and obtain a better algorithm. We show that for every $epsilon> 0$, there exists a polynomial-time algorithm that can solve Reachability in an $n$ vertex directed penny graph, using $O(n^{1/4+epsilon})$ space. We note that the method used to solve penny graphs does not extend naturally to the class of geometric intersection graphs that include arbitrary size cliques.
We show that for each single crossing graph $H$, a polynomially bounded weight function for all $H$-minor free graphs $G$ can be constructed in Logspace such that it gives nonzero weights to all the cycles in $G$. This class of graphs subsumes almost all classes of graphs for which such a weight function is known to be constructed in Logspace. As a consequence, we obtain that for the class of $H$-minor free graphs where $H$ is a single crossing graph, reachability can be solved in UL, and bipartite maximum matching can be solved in SPL, which are small subclasses of the parallel complexity class NC. In the restrictive case of bipartite graphs, our maximum matching result improves upon the recent result of Eppstein and Vazirani, where they show an NC bound for constructing perfect matching in general single crossing minor free graphs.
Many papers in the field of integer linear programming (ILP, for short) are devoted to problems of the type $max{c^top x colon A x = b,, x in mathbb{Z}^n_{geq 0}}$, where all the entries of $A,b,c$ are integer, parameterized by the number of rows of $A$ and $|A|_{max}$. This class of problems is known under the name of ILP problems in the standard form, adding the word bounded if $x leq u$, for some integer vector $u$. Recently, many new sparsity, proximity, and complexity results were obtained for bounded and unbounded ILP problems in the standard form. In this paper, we consider ILP problems in the canonical form $$max{c^top x colon b_l leq A x leq b_r,, x in mathbb{Z}^n},$$ where $b_l$ and $b_r$ are integer vectors. We assume that the integer matrix $A$ has the rank $n$, $(n + m)$ rows, $n$ columns, and parameterize the problem by $m$ and $Delta(A)$, where $Delta(A)$ is the maximum of $n times n$ sub-determinants of $A$, taken in the absolute value. We show that any ILP problem in the standard form can be polynomially reduced to some ILP problem in the canonical form, preserving $m$ and $Delta(A)$, but the reverse reduction is not always possible. More precisely, we define the class of generalized ILP problems in the standard form, which includes an additional group constraint, and prove the equivalence to ILP problems in the canonical form. We generalize known sparsity, proximity, and complexity bounds for ILP problems in the canonical form. Additionally, sometimes, we strengthen previously known results for ILP problems in the canonical form, and, sometimes, we give shorter proofs. Finally, we consider the special cases of $m in {0,1}$. By this way, we give specialised sparsity, proximity, and complexity bounds for the problems on simplices, Knapsack problems and Subset-Sum problems.
Butman, Hermelin, Lewenstein, and Rawitz proved that Clique in t-interval graphs is NP-hard for t >= 3. We strengthen this result to show that Clique in 3-track interval graphs is APX-hard.
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