scholarly journals (Star, k)-colourings of graphs with bounded treewidth

2020 ◽  
Author(s):  
C. A. Weffort-Santos ◽  
L. L. C. Pedrosa
Keyword(s):  
Open Problem ◽  
Time Algorithm ◽  
Graph Colouring ◽  
Bounded Treewidth ◽  
Colour Class

We study a generalization of graph colouring define as follows. Given a graph G, a (star, k)-colouring of G is a colouring c : V(G) → {1, ..., k} such that every colour class induces a star. We propose an O*(2^(O(tw))k^(tw)-time algorithm that decides whether a graph G of treewidth at most tw admits a (star, k)-colouring. This resolves an open problem posed by Angelini et al. in 2017. Our approach can be extended to other defective colouring models.

scholarly journals Metric Dimension Parameterized By Treewidth

Algorithmica ◽  
2021 ◽  
Author(s):  
Édouard Bonnet ◽  
Nidhi Purohit
Keyword(s):  
Computable Function ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Minimum Size ◽  
Metric Dimension ◽  
Resolving Set ◽  
Input Graph ◽  
Outerplanar Graphs ◽  
Bounded Treewidth ◽  
Fpt Algorithm

AbstractA resolving set S of a graph G is a subset of its vertices such that no two vertices of G have the same distance vector to S. The Metric Dimension problem asks for a resolving set of minimum size, and in its decision form, a resolving set of size at most some specified integer. This problem is NP-complete, and remains so in very restricted classes of graphs. It is also W[2]-complete with respect to the size of the solution. Metric Dimension has proven elusive on graphs of bounded treewidth. On the algorithmic side, a polynomial time algorithm is known for trees, and even for outerplanar graphs, but the general case of treewidth at most two is open. On the complexity side, no parameterized hardness is known. This has led several papers on the topic to ask for the parameterized complexity of Metric Dimension with respect to treewidth. We provide a first answer to the question. We show that Metric Dimension parameterized by the treewidth of the input graph is W[1]-hard. More refinedly we prove that, unless the Exponential Time Hypothesis fails, there is no algorithm solving Metric Dimension in time $$f(\text {pw})n^{o(\text {pw})}$$ f ( pw ) n o ( pw ) on n-vertex graphs of constant degree, with $$\text {pw}$$ pw the pathwidth of the input graph, and f any computable function. This is in stark contrast with an FPT algorithm of Belmonte et al. (SIAM J Discrete Math 31(2):1217–1243, 2017) with respect to the combined parameter $$\text {tl}+\Delta$$ tl + Δ , where $$\text {tl}$$ tl is the tree-length and $$\Delta$$ Δ the maximum-degree of the input graph.

DIAMETER FORMULAS FOR A CLASS OF UNDIRECTED DOUBLE-LOOP NETWORKS

2005 ◽  
Vol 06 (01) ◽  
pp. 1-15 ◽  
Author(s):  
BAOXING CHEN ◽  
WENJUN XIAO ◽  
BEHROOZ PARHAMI
Keyword(s):  
Closed Form ◽  
Open Problem ◽  
Time Algorithm ◽  
Double Loop ◽  
Network Diameter ◽  
Double Loop Network ◽  
Node Network ◽  
Loop Network ◽  
Double Loop Networks ◽  
Closed Form Formula

An n-node network, with nodes numbered 0 to n-1, is an undirected double-loop network with chord lengths 1 and s(2≤s<n/2) when each node i(0≤i<n) is connected to each of the four nodes i±1 and i±s via an undirected link; all node-index expressions are evaluated modulo n. Let n=qs+r, where r(0≤r<s) is the remainder of dividing n by s. Furthermore, let s=ar+b, where b(0≤b<r) is the remainder of dividing s by r. In this paper, we provide closed-form formulas for the diameter of a double-loop network for the case q>r and for a subcase of the case q≤r when b≤aq+1. In the complementary subcase of q≤r, when b>aq+1, network diameter can be derived by applying the O(log n)-time algorithm of Zerovnik and Pisanski (J. Algorithms, Vol. 14, pp. 226-243, 1993). Obtaining a closed-form formula for diameter of the double-loop network in the latter subcase remains an open problem.

scholarly journals Finding a Shortest Odd Hole

2021 ◽  
Vol 17 (2) ◽  
pp. 1-21
Author(s):  
Maria Chudnovsky ◽  
Alex Scott ◽  
Paul Seymour
Keyword(s):  
Polynomial Time ◽  
Open Problem ◽  
Polynomial Time Algorithm ◽  
Time Algorithm

