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Register a new userA generalization of extension complexity that captures $P$
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Added by
Hans Raj Tiwary
Publication date
2014
fields
Informatics Engineering
and research's language is
English
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In this paper we propose a generalization of the extension complexity of a polyhedron $Q$. On the one hand it is general enough so that all problems in $P$ can be formulated as linear programs with polynomial size extension complexity. On the other hand it still allows non-polynomial lower bounds to be proved for $NP$-hard problems independently of whether or not $P=NP$. The generalization, called $H$-free extension complexity, allows for a set of valid inequalities $H$ to be excluded in computing the extension complexity of $Q$. We give results on the $H$-free extension complexity of hard matching problems (when $H$ are the odd set inequalities) and the traveling salesman problem (when $H$ are the subtour elimination constraints).
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In this article we undertake a study of extension complexity from the perspective of formal languages. We define a natural way to associate a family of polytopes with binary languages. This allows us to define the notion of extension complexity of formal languages. We prove several closure properties of languages admitting compact extended formulations. Furthermore, we give a sufficient machine characterization of compact languages. We demonstrate the utility of this machine characterization by obtaining upper bounds for polytopes for problems in nondeterministic logspace; lower bounds in streaming models; and upper bounds on extension complexities of several polytopes.
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.
It is known that the extension complexity of the TSP polytope for the complete graph $K_n$ is exponential in $n$ even if the subtour inequalities are excluded. In this article we study the polytopes formed by removing other subsets $mathcal{H}$ of facet-defining inequalities of the TSP polytope. In particular, we consider the case when $mathcal{H}$ is either the set of blossom inequalities or the simple comb inequalities. These inequalities are routinely used in cutting plane algorithms for the TSP. We show that the extension complexity remains exponential even if we exclude these inequalities. In addition we show that the extension complexity of polytope formed by all comb inequalities is exponential. For our proofs, we introduce a subclass of comb inequalities, called $(h,t)$-uniform inequalities, which may be of independent interest.
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.
This paper is aimed to investigate some computational aspects of different isoperimetric problems on weighted trees. In this regard, we consider different connectivity parameters called {it minimum normalized cuts}/{it isoperimteric numbers} defined through taking minimum of the maximum or the mean of the normalized outgoing flows from a set of subdomains of vertices, where these subdomains constitute a {it partition}/{it subpartition}. Following the main result of [A. Daneshgar, {it et. al.}, {it On the isoperimetric spectrum of graphs and its approximations}, JCTB, (2010)], it is known that the isoperimetric number and the minimum normalized cut both can be described as ${0,1}$-optimization programs, where the latter one does {it not} admit a relaxation to the reals. We show that the decision problem for the case of taking $k$-partitions and the maximum (called the max normalized cut problem {rm NCP}$^M$) as well as the other two decision problems for the mean version (referred to as {rm IPP}$^m$ and {rm NCP}$^m$) are $NP$-complete problems. On the other hand, we show that the decision problem for the case of taking $k$-subpartitions and the maximum (called the max isoperimetric problem {rm IPP}$^M$) can be solved in {it linear time} for any weighted tree and any $k geq 2$. Based on this fact, we provide polynomial time $O(k)$-approximation algorithms for all differe
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