ترغب بنشر مسار تعليمي؟ اضغط هنا

Parameterized Extension Complexity of Independent Set and Related Problems

160   0   0.0 ( 0 )
 نشر من قبل Hans Raj Tiwary
 تاريخ النشر 2015
  مجال البحث الهندسة المعلوماتية
والبحث باللغة English




اسأل ChatGPT حول البحث

Let $G$ be a graph on $n$ vertices and $mathrm{STAB}_k(G)$ be the convex hull of characteristic vectors of its independent sets of size at most $k$. We study extension complexity of $mathrm{STAB}_k(G)$ with respect to a fixed parameter $k$ (analogously to, e.g., parameterized computational complexity of problems). We show that for graphs $G$ from a class of bounded expansion it holds that $mathrm{xc}(mathrm{STAB}_k(G))leqslant mathcal{O}(f(k)cdot n)$ where the function $f$ depends only on the class. This result can be extended in a simple way to a wide range of similarly defined graph polytopes. In case of general graphs we show that there is {em no function $f$} such that, for all values of the parameter $k$ and for all graphs on $n$ vertices, the extension complexity of $mathrm{STAB}_k(G)$ is at most $f(k)cdot n^{mathcal{O}(1)}.$ While such results are not surprising since it is known that optimizing over $mathrm{STAB}_k(G)$ is $FPT$ for graphs of bounded expansion and $W[1]$-hard in general, they are also not trivial and in both cases stronger than the corresponding computational complexity results.



قيم البحث

اقرأ أيضاً

In this paper we study the problem of finding a small safe set $S$ in a graph $G$, i.e. a non-empty set of vertices such that no connected component of $G[S]$ is adjacent to a larger component in $G - S$. We enhance our understanding of the problem f rom the viewpoint of parameterized complexity by showing that (1) the problem is W[2]-hard when parameterized by the pathwidth $pw$ and cannot be solved in time $n^{o(pw)}$ unless the ETH is false, (2) it admits no polynomial kernel parameterized by the vertex cover number $vc$ unless $mathrm{PH} = Sigma^{mathrm{p}}_{3}$, but (3) it is fixed-parameter tractable (FPT) when parameterized by the neighborhood diversity $nd$, and (4) it can be solved in time $n^{f(cw)}$ for some double exponential function $f$ where $cw$ is the clique-width. We also present (5) a faster FPT algorithm when parameterized by solution size.
We study the NP-hard textsc{$k$-Sparsest Cut} problem ($k$SC) in which, given an undirected graph $G = (V, E)$ and a parameter $k$, the objective is to partition vertex set into $k$ subsets whose maximum edge expansion is minimized. Herein, the edge expansion of a subset $S subseteq V$ is defined as the sum of the weights of edges exiting $S$ divided by the number of vertices in $S$. Another problem that has been investigated is textsc{$k$-Small-Set Expansion} problem ($k$SSE), which aims to find a subset with minimum edge expansion with a restriction on the size of the subset. We extend previous studies on $k$SC and $k$SSE by inspecting their parameterized complexity. On the positive side, we present two FPT algorithms for both $k$SSE and 2SC problems where in the first algorithm we consider the parameter treewidth of the input graph and uses exponential space, and in the second we consider the parameter vertex cover number of the input graph and uses polynomial space. Moreover, we consider the unweighted version of the $k$SC problem where $k geq 2$ is fixed and proposed two FPT algorithms with parameters treewidth and vertex cover number of the input graph. We also propose a randomized FPT algorithm for $k$SSE when parameterized by $k$ and the maximum degree of the input graph combined. Its derandomization is done efficiently. oindent On the negative side, first we prove that for every fixed integer $k,taugeq 3$, the problem $k$SC is NP-hard for graphs with vertex cover number at most $tau$. We also show that $k$SC is W[1]-hard when parameterized by the treewidth of the input graph and the number~$k$ of components combined using a reduction from textsc{Unary Bin Packing}. Furthermore, we prove that $k$SC remains NP-hard for graphs with maximum degree three and also graphs with degeneracy two. Finally, we prove that the unweighted $k$SSE is W[1]-hard for the parameter $k$.
The question if a given partial solution to a problem can be extended reasonably occurs in many algorithmic approaches for optimization problems. For instance, when enumerating minimal dominating sets of a graph $G=(V,E)$, one usually arrives at the problem to decide for a vertex set $U subseteq V$, if there exists a textit{minimal} dominating set $S$ with $Usubseteq S$. We propose a general, partial-order based formulation of such extension problems and study a number of specific problems which can be expressed in this framework. Possibly contradicting intuition, these problems tend to be NP-hard, even for problems where the underlying optimisation problem can be solved in polynomial time. This raises the question of how fixing a partial solution causes this increase in difficulty. In this regard, we study the parameterised complexity of extension problems with respect to parameters related to the partial solution, as well as the optimality of simple exact algorithms under the Exponential-Time Hypothesis. All complexity considerations are also carried out in very restricted scenarios, be it degree restrictions or topological restrictions (planarity) for graph problems or the size of the given partition for the considered extension variant of Bin Packing.
We consider extension variants of the classical graph problems Vertex Cover and Independent Set. Given a graph $G=(V,E)$ and a vertex set $U subseteq V$, it is asked if there exists a minimal vertex cover (resp. maximal independent set) $S$ with $Usu bseteq S$ (resp. $U supseteq S$). Possibly contradicting intuition, these problems tend to be NP-hard, even in graph classes where the classical problem can be solved in polynomial time. Yet, we exhibit some graph classes where the extension variant remains polynomial-time solvable. We also study the parameterized complexity of these problems, with parameter $|U|$, as well as the optimality of simple exact algorithms under the Exponential-Time Hypothesis. All these complexity considerations are also carried out in very restricted scenarios, be it degree or topological restrictions (bipartite, planar or chordal graphs). This also motivates presenting some explicit branching algorithms for degree-bounded instances. We further discuss the price of extension, measuring the distance of $U$ to the closest set that can be extended, which results in natural optimization problems related to extension problems for which we discuss polynomial-time approximability.
In this paper, we consider the Target Set Selection problem: given a graph and a threshold value $thr(v)$ for any vertex $v$ of the graph, find a minimum size vertex-subset to activate s.t. all the vertices of the graph are activated at the end of th e propagation process. A vertex $v$ is activated during the propagation process if at least $thr(v)$ of its neighbors are activated. This problem models several practical issues like faults in distributed networks or word-to-mouth recommendations in social networks. We show that for any functions $f$ and $rho$ this problem cannot be approximated within a factor of $rho(k)$ in $f(k) cdot n^{O(1)}$ time, unless FPT = W[P], even for restricted thresholds (namely constant and majority thresholds). We also study the cardinality constraint maximization and minimizati
التعليقات
جاري جلب التعليقات جاري جلب التعليقات
سجل دخول لتتمكن من متابعة معايير البحث التي قمت باختيارها
mircosoft-partner

هل ترغب بارسال اشعارات عن اخر التحديثات في شمرا-اكاديميا