An odd hole in a graph is an induced cycle with odd length greater than 3. In an earlier paper (with Sophie Spirkl), solving a longstanding open problem, we gave a polynomial-time algorithm to test if a graph has an odd hole. We subsequently showed that, for every t , there is a polynomial-time algorithm to test whether a graph contains an odd hole of length at least t . In this article, we give an algorithm that finds a shortest odd hole, if one exists.

scholarly journals Fast separation in a graph with an excluded minor

2005 ◽  
Vol DMTCS Proceedings vol. AE,... (Proceedings) ◽  
Author(s):  
Bruce Reed ◽  
David R. Wood
Keyword(s):  
Open Problem ◽  
Time Algorithm ◽  
Total Weight ◽  
Running Time ◽  
Fast Separation ◽  
Edge Graph ◽  
Excluded Minor ◽  
International Audience ◽  
Vertex Sets

International audience Let $G$ be an $n$-vertex $m$-edge graph with weighted vertices. A pair of vertex sets $A,B \subseteq V(G)$ is a $\frac{2}{3} - \textit{separation}$ of $\textit{order}$ $|A \cap B|$ if $A \cup B = V(G)$, there is no edge between $A \backslash B$ and $B \backslash A$, and both $A \backslash B$ and $B \backslash A$ have weight at most $\frac{2}{3}$ the total weight of $G$. Let $\ell \in \mathbb{Z}^+$ be fixed. Alon, Seymour and Thomas [$\textit{J. Amer. Math. Soc.}$ 1990] presented an algorithm that in $\mathcal{O}(n^{1/2}m)$ time, either outputs a $K_\ell$-minor of $G$, or a separation of $G$ of order $\mathcal{O}(n^{1/2})$. Whether there is a $\mathcal{O}(n+m)$ time algorithm for this theorem was left as open problem. In this paper, we obtain a $\mathcal{O}(n+m)$ time algorithm at the expense of $\mathcal{O}(n^{2/3})$ separator. Moreover, our algorithm exhibits a tradeoff between running time and the order of the separator. In particular, for any given $\epsilon \in [0,\frac{1}{2}]$, our algorithm either outputs a $K_\ell$-minor of $G$, or a separation of $G$ with order $\mathcal{O}(n^{(2-\epsilon )/3})$ in $\mathcal{O}(n^{1+\epsilon} +m)$ time.

scholarly journals The Tractability of the Shapley Value over Bounded Treewidth Matching Games

2017 ◽  
Author(s):  
Gianluigi Greco ◽  
Francesco Lupia ◽  
Francesco Scarcello
Keyword(s):  
Polynomial Time ◽  
Shapley Value ◽  
Open Problem ◽  
Positive Answer ◽  
Coalitional Games ◽  
Solution Concepts ◽  
Bounded Treewidth ◽  
The Shapley Value ◽  
Matching Games

Matching games form a class of coalitional games that attracted much attention in the literature. Indeed, several results are known about the complexity of computing over them {solution concepts}. In particular, it is known that computing the Shapley value is intractable in general, formally #P-hard, and feasible in polynomial time over games defined on trees. In fact, it was an open problem whether or not this tractability result holds over classes of graphs properly including acyclic ones. The main contribution of the paper is to provide a positive answer to this question, by showing that the Shapley value is tractable for matching games defined over graphs having bounded treewidth. The proposed technique has been implemented and tested on classes of graphs having different sizes and treewidth at most three.

scholarly journals Linear Index Coding via Semidefinite Programming

2013 ◽  
Vol 23 (2) ◽  
pp. 223-247 ◽  
Author(s):  
EDEN CHLAMTÁČ ◽  
ISHAY HAVIV
Keyword(s):  
Side Information ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Exact Expression ◽  
Graph Colouring ◽  
Semidefinite Program ◽  
Index Coding ◽  
Crucial Component ◽  
Linear Index ◽  
Combinatorial Result

