53 research outputs found

    Parameterized Complexity of Problems in Coalitional Resource Games

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    Coalition formation is a key topic in multi-agent systems. Coalitions enable agents to achieve goals that they may not have been able to achieve on their own. Previous work has shown problems in coalitional games to be computationally hard. Wooldridge and Dunne (Artificial Intelligence 2006) studied the classical computational complexity of several natural decision problems in Coalitional Resource Games (CRG) - games in which each agent is endowed with a set of resources and coalitions can bring about a set of goals if they are collectively endowed with the necessary amount of resources. The input of coalitional resource games bundles together several elements, e.g., the agent set Ag, the goal set G, the resource set R, etc. Shrot, Aumann and Kraus (AAMAS 2009) examine coalition formation problems in the CRG model using the theory of Parameterized Complexity. Their refined analysis shows that not all parts of input act equal - some instances of the problem are indeed tractable while others still remain intractable. We answer an important question left open by Shrot, Aumann and Kraus by showing that the SC Problem (checking whether a Coalition is Successful) is W[1]-hard when parameterized by the size of the coalition. Then via a single theme of reduction from SC, we are able to show that various problems related to resources, resource bounds and resource conflicts introduced by Wooldridge et al are 1. W[1]-hard or co-W[1]-hard when parameterized by the size of the coalition. 2. para-NP-hard or co-para-NP-hard when parameterized by |R|. 3. FPT when parameterized by either |G| or |Ag|+|R|.Comment: This is the full version of a paper that will appear in the proceedings of AAAI 201

    Refined Lower Bounds for Nearest Neighbor Condensation

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    Refined Lower Bounds for Nearest Neighbor Condensation

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    Fixed-Parameter Tractability of Directed Multiway Cut Parameterized by the Size of the Cutset

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    Given a directed graph GG, a set of kk terminals and an integer pp, the \textsc{Directed Vertex Multiway Cut} problem asks if there is a set SS of at most pp (nonterminal) vertices whose removal disconnects each terminal from all other terminals. \textsc{Directed Edge Multiway Cut} is the analogous problem where SS is a set of at most pp edges. These two problems indeed are known to be equivalent. A natural generalization of the multiway cut is the \emph{multicut} problem, in which we want to disconnect only a set of kk given pairs instead of all pairs. Marx (Theor. Comp. Sci. 2006) showed that in undirected graphs multiway cut is fixed-parameter tractable (FPT) parameterized by pp. Marx and Razgon (STOC 2011) showed that undirected multicut is FPT and directed multicut is W[1]-hard parameterized by pp. We complete the picture here by our main result which is that both \textsc{Directed Vertex Multiway Cut} and \textsc{Directed Edge Multiway Cut} can be solved in time 22O(p)nO(1)2^{2^{O(p)}}n^{O(1)}, i.e., FPT parameterized by size pp of the cutset of the solution. This answers an open question raised by Marx (Theor. Comp. Sci. 2006) and Marx and Razgon (STOC 2011). It follows from our result that \textsc{Directed Multicut} is FPT for the case of k=2k=2 terminal pairs, which answers another open problem raised in Marx and Razgon (STOC 2011)

    Tight Bounds for Gomory-Hu-like Cut Counting

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    By a classical result of Gomory and Hu (1961), in every edge-weighted graph G=(V,E,w)G=(V,E,w), the minimum stst-cut values, when ranging over all s,tVs,t\in V, take at most V1|V|-1 distinct values. That is, these (V2)\binom{|V|}{2} instances exhibit redundancy factor Ω(V)\Omega(|V|). They further showed how to construct from GG a tree (V,E,w)(V,E',w') that stores all minimum stst-cut values. Motivated by this result, we obtain tight bounds for the redundancy factor of several generalizations of the minimum stst-cut problem. 1. Group-Cut: Consider the minimum (A,B)(A,B)-cut, ranging over all subsets A,BVA,B\subseteq V of given sizes A=α|A|=\alpha and B=β|B|=\beta. The redundancy factor is Ωα,β(V)\Omega_{\alpha,\beta}(|V|). 2. Multiway-Cut: Consider the minimum cut separating every two vertices of SVS\subseteq V, ranging over all subsets of a given size S=k|S|=k. The redundancy factor is Ωk(V)\Omega_{k}(|V|). 3. Multicut: Consider the minimum cut separating every demand-pair in DV×VD\subseteq V\times V, ranging over collections of D=k|D|=k demand pairs. The redundancy factor is Ωk(Vk)\Omega_{k}(|V|^k). This result is a bit surprising, as the redundancy factor is much larger than in the first two problems. A natural application of these bounds is to construct small data structures that stores all relevant cut values, like the Gomory-Hu tree. We initiate this direction by giving some upper and lower bounds.Comment: This version contains additional references to previous work (which have some overlap with our results), see Bibliographic Update 1.

