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Depth first search is a natural algorithmic technique for constructing a closed route that visits all vertices of a graph. The length of such a route equals, in an edge-weighted tree, twice the total weight of all edges of the tree and this is asymptotically optimal over all exploration strategies. This paper considers a variant of such search strategies where the length of each route is bounded by a positive integer B (e.g. due to limited energy resources of the searcher). The objective is to cover all the edges of a tree T using the minimum number of routes, each starting and ending at the root and each being of length at most B. To this end, we analyze the following natural greedy tree traversal process that is based on decomposing a depth first search traversal into a sequence of limited length routes. Given any arbitrary depth first search traversal R of the tree T, we cover R with routes (R_1,ldots ,R_l), each of length at most B such that: (R_i) starts at the root, reaches directly the farthest point of R visited by (R_{i-1}), then (R_i) continues along the path R as far as possible, and finally (R_i) returns to the root. We call the above algorithm piecemeal-DFS and we prove that it achieves the asymptotically minimal number of routes l, regardless of the choice of R. Our analysis also shows that the total length of the traversal (and thus the traversal time) of piecemeal-DFS is asymptotically minimum over all energy-constrained exploration strategies. The fact that R can be chosen arbitrarily means that the exploration strategy can be constructed in an online fashion when the input tree T is not known in advance. Each route (R_i) can be constructed without any knowledge of the yet unvisited part of T. Surprisingly, our results show that depth first search is efficient for energy constrained exploration of trees, even though it is known that the same does not hold for energy constrained exploration of arbitrary graphs.

This work revisits quantum algorithms for the well-known welded tree problem, proposing a succinct quantum algorithm based on the simple coined quantum walks. It iterates the naturally defined coined quantum walk operator for a classically precomputed number of iterations, and measures. The number of iterations is linear in the depth of the tree. The success probability of this procedure is inversely linear in the depth of the tree. Moreover, it is the same for all instances of the problem of a fixed size, therefore, we can use the exact quantum amplitude amplification subroutine to answer with probability 1. This gives an exponential speedup over any classical algorithm for the same problem. The significance of the results may be seen as follows. (i) Our algorithm is rather simple compared with the one in (Jeffery and Zur, STOC’2023), which not only breaks the stereotype that coined quantum walks can only achieve quadratic speedups over classical algorithms, but also demonstrates the power of the simplest quantum walk model. (ii) Our algorithm achieves certainty of success for the first time. Thus, it becomes one of the few examples that exhibit exponential separation between exact quantum and randomized query complexities.

We study a new combinatorial problem based on the famous Minimum Common String Partition problem, which we call Permutation-constrained Common String Partition (PCSP for short). In PCSP, we are given two sequences/genomes s and t with the same length and a permutation (pi ) on ([ell ]), the question is to decide whether it is possible to decompose s and t into (ell ) blocks that can be matched according to some specified requirements, and that conform with the permutation (pi ). Our main result is that PCSP is FPT in parameter (ell + d), where d is the maximum number of occurrences that any symbol may have in s or t. We also study a variant where the input specifies whether each matched pair of block needs to be preserved as is, or reversed. With this result on PCSP, we show that a series of genome rearrangement problems are FPT (k + d), where k is the rearrangement distance between two genomes of interest.

In the allocation of indivisible goods, a prominent fairness notion is envy-freeness up to one good (EF1). We initiate the study of reachability problems in fair division by investigating the problem of whether one EF1 allocation can be reached from another EF1 allocation via a sequence of exchanges such that every intermediate allocation is also EF1. We show that two EF1 allocations may not be reachable from each other even in the case of two agents, and deciding their reachability is PSPACE-complete in general. On the other hand, we prove that reachability is guaranteed for two agents with identical or binary utilities as well as for any number of agents with identical binary utilities. We also examine the complexity of deciding whether there is an EF1 exchange sequence that is optimal in the number of exchanges required.

