Exploiting structure to cope with NP-hard graph problems: Polynomial and exponential time exact algorithms

An ideal algorithm for solving a particular problem always finds an optimal solution, finds such a solution for every possible instance, and finds it in polynomial time. When dealing with NP-hard problems, algorithms can only be expected to possess at most two out of these three desirable properties. All algorithms presented in this thesis are exact algorithms, which means that they always find an optimal solution. Demanding the solution to be optimal means that other concessions have to be made when designing an exact algorithm for an NP-hard problem: we either have to impose restrictions on the instances of the problem in order to achieve a polynomial time complexity, or we have to abandon the requirement that the worst-case running time has to be polynomial. In some cases, when the problem under consideration remains NP-hard on restricted input, we are even forced to do both. Most of the problems studied in this thesis deal with partitioning the vertex set of a given graph. In the other problems the task is to find certain types of paths and cycles in graphs. The problems all have in common that they are NP-hard on general graphs. We present several polynomial time algorithms for solving restrictions of these problems to specific graph classes, in particular graphs without long induced paths, chordal graphs and claw-free graphs. For problems that remain NP-hard even on restricted input we present exact exponential time algorithms. In the design of each of our algorithms, structural graph properties have been heavily exploited. Apart from using existing structural results, we prove new structural properties of certain types of graphs in order to obtain our algorithmic results

Round-robin tournaments with minimum number of breaks and two teams per club

We study round-robin tournaments for 2n teams. Here n is either interpreted as the number of clubs, each having two teams, or the number of strength groups with two teams each. For even n we give a construction of a single round-robin tournament for 2n teams with 2n - 2 breaks, where the teams of the same club have complementary home-away patterns and play against each other in the first round. If the pairs of teams are strength groups, then a cyclic permutation of the constructed schedule results in a group-balanced tournament

Constructing fair round robin tournaments with a minimum number of breaks

Given n clubs with two teams each, we show that, if n is even, it is possible to construct a schedule for a single round robin tournament satisfying the following properties: the number of breaks is 2n−2, teams of the same club never play at home simultaneously, and they play against each other in the first round. We also consider a fairness constraint related to different playing strengths of teams competing in the tournament

Editing to Eulerian Graphs

We investigate the problem of modifying a graph into a connected graph in which the degree of each vertex satisfies a prescribed parity constraint. Let ea, ed and vd denote the operations edge addition, edge deletion and vertex deletion respectively. For any S subseteq {ea,ed,vd}, we define Connected Degree Parity Editing (S) (CDPE(S)) to be the problem that takes as input a graph G, an integer k and a function delta: V(G) -> {0,1}, and asks whether G can be modified into a connected graph H with d_H(v) = delta(v)(mod 2) for each v in V(H), using at most k operations from S. We prove that (*) if S={ea} or S={ea,ed}, then CDPE(S) can be solved in polynomial time; (*) if {vd} subseteq S subseteq {ea,ed,vd}, then CDPE(S) is NP-complete and W-hard when parameterized by k, even if delta = 0. Together with known results by Cai and Yang and by Cygan, Marx, Pilipczuk, Pilipczuk and Schlotter, our results completely classify the classical and parameterized complexity of the CDPE(S) problem for all S subseteq {ea,ed,vd}. We obtain the same classification for a natural variant of the cdpe(S) problem on directed graphs, where the target is a weakly connected digraph in which the difference between the in- and out-degree of every vertex equals a prescribed value. As an important implication of our results, we obtain polynomial-time algorithms for Eulerian Editing problem and its directed variant. To the best of our knowledge, the only other natural non-trivial graph class H for which the H-Editing problem is known to be polynomial-time solvable is the class of split graphs.publishedVersio

