Approximation hardness of Travelling Salesman via weighted amplifiers

The expander graph constructions and their variants are the main tool used in gap preserving reductions to prove approximation lower bounds of combinatorial optimisation problems. In this paper we introduce the weighted amplifiers and weighted low occurrence of Constraint Satisfaction problems as intermediate steps in the NP-hard gap reductions. Allowing the weights in intermediate problems is rather natural for the edge-weighted problems as Travelling Salesman or Steiner Tree. We demonstrate the technique for Travelling Salesman and use the parametrised weighted amplifiers in the gap reductions to allow more flexibility in fine-tuning their expanding parameters. The purpose of this paper is to point out effectiveness of these ideas, rather than to optimise the expander’s parameters. Nevertheless, we show that already slight improvement of known expander values modestly improve the current best approximation hardness value for TSP from 123/122 ([9]) to 117/116 . This provides a new motivation for study of expanding properties of random graphs in order to improve approximation lower bounds of TSP and other edge-weighted optimisation problems

Tropical Dominating Sets in Vertex-Coloured Graphs

Given a vertex-coloured graph, a dominating set is said to be tropical if every colour of the graph appears at least once in the set. Here, we study minimum tropical dominating sets from structural and algorithmic points of view. First, we prove that the tropical dominating set problem is NP-complete even when restricted to a simple path. Then, we establish upper bounds related to various parameters of the graph such as minimum degree and number of edges. We also give upper bounds for random graphs. Last, we give approximability and inapproximability results for general and restricted classes of graphs, and establish a FPT algorithm for interval graphs.Comment: 19 pages, 4 figure

Inapproximability of maximal strip recovery

In comparative genomic, the first step of sequence analysis is usually to decompose two or more genomes into syntenic blocks that are segments of homologous chromosomes. For the reliable recovery of syntenic blocks, noise and ambiguities in the genomic maps need to be removed first. Maximal Strip Recovery (MSR) is an optimization problem proposed by Zheng, Zhu, and Sankoff for reliably recovering syntenic blocks from genomic maps in the midst of noise and ambiguities. Given $d$ genomic maps as sequences of gene markers, the objective of \msr{d} is to find $d$ subsequences, one subsequence of each genomic map, such that the total length of syntenic blocks in these subsequences is maximized. For any constant $d \ge 2$, a polynomial-time 2d-approximation for \msr{d} was previously known. In this paper, we show that for any $d \ge 2$, \msr{d} is APX-hard, even for the most basic version of the problem in which all gene markers are distinct and appear in positive orientation in each genomic map. Moreover, we provide the first explicit lower bounds on approximating \msr{d} for all $d \ge 2$. In particular, we show that \msr{d} is NP-hard to approximate within $\Omega(d/\log d)$. From the other direction, we show that the previous 2d-approximation for \msr{d} can be optimized into a polynomial-time algorithm even if $d$ is not a constant but is part of the input. We then extend our inapproximability results to several related problems including \cmsr{d}, \gapmsr{\delta}{d}, and \gapcmsr{\delta}{d}.Comment: A preliminary version of this paper appeared in two parts in the Proceedings of the 20th International Symposium on Algorithms and Computation (ISAAC 2009) and the Proceedings of the 4th International Frontiers of Algorithmics Workshop (FAW 2010

Algorithmic aspects of disjunctive domination in graphs

For a graph $G=(V,E)$, a set $D\subseteq V$ is called a \emph{disjunctive dominating set} of $G$ if for every vertex $v\in V\setminus D$, $v$ is either adjacent to a vertex of $D$ or has at least two vertices in $D$ at distance $2$ from it. The cardinality of a minimum disjunctive dominating set of $G$ is called the \emph{disjunctive domination number} of graph $G$, and is denoted by $\gamma_{2}^{d}(G)$. The \textsc{Minimum Disjunctive Domination Problem} (MDDP) is to find a disjunctive dominating set of cardinality $\gamma_{2}^{d}(G)$. Given a positive integer $k$ and a graph $G$, the \textsc{Disjunctive Domination Decision Problem} (DDDP) is to decide whether $G$ has a disjunctive dominating set of cardinality at most $k$. In this article, we first propose a linear time algorithm for MDDP in proper interval graphs. Next we tighten the NP-completeness of DDDP by showing that it remains NP-complete even in chordal graphs. We also propose a $(\ln(\Delta^{2}+\Delta+2)+1)$-approximation algorithm for MDDP in general graphs and prove that MDDP can not be approximated within $(1-\epsilon) \ln(|V|)$ for any $\epsilon>0$ unless NP $\subseteq$ DTIME$(|V|^{O(\log \log |V|)})$. Finally, we show that MDDP is APX-complete for bipartite graphs with maximum degree $3$

Approximating Clustering of Fingerprint Vectors with Missing Values

The problem of clustering fingerprint vectors is an interesting problem in Computational Biology that has been proposed in (Figureroa et al. 2004). In this paper we show some improvements in closing the gaps between the known lower bounds and upper bounds on the approximability of some variants of the biological problem. Namely we are able to prove that the problem is APX-hard even when each fingerprint contains only two unknown position. Moreover we have studied some variants of the orginal problem, and we give two 2-approximation algorithm for the IECMV and OECMV problems when the number of unknown entries for each vector is at most a constant.Comment: 13 pages, 4 figure

