Graph-theoretic models have come to the forefront as some of the most powerful and practical methods for sequence assembly. Simultaneously, the computational hardness of the underlying graph algorithms has remained open. Here we present two theoretical results about the complexity of these models for sequence assembly. In the first part, we show sequence assembly to be NP-hard under two different models: string graphs and de Bruijn graphs. Together with an earlier result on the NP-hardness of overlap graphs, this demonstrates that all of the popular graph-theoretic sequence assembly paradigms are NP-hard. In our second result, we give the first, to our knowledge, optimal polynomial time algorithm for genome assembly that explicitly models the double-strandedness of DNA. We solve the Chinese Postman Problem on bidirected graphs using bidirected flow techniques and show to how to use it to find the shortest doublestranded DNA sequence which contains a given set of k-long words. This algorithm has applications to sequencing by hybridization and short read assembly.

Recently Raghavendra and Tan (SODA 2012) gave a 0.85-approximation algorithm for the Max Bisection problem. We improve their algorithm to a 0.8776-approximation. As Max Bisection is hard to approximate within α GW + ≈ 0.8786 under the Unique Games Conjecture (UGC), our algorithm is nearly optimal. We conjecture that Max Bisection is approximable within α GW − , i.e., that the bisection constraint (essentially) does not make Max Cut harder.We also obtain an optimal algorithm (assuming the UGC) for the analogous variant of Max 2-Sat. Our approximation ratio for this problem exactly matches the optimal approximation ratio for Max 2-Sat, i.e., α LLZ + ≈ 0.9401, showing that the bisection constraint does not make Max 2-Sat harder. This improves on a 0.93-approximation for this problem due to Raghavendra and Tan.

Assume that two robots are located at the centre of a unit disk. Their goal is to evacuate from the disk through an exit at an unknown location on the boundary of the disk. At any time the robots can move anywhere they choose on the disk, independently of each other, with maximum speed 1. The robots can cooperate by exchanging information whenever they meet. We study algorithms for the two robots to minimize the evacuation time: the time when both robots reach the exit. In [9] the authors gave an algorithm defining trajectories for the two robots yielding evacuation time at most 5.740 and also proved that any algorithm has evacuation time at least 3 + π 4 + √ 2 ≈ 5.199. We improve both the upper and lower bounds on the evacuation time of a unit disk. Namely, we present a new non-trivial algorithm whose evacuation time is at most 5.628 and show that any algorithm has evacuation time at least 3 + π 6 + √ 3 ≈ 5.255. To achieve the upper bound, we designed an algorithm which non-intuitively proposes a forced meeting between the two robots, even if the exit has not been found by either of them.We study the problem of two robots that start at the same time at the centre of a unit disk and attempt to reach an exit placed at an unknown location on the boundary of the disk. At any time the robots can move anywhere they choose within the disk. Indeed, they can take short-cuts by moving in the interior of the disk if desired. We assume their maximum speed is 1. The robots can communicate with each other only if they are at the same point at the same time: we call this communication model face-to-face communication. Our goal is to schedule the trajectories of the robots so as to minimize the evacuation time, which is the time it takes both robots to reach the exit (for the worst case location of the exit).

We introduce and study a new problem concerning the exploration of a geometric domain by mobile robots. Consider a line segment [0, I] and a set of n mobile robots r1, r2, . . . , rn placed at one of its endpoints. Each robot has a searching speed si and a walking speed wi, where si < wi. We assume that each robot is aware of the number of robots of the collection and their corresponding speeds. At each time moment a robot ri either walks along a portion of the segment not exceeding its walking speed wi or searches a portion of the segment with the speed not exceeding si. A search of segment [0, I] is completed at the time when each of its points have been searched by at least one of the n robots. We want to develop mobility schedules (algorithms) for the robots which complete the search of the segment as fast as possible. More exactly we want to maximize the speed of the mobility schedule (equal to the ratio of the segment length versus the time of the completion of the schedule). We analyze first the offline scenario when the robots know the length of the segment that is to be searched. We give an algorithm producing a mobility schedule for arbitrary walking and searching speeds and prove its optimality. Then we propose an online algorithm, when the robots do not know in advance the actual length of the segment to be searched. The speed S of such algorithm is defined aswhere S(IL) denotes the speed of searching of segment IL = [0, L]. We prove that the proposed online algorithm is 2-competitive. The competitive ratio is shown to be better in the case when the robots' walking speeds are all the same.

Abstract