Editing to Eulerian Graphs

We investigate the problem of modifying a graph into a connected graph in which the degree of each vertex satisfies a prescribed parity constraint. Let $ea$, $ed$ and $vd$ denote the operations edge addition, edge deletion and vertex deletion respectively. For any $S\subseteq \{ea,ed,vd\}$, we define Connected Degree Parity Editing$(S)$ (CDPE($S$)) to be the problem that takes as input a graph $G$, an integer $k$ and a function $\delta\colon V(G)\rightarrow\{0,1\}$, and asks whether $G$ can be modified into a connected graph $H$ with $d_{H}(v)\equiv\delta(v)~(\bmod~2)$ for each $v\in V(H)$, using at most $k$ operations from $S$. We prove that 1. if $S=\{ea\}$ or $S=\{ea,ed\}$, then CDPE($S$) can be solved in polynomial time; 2. if $\{vd\} \subseteq S\subseteq \{ea,ed,vd\}$, then CDPE($S$) is NP-complete and W[1]-hard when parameterized by $k$, even if $\delta\equiv 0$. Together with known results by Cai and Yang and by Cygan, Marx, Pilipczuk, Pilipczuk and Schlotter, our results completely classify the classical and parameterized complexity of the CDPE($S$) problem for all $S\subseteq \{ea,ed,vd\}$. We obtain the same classification for a natural variant of the CDPE($S$) problem on directed graphs, where the target is a weakly connected digraph in which the difference between the in- and out-degree of every vertex equals a prescribed value. As an important implication of our results, we obtain polynomial-time algorithms for the Eulerian Editing problem and its directed variant.Comment: 33 pages. An extended abstract of this paper will appear in the proceedings of FSTTCS 201

Editing to a planar graph of given degrees.

We consider the following graph modification problem. Let the input consist of a graph G=(V,E), a weight function w:V∪E→N, a cost function c:V∪E→N0 and a degree function δ:V→N0, together with three integers kv,ke and C . The question is whether we can delete a set of vertices of total weight at most kv and a set of edges of total weight at most ke so that the total cost of the deleted elements is at most C and every non-deleted vertex v has degree δ(v) in the resulting graph G′. We also consider the variant in which G′ must be connected. Both problems are known to be NP-complete and W[1]-hard when parameterized by kv+ke. We prove that, when restricted to planar graphs, they stay NP-complete but have polynomial kernels when parameterized by kv+ke

Cost Effectiveness of a CYP2C19 Genotype-Guided Strategy in Patients with Acute Myocardial Infarction:Results from the POPular Genetics Trial

INTRODUCTION: The POPular Genetics trial demonstrated that a CYP2C19 genotype-guided P2Y12 inhibitor strategy reduced bleeding rates compared with standard treatment with ticagrelor or prasugrel without increasing thrombotic event rates after primary percutaneous coronary intervention (PCI). OBJECTIVE: In this analysis, we aimed to evaluate the cost effectiveness of a genotype-guided strategy compared with standard treatment with ticagrelor or prasugrel. METHODS: A 1-year decision tree based on the POPular Genetics trial in combination with a lifelong Markov model was developed to compare costs and quality-adjusted life-years (QALYs) between a genotype-guided and a standard P2Y12 inhibitor strategy in patients with myocardial infarction undergoing primary PCI. The cost-effectiveness analysis was conducted from a Dutch healthcare system perspective. Within-trial survival and utility data were combined with lifetime projections to evaluate lifetime cost effectiveness for a cohort of 1000 patients. Costs and utilities were discounted at 4 and 1.5%, respectively, according to Dutch guidelines for health economic studies. Besides deterministic and probabilistic sensitivity analyses, several scenario analyses were also conducted (different time horizons, different discount rates, equal prices for P2Y12 inhibitors, and equal distribution of thrombotic events between the two strategies). RESULTS: Base-case analysis with a hypothetical cohort of 1000 subjects demonstrated 8.98 QALYs gained and €725,550.69 in cost savings for the genotype-guided strategy (dominant). The deterministic and probabilistic sensitivity analysis confirmed the robustness of the model and the cost-effectiveness results. In scenario analyses, the genotype-guided strategy remained dominant. CONCLUSION: In patients undergoing primary PCI, a CYP2C19 genotype-guided strategy compared with standard treatment with ticagrelor or prasugrel resulted in QALYs gained and cost savings. TRIAL REGISTRATION: Clinicaltrials.gov number: NCT01761786, Netherlands trial register number: NL2872

Internet-based, culturally sensitive, problem-solving therapy for turkish migrants with depression: Randomized controlled trial