In theindex codingproblem, introduced by Birk and Kol (INFOCOM, 1998), the goal is to broadcast ann-bit word tonreceivers (one bit per receiver), where the receivers haveside informationrepresented by a graphG. The objective is to minimize the length of a codeword sent to all receivers which allows each receiver to learn its bit. Forlinearindex coding, the minimum possible length is known to be equal to a graph parameter calledminrank(Bar-Yossef, Birk, Jayram and Kol,IEEE Trans. Inform. Theory, 2011).We show a polynomial-time algorithm that, given ann-vertex graphGwith minrankk, finds a linear index code forGof lengthÕ(nf(k)), wheref(k) depends only onk. For example, fork= 3 we obtainf(3) ≈ 0.2574. Our algorithm employs a semidefinite program (SDP) introduced by Karger, Motwani and Sudan for graph colouring (J. Assoc. Comput. Mach., 1998) and its refined analysis due to Arora, Chlamtac and Charikar (STOC, 2006). Since the SDP we use is not a relaxation of the minimization problem we consider, a crucial component of our analysis is anupper boundon the objective value of the SDP in terms of the minrank.At the heart of our analysis lies a combinatorial result which may be of independent interest. Namely, we show an exact expression for the maximum possible value of the Lovász ϑ-function of a graph with minrankk. This yields a tight gap between two classical upper bounds on the Shannon capacity of a graph.

scholarly journals Efficient Inference of Partial Types

1992 ◽  
Vol 21 (394) ◽  
Author(s):  
Dexter Kozen ◽  
Jens Palsberg ◽  
Michael I. Schwartzbach
Keyword(s):  
Open Problem ◽  
Finite Automaton ◽  
Lambda Calculus ◽  
Time Algorithm ◽  
Type Inference ◽  
Exponential Time ◽  
Canonical Solution ◽  
Persistent Data ◽  
Exponential Time Algorithm ◽  
Construction Works

Partial types for the lambda-calculus were introduced by Thatte in 1988 as a means of typing objects that are not typable with simple types, such as heterogeneous lists and persistent data In that paper he showed that type inference for partial types was semidecidable. Decidability remained open until quite recently, when O'Keefe and Wand gave an exponential time algorithm for type inference.<p>In this paper we give an O(n^3) algorithm. Our algorithm constructs a certain finite automaton that represents a canonical solution to a given set of type constraints. Moreover, the construction works equally well for recursive types; this solves an open problem stated by O'Keefe and Wand.</p>

A Note on the Minimum H-Subgraph Edge Deletion

2015 ◽  
Vol 26 (03) ◽  
pp. 399-411
Author(s):  
Alexander Grigoriev ◽  
Bert Marchal ◽  
Natalya Usotskaya
Keyword(s):  
Planar Graphs ◽  
General Framework ◽  
Approximation Scheme ◽  
Linear Time ◽  
Time Algorithm ◽  
Polynomial Time Approximation Scheme ◽  
Edge Deletion ◽  
Time Dynamic ◽  
Bounded Treewidth ◽  
Programming Algorithms

In this note we consider the following problem: Given a graph [Formula: see text] and a subgraph [Formula: see text], what is the smallest subset [Formula: see text] of edges in [Formula: see text] that needs to be deleted from the graph to make it [Formula: see text]-free? Several algorithmic results are presented. First, using the general framework of Courcelle [9], we show that, for a fixed subgraph [Formula: see text], the problem can be solved in linear time on graphs of bounded treewidth. It is known that the constant hidden in the big-O notation of Courcelle algorithm is big which makes the approach impractical. Thus, we present two explicit linear time dynamic programming algorithms on graphs of bounded treewidth for restricted settings of the problem with reasonable constants. Third, using the linear time algorithm for graphs of bounded treewidth, we design a Baker's type polynomial time approximation scheme for the problem on planar graphs.

scholarly journals Object Reachability via Swaps along a Line

2019 ◽  
Vol 33 ◽  
pp. 2037-2044 ◽  
Author(s):  
Sen Huang ◽  
Mingyu Xiao
Keyword(s):  
Social Network ◽  
Open Problem ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Resources Allocation ◽  
Weak Ties ◽  
Initial Endowment ◽  
Single Object ◽  
Star Network ◽  
Polynomially Solvable

The HOUSING MARKET problem is a widely studied resources allocation problem. In this problem, each agent can only receive a single object and has preferences over all objects. Starting from an initial endowment, we want to reach a certain assignment via a sequence of rational trades. We consider the problem whether an object is reachable for a given agent under a social network, where a trade between two agents is allowed if they are neighbors in the network and no participant has a deficit from the trade. Assume that the preferences of the agents are strict (no tie is allowed). This problem is polynomially solvable in a star-network and NPcomplete in a tree-network. It is left as a challenging open problem whether the problem is polynomially solvable when the network is a path. We answer this open problem positively by giving a polynomial-time algorithm. Furthermore, we show that the problem on a path will become NP-hard when the preferences of the agents are weak (ties are allowed).

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