53 research outputs found

### Parameterized Complexity of Problems in Coalitional Resource Games

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

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

Given a directed graph $G$, a set of $k$ terminals and an integer $p$, the
\textsc{Directed Vertex Multiway Cut} problem asks if there is a set $S$ of at
most $p$ (nonterminal) vertices whose removal disconnects each terminal from
all other terminals. \textsc{Directed Edge Multiway Cut} is the analogous
problem where $S$ is a set of at most $p$ 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 $k$ 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 $p$. Marx and Razgon (STOC 2011) showed that undirected multicut is FPT and
directed multicut is W[1]-hard parameterized by $p$. 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
$2^{2^{O(p)}}n^{O(1)}$, i.e., FPT parameterized by size $p$ 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 $k=2$ terminal pairs, which
answers another open problem raised in Marx and Razgon (STOC 2011)

### Tight Bounds for Gomory-Hu-like Cut Counting

By a classical result of Gomory and Hu (1961), in every edge-weighted graph
$G=(V,E,w)$, the minimum $st$-cut values, when ranging over all $s,t\in V$,
take at most $|V|-1$ distinct values. That is, these $\binom{|V|}{2}$ instances
exhibit redundancy factor $\Omega(|V|)$. They further showed how to construct
from $G$ a tree $(V,E',w')$ that stores all minimum $st$-cut values. Motivated
by this result, we obtain tight bounds for the redundancy factor of several
generalizations of the minimum $st$-cut problem.
1. Group-Cut: Consider the minimum $(A,B)$-cut, ranging over all subsets
$A,B\subseteq V$ of given sizes $|A|=\alpha$ and $|B|=\beta$. The redundancy
factor is $\Omega_{\alpha,\beta}(|V|)$.
2. Multiway-Cut: Consider the minimum cut separating every two vertices of
$S\subseteq V$, ranging over all subsets of a given size $|S|=k$. The
redundancy factor is $\Omega_{k}(|V|)$.
3. Multicut: Consider the minimum cut separating every demand-pair in
$D\subseteq V\times V$, ranging over collections of $|D|=k$ demand pairs. The
redundancy factor is $\Omega_{k}(|V|^k)$. 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

The Directed Steiner Network (DSN) problem takes as input a directed
edge-weighted graph $G=(V,E)$ and a set $\mathcal{D}\subseteq V\times V$ of $k$
demand pairs. The aim is to compute the cheapest network $N\subseteq G$ for
which there is an $s\to t$ path for each $(s,t)\in\mathcal{D}$. It is known
that this problem is notoriously hard as there is no
$k^{1/4-o(1)}$-approximation algorithm under Gap-ETH, even when parametrizing
the runtime by $k$ [Dinur & Manurangsi, ITCS 2018]. In light of this, we
systematically study several special cases of DSN and determine their
parameterized approximability for the parameter $k$.
For the bi-DSN$_\text{Planar}$ problem, the aim is to compute a planar
optimum solution $N\subseteq G$ in a bidirected graph $G$, i.e., for every edge
$uv$ of $G$ the reverse edge $vu$ 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 $k$. 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-DSN$_\text{Planar}$, unless FPT=W[1].
One important special case of DSN is the Strongly Connected Steiner Subgraph
(SCSS) problem, for which the solution network $N\subseteq G$ needs to strongly
connect a given set of $k$ terminals. It has been observed before that for SCSS
a parameterized $2$-approximation exists when parameterized by $k$ [Chitnis et
al., IPEC 2013]. We give a tight inapproximability result by showing that for
$k$ no parameterized $(2-\varepsilon)$-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 $k$

### Preventing Unraveling in Social Networks Gets Harder

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 $k$ the equilibrium for
this model is a maximal subgraph whose minimum degree is $\geq k$. However the
dropping out of individuals with degrees less than $k$ 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 $k$-Core}
problem: Given a graph $G$ and integers $b, k$ and $p$ do there exist a set of
vertices $B\subseteq H\subseteq V(G)$ such that $|B|\leq b, |H|\geq p$ and
every vertex $v\in H\setminus B$ has degree at least $k$ is the induced
subgraph $G[H]$. They showed that the problem is NP-hard for $k\geq 2$ 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 $k$-Core} problem is W[1]-hard parameterized by $p$,
even for $k=3$. This improves the result of Bhawalkar et al.
\cite{bhawalkar-icalp} (who show W[2]-hardness parameterized by $b$) as our
parameter is always bigger since $p\geq b$. Then we answer a question of
Bhawalkar et al. \cite{bhawalkar-icalp} by showing that the \textsc{Anchored
$k$-Core} problem remains NP-hard on planar graphs for all $k\geq 3$, even if
the maximum degree of the graph is $k+2$. Finally we show that the problem is
FPT on planar graphs parameterized by $b$ for all $k\geq 7$.Comment: To appear in AAAI 201

### Parameterized Streaming Algorithms for Vertex Cover

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 $k$ and the objective is to decide whether there
is a solution bounded by $k$. 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 $n$ 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 $k$.
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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