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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.

Motivated by an application from geodesy, we study the connected k-center problem and the connected k-diameter problem. The former problem has been introduced by Ge et al. (ACM Trans Knowl Discov Data 2(2):1–35, 2008. https://doi.org/10.1145/1376815.1376816) to model clustering of data sets with both attribute and relationship data. These problems arise from the classical k-center and k-diameter problems by adding a side constraint. For the side constraint, we are given an undirected connectivity graph G on the input points, and a clustering is now only feasible if every cluster induces a connected subgraph in G. Usually in clustering problems one assumes that the clusters are pairwise disjoint. We study this case but additionally also the case that clusters are allowed to be non-disjoint. This can help to satisfy the connectivity constraints. Our main result is an (O(log ^2k))-approximation algorithm for the disjoint connected k-center and k-diameter problem. For Euclidean spaces of constant dimension and for metrics with constant doubling dimension, the approximation factor improves to O(1). Our algorithm works by computing a non-disjoint connected clustering first and transforming it into a disjoint connected clustering. We complement these upper bounds by several upper and lower bounds for variations and special cases of the model.

Given a graph and two vertex sets satisfying a certain feasibility condition, a reconfiguration problem asks whether we can reach one vertex set from the other by repeating prescribed modification steps while maintaining feasibility. In this setting, as reported by Mouawad et al. (IPEC, Springer, Berlin, 2014) presented an algorithmic meta-theorem for reconfiguration problems that says if the feasibility can be expressed in monadic second-order logic (MSO), then the problem is fixed-parameter tractable parameterized by (text {treewidth} + ell ), where (ell ) is the number of steps allowed to reach the target set. On the other hand, it is shown by Wrochna (J Comput Syst Sci 93:1–10, 2018). https://doi.org/10.1016/j.jcss.2017.11.003) that if (ell ) is not part of the parameter, then the problem is PSPACE-complete even on graphs of constant bandwidth. In this paper, we present the first algorithmic meta-theorems for the case where (ell ) is not part of the parameter, using some structural graph parameters incomparable with bandwidth. We show that if the feasibility is defined in MSO, then the reconfiguration problem under the so-called token jumping rule is fixed-parameter tractable parameterized by neighborhood diversity. We also show that the problem is fixed-parameter tractable parameterized by (text {treedepth} + k), where k is the size of sets being transformed. We finally complement the positive result for treedepth by showing that the problem is PSPACE-complete on forests of depth 3.

Classical streaming algorithms operate under the (not always reasonable) assumption that the input stream is fixed in advance. Recently, there is a growing interest in designing robust streaming algorithms that provide provable guarantees even when the input stream is chosen adaptively as the execution progresses. We propose a new framework for robust streaming that combines techniques from two recently suggested frameworks by Hassidim et al. (NeurIPS 2020) and by Woodruff and Zhou (FOCS 2021). These recently suggested frameworks rely on very different ideas, each with its own strengths and weaknesses. We combine these two frameworks into a single hybrid framework that obtains the “best of both worlds”, thereby solving a question left open by Woodruff and Zhou.

Reconfiguring two shortest paths in a graph means modifying one shortest path to the other by changing one vertex at a time so that all the intermediate paths are also shortest paths. This problem has several natural applications, namely: (a) repaving road networks, (b) rerouting data packets in a synchronous multiprocessing setting, (c) the shipping container stowage problem, and (d) the train marshalling problem. When modelled as graph problems, (a) is the most general case while (b), (c), (d) are restrictions to different graph classes. We show that (a) does not admit polynomial-time algorithms (assuming ({{,mathrm{texttt {P}},}}ne {{,mathrm{texttt {NP}},}})), even for relaxed variants of the problem (assuming ({{,mathrm{texttt {P}},}}ne {{,mathrm{texttt {PSPACE}},}})). For (b), (c), (d), we present polynomial-time algorithms to solve the respective problems. We also generalize the problem to when at most k (for a fixed integer (kge 2)) contiguous vertices on a shortest path can be changed at a time.

Our work is motivated by the challenges presented in preparing arrays of atoms for use in quantum simulation. The recently-developed process of loading atoms into traps results in approximately half of the traps being filled. To consolidate the atoms so that they form a dense and regular arrangement, such as all locations in a grid, atoms are rearranged using moving optical tweezers. Time is of the essence, as the longer that the process takes and the more that atoms are moved, the higher the chance that atoms will be lost in the process. Viewed as a problem on graphs, we wish to solve the problem of reconfiguring one arrangement of tokens (representing atoms) to another using as few moves as possible. Because the problem is NP-complete on general graphs as well as on grids, we focus on the parameterized complexity for various parameters, considering both undirected and directed graphs, and tokens with and without labels. For unlabelled tokens, the problem is fixed-parameter tractable when parameterized by the number of tokens, the number of moves, or the number of moves plus the number of vertices without tokens in either the source or target configuration, but intractable when parameterized by the difference between the number of moves and the number of differences in the placement of tokens in the source and target configurations. When labels are added to tokens, however, most of the tractability results are replaced by hardness results.