The classical and extensively-studied Fréchet distance between two curves is defined as an inf max, where the infimum is over all traversals of the curves, and the maximum is over all concurrent positions of the two agents. In this article we investigate a “flipped” Fréchet measure defined by a sup min – the supremum is over all traversals of the curves, and the minimum is over all concurrent positions of the two agents. This measure produces a notion of “social distance” between two curves (or general domains), where agents traverse curves while trying to stay as far apart as possible. We first study the flipped Fréchet measure between two polygonal curves in one and two dimensions, providing conditional lower bounds and matching algorithms. We then consider this measure on polygons, where it denotes the minimum distance that two agents can maintain while restricted to travel in or on the boundary of the same polygon. We investigate several variants of the problem in this setting, for some of which we provide linear-time algorithms. We draw connections between our proposed flipped Fréchet measure and existing related work in computational geometry, hoping that our new measure may spawn investigations akin to those performed for the Fréchet distance, and into further interesting problems that arise.

We consider the problem of maximizing a non-negative submodular function under the b-matching constraint, in the semi-streaming model. When the function is linear, monotone, and non-monotone, we obtain the approximation ratios of (2+varepsilon ), (3 + 2 sqrt{2} approx 5.828), and (4 + 2 sqrt{3} approx 7.464), respectively. We also consider a generalized problem, where a k-uniform hypergraph is given, along with an extra matroid or a (k')-matchoid constraint imposed on the edges, with the same goal of finding a b-matching that maximizes a submodular function. When the extra constraint is a matroid, we obtain the approximation ratios of (k + 1 + varepsilon ), (k + 2sqrt{k+1} + 2), and (k + 2sqrt{k + 2} + 3) for linear, monotone and non-monotone submodular functions, respectively. When the extra constraint is a (k')-matchoid, we attain the approximation ratio (frac{8}{3}k+ frac{64}{9}k' + O(1)) for general submodular functions.

The Set Packing problem is, given a collection of sets (mathcal {S}) over a ground set U, to find a maximum collection of sets that are pairwise disjoint. The problem is among the most fundamental NP-hard optimization problems that have been studied extensively in various computational regimes. The focus of this work is on parameterized complexity, Parameterized Set Packing (PSP): Given parameter (r in {mathbb N}), is there a collection ( mathcal {S}' subseteq mathcal {S}: |mathcal {S}'| = r) such that the sets in (mathcal {S}') are pairwise disjoint? Unfortunately, the problem is not fixed parameter tractable unless (textsf {W[1]} = textsf {FPT} ), and, in fact, an “enumerative” running time of (|mathcal {S}|^{Omega (r)}) is required unless the exponential time hypothesis (ETH) fails. This paper is a quest for tractable instances of Set Packing from parameterized complexity perspectives. We say that the input (({U},mathcal {S})) is “compact” if (|{U}| = f(r)cdot textsf {poly} ( log |mathcal {S}|)), for some (f(r) ge r). In the Compact PSP problem, we are given a compact instance of PSP. In this direction, we present a “dichotomy” result of PSP: When (|{U}| = f(r)cdot o(log |mathcal {S}|)), PSP is in FPT, while for (|{U}| = rcdot Theta (log (|mathcal {S}|))), the problem is W[1]-hard; moreover, assuming ETH, Compact PSP does not admit (|mathcal {S}|^{o(r/log r)}) time algorithm even when (|{U}| = rcdot Theta (log (|mathcal {S}|))). Although certain results in the literature imply hardness of compact versions of related problems such as Set (r)-Covering and Exact (r)-Covering, these constructions fail to extend to Compact PSP. A novel contribution of our work is the identification and construction of a gadget, which we call Compatible Intersecting Set System pair, that is crucial in obtaining the hardness result for Compact PSP. Finally, our framework can be extended to obtain improved running time lower bounds for Compact (r)-VectorSum.

