Alexandru Popa

Synthesizing Minimal Tile Sets for Complex Patterns in the Framework of Patterned DNA Self-Assembly

Ma and Lombardi (2009) introduce and study the Pattern self-Assembly Tile set Synthesis (PATS) problem. In particular they show that the optimization version of the PATS problem is NP-hard. However, their NP-hardness proof turns out to be incorrect. Our main result is to give a correct NP-hardness proof via a reduction from the 3SAT. By definition, the PATS problem assumes that the assembly of a pattern starts always from an “L”-shaped seed structure, fixing the borders of the pattern. In this context, we study the assembly complexity of various pattern families and we show how to construct families of patterns which require a non-constant number of tiles to be assembled.

Generalised Matching

Given a pattern p over an alphabet Σ p and a text t over an alphabet Σ t , we consider the problem of determining a mapping f from Σ p to

Modelling the Power Supply Network - Hardness and Approximation

{\Sigma}_{t}^{+}

Restricted common superstring and restricted common supersequence

such that t = f (p 1 )f (p 2 )...f (p m ). This class of problems, which was first introduced by Amir and Nor in 2004, is defined by different constraints on the mapping f . We give NP-Completeness results for a wide range of conditions. These include when f is either many-to-one or one-to-one, when Σ t is binary and when the range of f is limited to strings of constant length. We then introduce a related problem we term pattern matching with string classes which we show to be solvable efficiently. Finally, we discuss an optimisation variant of generalised matching and give a polynomial-time min

On shortest common superstring and swap permutations

(1,\sqrt{k/OPT})

Fully polynomial FPT algorithms for some classes of bounded clique-width graphs

-approximation algorithm for fixed k .

Approximation and Hardness Results for the Maximum Edge q -coloring Problem

In this paper we study a problem named graph partitioning with supply and demand (GPSD), motivated by applications in energy transmission. The input consists of an undirected graph G with the nodes partitioned into two sets: suppliers and consumers. Each supply node has associated a capacity and each consumer node has associated a demand. The goal is to find a subgraph of G and to partition it into trees, such that in each tree: (i) there is precisely one supplier and (ii) the total demand of the consumers is less than or equal to the capacity of the supplier. Moreover, we want to maximize the demand of all the consumers in such a partition.

Undecidability Results for Finite Interactive Systems

The shortest common superstring and the shortest common supersequence are two well studied problems having a wide range of applications. In this paper we consider both problems with resource constraints, denoted as the Restricted Common Superstring (shortly RCSstr) problem and the Restricted Common Supersequence (shortly RCSseq). In the RCSstr (RCSseq) problem we are given a set S of n strings, s1, s2, ..., sn, and a multiset t = {t1, t2, ..., tm}, and the goal is to find a permutation π : {1,..., m} → {1, ..., m} to maximize the number of strings in S that are substrings (subsequences) of π(t) = tπ(1) tπ(2) ... tπ(m) (we call this ordering of the multiset, π(t), a permutation of t). We first show that in its most general setting the RC-Sstr problem is NP-complete and hard to approximate within a factor of n1-e, for any e > 0, unless P = NP. Afterwards, we present two separate reductions to show that the RCSstr problem remains NP-Hard even in the case where the elements of t are drawn from a binary alphabet or for the case where all input strings are of length two. We then present some approximation results for several variants of the RCSstr problem. In the second part of this paper, we turn to the RCSseq problem, where we present some hardness results, tight lower bounds and approximation algorithms.

Approximation and Hardness Results for the Maximum Edges in Transitive Closure Problem

The Shortest Common Superstring (SCS) is a well studied problem, having a wide range of applications. In this paper we consider two problems closely related to it. First we define the Swapped Restricted Superstring(SRS) problem, where we are given a set S of n strings, s1, s2, . . . , sn, and a text T = t1t2 . . . tm, and our goal is to find a swap permutation π : {1, . . . ,m} → {1, . . . , m} to maximize the number of strings in S that are substrings of tπ(1)tπ(2) . . . tπ(m). We then show that the SRS problem is NP-Complete. Afterwards, we consider a similar variant denoted SRSR, where our goal is to find a swap permutation π : {1, . . . , m} → {1, . . . , m} to maximize the total number of times that the strings of S appear in tπ(1)tπ(2) . . . tπ(m) (we can count the same string si as a substring of tπ(1)tπ(2) . . . tπ(m) more than once). For this problem, we present a polynomial time exact algorithm.

Better lower and upper bounds for the minimum rainbow subgraph problem

Parameterized complexity theory has enabled a refined classification of the difficulty of NP-hard optimization problems on graphs with respect to key structural properties, and so to a better understanding of their true difficulties. More recently, hardness results for problems in P were achieved using reasonable complexity theoretic assumptions such as: Strong Exponential Time Hypothesis (SETH), 3SUM and All-Pairs Shortest-Paths (APSP). According to these assumptions, many graph theoretic problems do not admit truly subquadratic algorithms, nor even truly subcubic algorithms (Williams and Williams, FOCS 2010 and Abboud, Grandoni, Williams, SODA 2015). A central technique used to tackle the difficulty of the above mentioned problems is fixed-parameter algorithms for polynomial-time problems with polynomial dependency in the fixed parameter (P-FPT). This technique was introduced by Abboud, Williams and Wang in SODA 2016 and continued by Husfeldt (IPEC 2016) and Fomin et al. (SODA 2017), using the treewidth as a parameter. Applying this technique to clique-width, another important graph parameter, remained to be done. In this paper we study several graph theoretic problems for which hardness results exist such as cycle problems (triangle detection, triangle counting, girth, diameter), distance problems (diameter, eccentricities, Gromov hyperbolicity, betweenness centrality) and maximum matching. We provide hardness results and fully polynomial FPT algorithms, using clique-width and some of its upper-bounds as parameters (split-width, modular-width and