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Register a new userRestricted Max-Min Fair Allocation
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Added by
Yuchen Mao
Publication date
2018
fields
Informatics Engineering
and research's language is
English
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The restricted max-min fair allocation problem seeks an allocation of resources to players that maximizes the minimum total value obtained by any player. It is NP-hard to approximate the problem to a ratio less than 2. Comparing the current best algorithm for estimating the optimal value with the current best for constructing an allocation, there is quite a gap between the ratios that can be achieved in polynomial time: roughly 4 for estimation and roughly $6 + 2sqrt{10}$ for construction. We propose an algorithm that constructs an allocation with value within a factor of $6 + delta$ from the optimum for any constant $delta > 0$. The running time is polynomial in the input size for any constant $delta$ chosen.
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The max-min fair allocation problem seeks an allocation of resources to players that maximizes the minimum total value obtained by any player. Each player $p$ has a non-negative value $v_{pr}$ on resource $r$. In the restricted case, we have $v_{pr}in {v_r, 0}$. That is, a resource $r$ is worth value $v_r$ for the players who desire it and value 0 for the other players. In this paper, we consider the configuration LP, a linear programming relaxation for the restricted problem. The integrality gap of the configuration LP is at least $2$. Asadpour, Feige, and Saberi proved an upper bound of $4$. We improve the upper bound to $23/6$ using the dual of the configuration LP. Since the configuration LP can be solved to any desired accuracy $delta$ in polynomial time, our result leads to a polynomial-time algorithm which estimates the optimal value within a factor of $23/6+delta$.
Asadpour, Feige, and Saberi proved that the integrality gap of the configuration LP for the restricted max-min allocation problem is at most $4$. However, their proof does not give a polynomial-time approximation algorithm. A lot of efforts have been devoted to designing an efficient algorithm whose approximation ratio can match this upper bound for the integrality gap. In ICALP 2018, we present a $(6 + delta)$-approximation algorithm where $delta$ can be any positive constant, and there is still a gap of roughly $2$. In this paper, we narrow the gap significantly by proposing a $(4+delta)$-approximation algorithm where $delta$ can be any positive constant. The approximation ratio is with respect to the optimal value of the configuration LP, and the running time is $mathit{poly}(m,n)cdot n^{mathit{poly}(frac{1}{delta})}$ where $n$ is the number of players and $m$ is the number of resources. We also improve the upper bound for the integrality gap of the configuration LP to $3 + frac{21}{26} approx 3.808$.
In this paper we present a new data structure for double ended priority queue, called min-max fine heap, which combines the techniques used in fine heap and traditional min-max heap. The standard operations on this proposed structure are also presented, and their analysis indicates that the new structure outperforms the traditional one.
In the ${-1,0,1}$-APSP problem the goal is to compute all-pairs shortest paths (APSP) on a directed graph whose edge weights are all from ${-1,0,1}$. In the (min,max)-product problem the input is two $ntimes n$ matrices $A$ and $B$, and the goal is to output the (min,max)-product of $A$ and $B$. This paper provides a new algorithm for the ${-1,0,1}$-APSP problem via a simple reduction to the target-(min,max)-product problem where the input is three $ntimes n$ matrices $A,B$, and $T$, and the goal is to output a Boolean $ntimes n$ matrix $C$ such that the $(i,j)$ entry of $C$ is 1 if and only if the $(i,j)$ entry of the (min,max)-product of $A$ and $B$ is exactly the $(i,j)$ entry of the target matrix $T$. If (min,max)-product can be solved in $T_{MM}(n) = Omega(n^2)$ time then it is straightforward to solve target-(min,max)-product in $O(T_{MM}(n))$ time. Thus, given the recent result of Bringmann, Kunnemann, and Wegrzycki [STOC 2019], the ${-1,0,1}$-APSP problem can be solved in the same time needed for solving approximate APSP on graphs with positive weights. Moreover, we design a simple algorithm for target-(min,max)-product when the inputs are restricted to the family of inputs generated by our reduction. Using fast rectangular matrix multiplication, the new algorithm is faster than the current best known algorithm for (min,max)-product.
We consider high dimensional variants of the maximum flow and minimum cut problems in the setting of simplicial complexes and provide both algorithmic and hardness results. By viewing flows and cuts topologically in terms of the simplicial (co)boundary operator we can state these problems as linear programs and show that they are dual to one another. Unlike graphs, complexes with integral capacity constraints may have fractional max-flows. We show that computing a maximum integral flow is NP-hard. Moreover, we give a combinatorial definition of a simplicial cut that seems more natural in the context of optimization problems and show that computing such a cut is NP-hard. However, we provide conditions on the simplicial complex for when the cut found by the linear program is a combinatorial cut. For $d$-dimensional simplicial complexes embedded into $mathbb{R}^{d+1}$ we provide algorithms operating on the dual graph: computing a maximum flow is dual to computing a shortest path and computing a minimum cut is dual to computing a minimum cost circulation. Finally, we investigate the Ford-Fulkerson algorithm on simplicial complexes, prove its correctness, and provide a heuristic which guarantees it to halt.
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