We study the approximability of the maximum size independent set (MIS) problem in bounded degree graphs. This is one of the most classic and widely studied NP-hard optimization problems. It is known for its inherent hardness of approximation. We focus on the well known minimum-degree greedy algorithm for this problem. This algorithm iteratively chooses a minimum degree vertex in the graph, adds it to the solution and removes its neighbors, until the remaining graph is empty. The approximation ratios of this algorithm have been widely studied, where it is augmented with an advice that tells the greedy algorithm which minimum degree vertex to choose if it is not unique. Our main contribution is a new mathematical theory for the design of such greedy algorithms for MIS with efficiently computable advice and for the analysis of their approximation ratios. Using this theory we obtain the ultimate approximation ratio of 5/4 for greedy algorithms on graphs with maximum degree 3, which completely solves an open problem from the paper by Halldórsson and Yoshihara (in: Staples, Eades, Katoh, Moffat (eds) Algorithms and computations—ISAAC ’95, in 2026 LNCS, Springer, Berlin, Heidelberg, 1995) . Our algorithm is the fastest currently known algorithm for MIS with this approximation ratio on such graphs. We also obtain a simple and short proof of the ((Delta +2)/3)-approximation ratio of any greedy algorithms on graphs with maximum degree (Delta ), the result proved previously by Halldórsson and Radhakrishnan (Nord J Comput 1:475–492, 1994) . We almost match this ratio by showing a lower bound of ((Delta +1)/3) on the ratio of a greedy algorithm that can use an advice. We apply our new algorithm to the minimum vertex cover problem on graphs with maximum degree 3 to obtain a substantially faster 6/5-approximation algorithm than the one currently known. We complement our algorithmic, upper bound results with lower bound results, which show that the problem of designing good advice for greedy algorithms for MIS is computationally hard and even hard to approximate on various classes of graphs. These results significantly improve on the previously known hardness results. Moreover, these results suggest that obtaining the upper bound results on the design and analysis of the greedy advice is non-trivial.

We investigate algorithms for testing whether an image is connected. Given a proximity parameter ({epsilon }in (0,1)) and query access to a black-and-white image represented by an (ntimes n) matrix of Boolean pixel values, a (1-sided error) connectedness tester accepts if the image is connected and rejects with probability at least 2/3 if the image is ({epsilon })-far from connected. We show that connectedness can be tested nonadaptively with (OBig (frac{1}{{epsilon }^2}Big )) queries and adaptively with (OBig (frac{1}{{epsilon }^{3/2}} sqrt{log frac{1}{{epsilon }}}Big )) queries. The best connectedness tester to date, by Berman, Raskhodnikova, and Yaroslavtsev (STOC 2014) had query complexity (OBig (frac{1}{{epsilon }^2}log frac{1}{{epsilon }}Big )) and was adaptive. We also prove that every nonadaptive, 1-sided error tester for connectedness must make (Omega Big (frac{1}{{epsilon }}log frac{1}{{epsilon }}Big )) queries.

The game of rendezvous with adversaries is a game on a graph played by two players: Facilitator and Divider. Facilitator has two agents and Divider has a team of (k ge 1) agents. While the initial positions of Facilitator’s agents are fixed, Divider gets to select the initial positions of his agents. Then, they take turns to move their agents to adjacent vertices (or stay put) with Facilitator’s goal to bring both her agents at same vertex and Divider’s goal to prevent it. The computational question of interest is to determine if Facilitator has a winning strategy against Divider with k agents. Fomin, Golovach, and Thilikos [WG, 2021] introduced this game and proved that it is PSPACE-hard and co-W[2]-hard parameterized by the number of agents. This hardness naturally motivates the structural parameterization of the problem. The authors proved that it admits an FPT algorithm when parameterized by the modular width and the number of allowed rounds. However, they left open the complexity of the problem from the perspective of other structural parameters. In particular, they explicitly asked whether the problem admits an FPT or XP-algorithm with respect to the treewidth of the input graph. We answer this question in the negative and show that Rendezvous is co-NP-hard even for graphs of constant treewidth. Further, we show that the problem is co-W[1]-hard when parameterized by the feedback vertex set number and the number of agents, and is unlikely to admit a polynomial kernel when parameterized by the vertex cover number and the number of agents. Complementing these hardness results, we show that the Rendezvous is FPT when parameterized by both the vertex cover number and the solution size. Finally, for graphs of treewidth at most two and girds, we show that the problem can be solved in polynomial time.