Hypergraphic LP Relaxations for Steiner Trees

We investigate hypergraphic LP relaxations for the Steiner tree problem, primarily the partition LP relaxation introduced by Koenemann et al. [Math. Programming, 2009]. Specifically, we are interested in proving upper bounds on the integrality gap of this LP, and studying its relation to other linear relaxations. Our results are the following. Structural results: We extend the technique of uncrossing, usually applied to families of sets, to families of partitions. As a consequence we show that any basic feasible solution to the partition LP formulation has sparse support. Although the number of variables could be exponential, the number of positive variables is at most the number of terminals. Relations with other relaxations: We show the equivalence of the partition LP relaxation with other known hypergraphic relaxations. We also show that these hypergraphic relaxations are equivalent to the well studied bidirected cut relaxation, if the instance is quasibipartite. Integrality gap upper bounds: We show an upper bound of sqrt(3) ~ 1.729 on the integrality gap of these hypergraph relaxations in general graphs. In the special case of uniformly quasibipartite instances, we show an improved upper bound of 73/60 ~ 1.216. By our equivalence theorem, the latter result implies an improved upper bound for the bidirected cut relaxation as well.Comment: Revised full version; a shorter version will appear at IPCO 2010

Matching techniques ride to rescue OLED displays

Combinatorial optimization problems have recently emerged in the design of controllers for OLED displays. The objective is to decompose an image into subframes minimizing the addressing time and thereby also the amplitude of the electrical current through the diodes, which has a direct impact on the lifetime of such a display. To this end, we model this problem as an integer linear program. Subsequently, we refine this formulation by exploiting the combinatorial structure of the problem. We propose a fully combinatorial separation routine for the LP-relaxation based on matching techniques. It can be used as an oracle in various frameworks to derive approximation algorithms or heuristics. We establish NP-hardness and hardness of approximation. Nevertheless, we are able to work around this issue by only focusing on a subsets of the variables and provide experimental evidence that they are sufficient to come up with near optimal solutions in practice. On this basis, one can derive custom-tailored solutions adapting to technical constraints such as memory requirements. By allowing the addressing of distributed doublelines, we improve the addressing time in cases where previous approaches fall short due to their restriction to consecutive doublelines

The Complexity of Flow Expansion and Electrical Flow Expansion

FlowExpansion is a network design problem, in which the input consists of a flow network and a set of candidate edges, which may be added to the network. Adding a candidate incurs given costs. The goal is to determine the cheapest set of candidate edges that, if added, allow the demands to be satisfied. FlowExpansion is a variant of the Minimum-Cost Flow problem with non-linear edge costs. We study FlowExpansion for both graph-theoretical and electrical flow networks. In the latter case this problem is also known as the Transmission Network Expansion Planning problem. We give a structured view over the complexity of the variants of FlowExpansion that arise from restricting, e.g., the graph classes, the capacities, or the number of sources and sinks. Our goal is to determine which restrictions have a crucial impact on the computational complexity. The results in this paper range from polynomial-time algorithms for the more restricted variants over NP-hardness proofs to proofs that certain variants are NP-hard to approximate even within a logarithmic factor of the optimal solution

Polynomial Kernels for Weighted Problems

Kernelization is a formalization of efficient preprocessing for NP-hard problems using the framework of parameterized complexity. Among open problems in kernelization it has been asked many times whether there are deterministic polynomial kernelizations for Subset Sum and Knapsack when parameterized by the number $n$ of items. We answer both questions affirmatively by using an algorithm for compressing numbers due to Frank and Tardos (Combinatorica 1987). This result had been first used by Marx and V\'egh (ICALP 2013) in the context of kernelization. We further illustrate its applicability by giving polynomial kernels also for weighted versions of several well-studied parameterized problems. Furthermore, when parameterized by the different item sizes we obtain a polynomial kernelization for Subset Sum and an exponential kernelization for Knapsack. Finally, we also obtain kernelization results for polynomial integer programs

Travelling on Graphs with Small Highway Dimension

We study the Travelling Salesperson (TSP) and the Steiner Tree problem (STP) in graphs of low highway dimension. This graph parameter was introduced by Abraham et al. [SODA 2010] as a model for transportation networks, on which TSP and STP naturally occur for various applications in logistics. It was previously shown [Feldmann et al. ICALP 2015] that these problems admit a quasi-polynomial time approximation scheme (QPTAS) on graphs of constant highway dimension. We demonstrate that a significant improvement is possible in the special case when the highway dimension is 1, for which we present a fully-polynomial time approximation scheme (FPTAS). We also prove that STP is weakly NP-hard for these restricted graphs. For TSP we show NP-hardness for graphs of highway dimension 6, which answers an open problem posed in [Feldmann et al. ICALP 2015]