    Parameterized Approximation Algorithms for Bidirected Steiner Network Problems

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    The Directed Steiner Network (DSN) problem takes as input a directed edge-weighted graph G=(V,E)G=(V,E) and a set DV×V\mathcal{D}\subseteq V\times V of kk demand pairs. The aim is to compute the cheapest network NGN\subseteq G for which there is an sts\to t path for each (s,t)D(s,t)\in\mathcal{D}. It is known that this problem is notoriously hard as there is no k1/4o(1)k^{1/4-o(1)}-approximation algorithm under Gap-ETH, even when parametrizing the runtime by kk [Dinur & Manurangsi, ITCS 2018]. In light of this, we systematically study several special cases of DSN and determine their parameterized approximability for the parameter kk. For the bi-DSNPlanar_\text{Planar} problem, the aim is to compute a planar optimum solution NGN\subseteq G in a bidirected graph GG, i.e., for every edge uvuv of GG the reverse edge vuvu exists and has the same weight. This problem is a generalization of several well-studied special cases. Our main result is that this problem admits a parameterized approximation scheme (PAS) for kk. We also prove that our result is tight in the sense that (a) the runtime of our PAS cannot be significantly improved, and (b) it is unlikely that a PAS exists for any generalization of bi-DSNPlanar_\text{Planar}, unless FPT=W[1]. One important special case of DSN is the Strongly Connected Steiner Subgraph (SCSS) problem, for which the solution network NGN\subseteq G needs to strongly connect a given set of kk terminals. It has been observed before that for SCSS a parameterized 22-approximation exists when parameterized by kk [Chitnis et al., IPEC 2013]. We give a tight inapproximability result by showing that for kk no parameterized (2ε)(2-\varepsilon)-approximation algorithm exists under Gap-ETH. Additionally we show that when restricting the input of SCSS to bidirected graphs, the problem remains NP-hard but becomes FPT for kk

    Preventing Unraveling in Social Networks Gets Harder

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    The behavior of users in social networks is often observed to be affected by the actions of their friends. Bhawalkar et al. \cite{bhawalkar-icalp} introduced a formal mathematical model for user engagement in social networks where each individual derives a benefit proportional to the number of its friends which are engaged. Given a threshold degree kk the equilibrium for this model is a maximal subgraph whose minimum degree is k\geq k. However the dropping out of individuals with degrees less than kk might lead to a cascading effect of iterated withdrawals such that the size of equilibrium subgraph becomes very small. To overcome this some special vertices called "anchors" are introduced: these vertices need not have large degree. Bhawalkar et al. \cite{bhawalkar-icalp} considered the \textsc{Anchored kk-Core} problem: Given a graph GG and integers b,kb, k and pp do there exist a set of vertices BHV(G)B\subseteq H\subseteq V(G) such that Bb,Hp|B|\leq b, |H|\geq p and every vertex vHBv\in H\setminus B has degree at least kk is the induced subgraph G[H]G[H]. They showed that the problem is NP-hard for k2k\geq 2 and gave some inapproximability and fixed-parameter intractability results. In this paper we give improved hardness results for this problem. In particular we show that the \textsc{Anchored kk-Core} problem is W[1]-hard parameterized by pp, even for k=3k=3. This improves the result of Bhawalkar et al. \cite{bhawalkar-icalp} (who show W[2]-hardness parameterized by bb) as our parameter is always bigger since pbp\geq b. Then we answer a question of Bhawalkar et al. \cite{bhawalkar-icalp} by showing that the \textsc{Anchored kk-Core} problem remains NP-hard on planar graphs for all k3k\geq 3, even if the maximum degree of the graph is k+2k+2. Finally we show that the problem is FPT on planar graphs parameterized by bb for all k7k\geq 7.Comment: To appear in AAAI 201

    Parameterized Streaming Algorithms for Vertex Cover

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    As graphs continue to grow in size, we seek ways to effectively process such data at scale. The model of streaming graph processing, in which a compact summary is maintained as each edge insertion/deletion is observed, is an attractive one. However, few results are known for optimization problems over such dynamic graph streams. In this paper, we introduce a new approach to handling graph streams, by instead seeking solutions for the parameterized versions of these problems where we are given a parameter kk and the objective is to decide whether there is a solution bounded by kk. By combining kernelization techniques with randomized sketch structures, we obtain the first streaming algorithms for the parameterized versions of the Vertex Cover problem. We consider the following three models for a graph stream on nn nodes: 1. The insertion-only model where the edges can only be added. 2. The dynamic model where edges can be both inserted and deleted. 3. The \emph{promised} dynamic model where we are guaranteed that at each timestamp there is a solution of size at most kk. In each of these three models we are able to design parameterized streaming algorithms for the Vertex Cover problem. We are also able to show matching lower bound for the space complexity of our algorithms. (Due to the arXiv limit of 1920 characters for abstract field, please see the abstract in the paper for detailed description of our results)Comment: Fixed some typo
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