Background: Turkish migrants living in the Netherlands have a high prevalence of depressive disorders, but experience considerable obstacles to accessing professional help. Providing easily accessible Internet treatments may help to overcome these barriers. Objective: The aim of this study was to evaluate the effectiveness of a culturally sensitive, guided, self-help, problem-solving intervention through the Internet for reducing depressive symptoms in Turkish migrants. Methods: A two-armed randomized controlled trial was conducted. The primary outcome measure was the severity of depressive symptoms; secondary outcome measures were somatic symptoms, anxiety, quality of life, and satisfaction with the treatment. Participants were assessed online at baseline, posttest (6 weeks after baseline), and 4 months after baseline. Posttest results were analyzed on the intention-to-treat sample. Missing values were estimated by means of multiple imputation. Differences in clinical outcome between groups were analyzed with a t test. Cohen's d was used to determine the between-groups effect size at posttreatment and follow-up. Results: Turkish adults (N=96) with depressive symptoms were randomized to the experimental group (n=49) or to a waitlist control group (n=47). High attrition rates were found among the 96 participants of which 42% (40/96) did not complete the posttest (6 weeks) and 62% (59/96) participants did not complete the follow-up assessment at 4 months. No significant difference between the experimental group and the control group was found for depression at posttest. Recovery occurred significantly more often in the experimental group (33%, 16/49) than in the control group (9%, 4/47) at posttest (P=.02). Because of the high attrition rate, a completers-only analysis was conducted at follow-up. The experimental group showed significant improvement in depression compared to the control group both at posttest (P=.01) and follow-up (P=.01). Conclusions: The results of this study did not show a significant effect on the reduction of depressive symptoms. However, the effect size at posttest was high, which might be an indicator of the possible effectiveness of the intervention when assessed in a larger sample and robust trial. Future research should replicate our study with adequately powered samples. © Burçin Ünlü Ince, Pim Cuijpers, Edith van 't Hof, Wouter van Ballegooijen, Helen Christensen, Heleen Riper

Aspirin with or without Clopidogrel after Transcatheter Aortic-Valve Implantation

BACKGROUND The effect of single as compared with dual antiplatelet treatment on bleeding and thromboembolic events after transcatheter aortic-valve implantation (TAVI) in patients who do not have an indication for long-term anticoagulation has not been well studied. METHODS In a randomized, controlled trial, we assigned a subgroup of patients who were undergoing TAVI and did not have an indication for long-term anticoagulation, in a 1:1 ratio, to receive aspirin alone or aspirin plus clopidogrel for 3 months. The two primary outcomes were all bleeding (including minor, major, and life-threatening or disabling bleeding) and non-procedure-related bleeding over a period of 12 months. Most bleeding at the TAVI puncture site was counted as non-procedure-related. The two secondary outcomes were a composite of death from cardiovascular causes, non-procedure-related bleeding, stroke, or myocardial infarction (secondary composite 1) and a composite of death from cardiovascular causes, ischemic stroke, or myocardial infarction (secondary composite 2) at 1 year, with both outcomes tested sequentially for noninferiority (noninferiority margin, 7.5 percentage points) and superiority. RESULTS A total of 331 patients were assigned to receive aspirin alone and 334 were assigned to receive aspirin plus clopidogrel. A bleeding event occurred in 50 patients (15.1%) receiving aspirin alone and in 89 (26.6%) receiving aspirin plus clopidogrel (risk ratio, 0.57; 95% confidence interval [CI], 0.42 to 0.77; P=0.001). Non-procedure-related bleeding occurred in 50 patients (15.1%) and 83 patients (24.9%), respectively (risk ratio, 0.61; 95% CI, 0.44 to 0.83; P=0.005). A secondary composite 1 event occurred in 76 patients (23.0%) receiving aspirin alone and in 104 (31.1%) receiving aspirin plus clopidogrel (difference, −8.2 percentage points; 95% CI for noninferiority, −14.9 to −1.5; P<0.001; risk ratio, 0.74; 95% CI for superiority, 0.57 to 0.95; P=0.04). A secondary composite 2 event occurred in 32 patients (9.7%) and 33 patients (9.9%), respectively (difference, −0.2 percentage points; 95% CI for noninferiority, −4.7 to 4.3; P=0.004; risk ratio, 0.98; 95% CI for superiority, 0.62 to 1.55; P=0.93). A total of 44 patients (13.3%) and 32 (9.6%), respectively, received oral anticoagulation during the trial. CONCLUSIONS Among patients undergoing TAVI who did not have an indication for oral anticoagulation, the incidence of bleeding and the composite of bleeding or thromboembolic events at 1 year were significantly less frequent with aspirin than with aspirin plus clopidogrel administered for 3 months